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Age-dependent maintenance strategies of medium-voltage circuit-breakers and transformers
Electric Power Systems Research 81 (2011) 1709–1714
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Age-dependent maintenance strategies of medium-voltage circuit-breakers and transformers夽 Xiang Zhang ∗ , Ernst Gockenbach Institute of Electric Power Systems, Division of High Voltage Engineering, Leibniz University of Hannover, Callinstrasse 25A, Hanover 30167, Germany
a r t i c l e
i n f o
Article history: Received 14 October 2008 Received in revised form 13 February 2011 Accepted 10 March 2011 Available online 7 May 2011 Keywords: Age Circuit-breaker Cost Failure rate Maintenance strategy Transformer
a b s t r a c t The general life and reliability models of electrical equipment are essential to evaluate their actual conditions because of the degradation of equipment. To optimize the maintenance strategies for maximal reliability and minimal cost in a quantitative way, available maintenance models of ageing equipment shall be found to describe actual maintenance actions in time-series processes. In particular, a functional relationship between failure rate and maintenance measures is to be developed for electrical equipment. This paper demonstrates some actual examples by applying these models and the results show the value of using a systematic quantitative approach to investigate the effect of different maintenance strategies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Deregulation of the power system market has forced electric utilities to scrutinize investment and maintenance expenditures much more rigorously than in the past. Many focus their investments on maintaining the competitive position of power generating assets, while trying to squeeze more performance from ageing power delivery assets with less expenditure. However, in the power transmission and distribution systems, the annual expenditures for maintenance and replacement average only 1% or less (equal to 20 billion Euro for distribution systems in Germany and 400 billion Dollar for transmission systems in USA), which corresponds to an expected lifetime for more than 100 years of electrical equipment [1]. Owing to the limited reinvestments, the age of electrical equipment will increase so that the utilities have to face various market requirements. On the one hand, customers are paying for a service and the authorities are imposed regulation, supervision, and compensation depending on the degree to which contracts and other obligations are fulfilled. On the other hand, utilities must ensure that their expenditure is cost-effective.
夽 This work was supported by the National Research Council of Germany under the Contract SPP 1101. ∗ Corresponding author. E-mail address: [email protected] (X. Zhang). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.03.018
The easiest way and the most widely used strategy today is the time-based maintenance (TBM) [2]. There are fixed time intervals for inspections and for certain maintenance works. However, it seems that the time intervals chosen are far from the safe side and the total replacement expenditure is extremely expensive, as there are many inspections revealing no problems at all. So, the time intervals obviously can be extended – the question is from which point of time the occurrence of failures will increase significantly. In order to obtain useful information about the actual conditions of equipment, condition monitoring technique and conditionbased maintenance (CBM) has been well developed [3–5]. Condition monitoring technique means mainly sensor development, data acquisition, data analysis, and development of methods for determination of equipment condition and early fault recognition. The importance of monitoring methods can recognize which measured parameter affects the ageing of equipment to a greater or lesser degree. It should support the introduction of conditionbased maintenance and help to avoid unexpected outages. The cost of sufficient instrumentation can often be quite large and off-line measurement even causes more significant outage. In the majority of cases, there are neither communication links nor suitable sensors available for monitoring from remote. In terms of numerical protection devices and control systems, those engineers are suffering from data overload. Also, from the technical side, it is not always as simple as possible. Even if some types of equipment can easily be observed by measuring simple values as temperature or
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X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
pressure, it is not trivial to turn this measured data into actionable knowledge about health of the equipment. In recent years, the reliability centered maintenance (RCM) has been established [6–8]. RCM is an industrial improvement approach focused on identifying and establishing the operational, maintenance, and capital improvement policies that will manage the risks (combination of severity and frequency) of equipment failure most effectively. As an economic and reliable life cycle management, the condition and the importance of the equipment must be combined and evaluated in order to create a cost-effective maintenance strategy, addressing dominant modes and causes of equipment failure. The artificial neutral network which exhibits the non-linear input–output relationships between equipment reliability and system maintenance is integrated into maintenance management system (expert system) via a commercial software tool. The reliability centered maintenance based on the analysis of “condition-importance” and the use of “weighting factors”, depends heavily on practical experiences in diagnostic analysis and maintenance measures. That has followed so-called reliability centered asset management (RCAM) since the last years [9–12]. The research approach deals with the life assessment (e.g. failure rate) and the maintenance tasks (e.g. inspection rate) based on several discrete exponential distributions. The assumption of exponential distribution spent in each stage implies constant failure/maintenance rate which is assigning to each operation state. In that way, the deterioration processes of equipment consist of a chain of discrete Markov models. Therefore, this parameter evaluated by using time between failures as input data for special network facility, is of the most concern in the analysis. These chosen parameters represent past experience by single facts or numbers, without assigning any degree of likelihood to future expectation. Therefore the approach does not provide enduring analytical models of the deterioration processes, by which the consequences of each failure can be effectively used on a predictive basis for the future. On the basis of the exponential distribution, the fitted curves are only a mathematical simulation that is not concerned with any information about the technical parameters and the operating conditions of electrical equipment, thus they can not give a complete understanding of the physical implications in failures. Up to now, the impacts of maintenance on reliability and cost can be analyzed only for those selected strategies. The reason is a lack of an available reliability model for the deteriorating electrical equipment and a theoretical maintenance model put into practice for power systems. In fact, only a continuous quantitative description (mathematical models) can lead to an optimal solution of the systematic maintenance strategies. Therefore, this broad perspective of asset management is recognized as “relating maintenance effort to system availability and total cost, with the aim to reach an optimal maintenance management”. This new research topic is world-wide challenge for all researchers on asset management of the power transmission and distribution systems.
2. Age-dependent reliability model The failure causes may identify the origin of detrimental effects, contributing to the failure of electrical equipment. The influence factors producing an ageing change of equipment are divided in different stresses such as electrical, thermal, mechanical and ambient stresses. The stresses of electric ageing are partial discharges, tracking, treeing, electrolysis and space charges. The intensity and progress of electrical ageing depends on the electric field strength in the insulation. The consequences of thermal ageing comprise chemical, physical and thermal–dynamic changes by chemical degradation,
polymerization, depolymerization of organic materials, diffusion and thermal–mechanical effects as a result of thermal extension or contraction. A mechanical ageing of equipment can only occur by solid material as a result of mechanical tractive and shear forces during transportation, installation and operation. The substantial reasons of mechanical ageing are electrodynamic, electromagnetic and thermal forces. Ageing caused by ambient conditions comes from the reaction of material to humidity, air, chemical, biological substances, weathering, pollution and radiation. But more or less the electrical equipment is simultaneously stressed by various influence factors. A thorough electro-thermo-mechanical life model of the electrical component can be established according to the Inverse Power Law and the Arrhenius Model. This can simply be done by assuming that ageing rate under these combined stresses is the product of ageing rates under each single stress [13,14]: L = L0
E −(n−bT ) M −m ·
E0
M0
· e−BT ,
T=
1 1 − ϑ0 ϑ
(1)
where E, M, T and L are the electrical, mechanical, thermal stresses and lifetime, respectively. E0 and M0 are the scale-parameters for the lower limit of electrical and mechanical stresses respectively (below which the ageing can be neglected) and L0 is the corresponding lifetime. n, m and B are the voltage-endurance coefficient, the mechanical stress-endurance coefficient and the activation energy of thermal degradation reaction, respectively. b is the correct coefficient which takes into account the reaction of materials due to combined stress application. ϑ and ϑ0 are the absolute temperature and the reference temperature. The failure of an electrical component may occur if over-voltage or mechanical stress is applied, or if the electrical component is aged by temperature or time. A criterion for the ageing of electrical components is consistent with the electrical, thermal or mechanical stress. Thus the relationship between lifetime and failure probability of electrical equipment may be determined by:
˛(n−bT ) m˛ ˛ E M L
Pr(L) = 1 − exp −
·
E0
M0
·
L0
·e
˛BT
(2)
Therefore, the lifetime L of an ageing component as a random variable t has a cumulative probability distribution F(t) ≡ Pr{L ≤ t} with right continuous, and a probability density function f(t) = dF(t)/dt
˛[n−b((1/ϑ )−(1/ϑ))] m˛ 0 E M
F(t) ≡ Pr{L ≤ t} = 1 − exp −
×
t ˛ L0
· e˛B((1/ϑ0 )−(1/ϑ))
E0
·
M0 (3)
Ageing is usually measured based on the term of a failure rate function. Failure rate is the most important quantity in maintenance theory. The instant failure rate function h(t) is defined as h(t) =
f (t) 1 dF(t) = · ¯ ¯ dt F(t) F(t)
(4)
The failure rate is the most important reliability items for electrical equipment. The failure rate allows electrical equipment in different asset classes to be compared with each other, and to make reference to several criteria like age, number of maintenance, time between events, etc. Whatever criteria for assessment are chosen, proper maintenance activities can be performed.
X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
3. Modeling risk and costs
=
cr
¯ ¯ S(x)h(x) dx − S() + cc S() + cp S()
0
3.1. Defining strategies and their contributions
0
During its life, the equipment might undergo several maintenance activities in order to improve its reliability, which is achieved by knowing its operability or by reducing its age. For circuitbreakers or transformers, all the typical activities are grouped in two strategies:
Strategy 2.
Maintenance with periodic preventive maintenance
The component is operated under a block-based periodic replacement. Imperfect repair between successive replacements is made after each failure so that the failure rate remains below a certain threshold level. Assumption.
1 S()
E{Z |L < } =
Notation.
Imperfect repair
When using a replacement, the system is “as good as new” after the repair action is completed. When the minimal repair is used, the component is “as bad as old” after the repair action. The replacement and minimal repair may thus be considered as two extreme cases, and components subject to imperfect repair will be somewhere between these two extremes [15]. Hence, an imperfect repair model is suggested for a component: the repair is a perfect repair with a certain failure probability Pr(t); the repair is a minimal one with 1 − Pr(t). In the case, the successive perfect repair times form a renewal process with interarrival time distribution
S(t) = 1 − exp
−
t
Pr(x)h(x) dx
(5)
y
¯ Pr(x)h(x) dx dS(y) 0
(8)
0
or the expected number of repairs when L ≥ [16]
¯ Pr(x)h(x) dx
E{Z |L ≥ } =
(9)
0
For Strategy 2. By definition the total expected cost C() of one cycle is given by the sum of preventive replacement cost, corrective replacement cost and repair cost within a preventive replacement:
C() =
All maintenance time is negligible.
(7)
¯ S(x) dx
Proof 1. Assume that Z represents the number of minimal repairs during the time interval (0, min {, L}), we have the expected number of repairs when L < [16]
Strategy 1. Maintenance with sequential preventive maintenance Consider a process where the component is subject to age-based sequential replacement. Imperfect repair between successive replacements is made after each failure so that the failure rate remains below a certain threshold level.
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[cr · (expected number of repairs during a preventive replacement) + cc · (expected number of corrective replacements during a preventive replacement) + cp ] one cycle of a preventive replacment
(10)
The total expected maintenance cost rate C() when using the interval time is equal to C() =
cr Nr () + cc Nc () + cp
(11)
Proof 2. If N() represents the number of failures during (0, ], N() is obviously the renewal function for the renewal process with the interarrival time distribution S(t) and can be determined by the solution method to the renewal function in renewal theory [16]. Therefore, the expected number of corrective replacements can be gotten.
Nc () = S() +
Nc ( − x) dS(x)
(12)
0
0
and the corresponding failure rate Pr(t)·h(t).
and the expected number of repairs
3.2. Modeling strategies
Nr () = expected number of repairs during a corrective replacement · cumulative probability of a corrective replacement
Within each strategy there are several component states and each one contributes to component risk. To calculate the various contributions to the total cost for each strategy, the renewal theory is used in time-series processes. In our maintenance model, two maintenance strategies are performed by the different combinations of preventive maintenance, repair or corrective replacement. For Strategy 1. Let C denotes the total expected maintenance cost per unit of time over an indefinitely long period. By definition the total expected cost C() of one cycle is given by the sum of cost rates of preventive maintenance and repair or corrective replacement and repair:
C() =
+ expected number of repairs during a preventive replacement · cumulative probability of a preventive replacement
¯ Nr ( − x) dS(x) = E{Z |L < }S() + E{Z |L ≥ }S()
+ 0
+
0
¯ S(x) h(x) dx − S() +
Nr ( − x) dS(x) = 0
Nr ( − x) dS(x) 0
(13)
{[cr · (expected number of repairs during a corrective replacement) + cc ] · cumulative probability of a corrective replacement + [cr · (expected number of repairs during a preventive replacement) + cp ] · cumulative probability of a preventive replacement} one cycle of a replacement (6)
where cc , cp and cr are the costs of the corrective- and consequential replacement, preventive replacement and imperfect repair respectively. cc includes unplanned replacement cost and unplanned unavailability cost. Thus, the total expected maintenance cost rate C() is equal to C() =
¯ {[cr E{Z |L < } + cc ] · S() + [cr E{Z |L ≥ } + cp ] · S()}
0
¯ S(x) dx
3.3. Optimizing strategies It is expected to decide at which intervals to maintain the component and whether to keep the component in an old state or replace the component with an identical component. These decisions are made by comparing the annual cost of the component and
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X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714 Table 3 Costs of different components (D ).
total cost rate
Component
cost rate
Optimum frequency preventive maintenance corrective-&consequential maintenance
Transformer Circuit-breaker
cc
3535 260
4771 6270
cr 172 55
0.3
maintenance frequency Fig. 1. Optimization of maintenance frequency.
Table 1 The parameters of Eq. (3). n
m
b (K)
B (K)
7.0
2.3
6000
17,000
L0 (year)
ϑ0 (K)
E0 (kV/mm)
M0 (N/mm2 )
4.5 × 10
298
5.0
2.4 × 10−4
failure rate (1/year)
repair
4
cp
0.2
0.1
0
0
7
14
21
28
35
year
dC() =0 d
(14)
Fig. 1 shows that the cost rate of preventive maintenance generally increases with higher maintenance frequency. On the other hand, the cost rates of corrective and consequential maintenance (including repair) associated with failures decrease as the maintenance frequency increases. The total cost rate, the sum of these three individual cost rates, exhibits a minimum, and so an “optimum” or target level of reliability is achieved. Thus the integrated cost benefit analyses involving customers and their decisions, in an asset management’s view, could be a more rewarding and applicable approach. 4. Simulation results from application studies The significant requirement in applying reliability models is a choice of the model parameters. This research work is initiated to collect nearly 120,000 failure data of historical events of electrical equipment in the special failure statistic from the year 1920 to 2005 [1]. These failure record data is dependent more on the component age than the calendar year. Furthermore, certain information on specific damage data, especially without disturbance of network operation, has been acquired. Therefore, the detailed failure statistic can provide information about the actual conditions and economic assessment of these increasingly old components as shown in Tables 1–3. In light of this goal, this work attempts to integrate so sufficient failure data into the reliability models that failure rate of electrical equipment can be modeled as a function of operating Table 2 The parameters of Eq. (3). Component
Transformer Circuit-breaker
Fig. 2. Calculated failure rates for circuit-breakers and transformers (dotted).
history, operational stresses and component type (Fig. 2). The failure rates increase with operation year and their increases become larger after a certain operation year. When a circuit-breaker is overhauled, the value of failure rate decreases to the value of the more new equipment. However, as wear parts without exchange may remain, the value of failure rate will increase like the shift curve. Under the maintenance policy of strategy 2, repair and corrective replacement are likely to be performed several times during the lifetime of equipment. Figs. 3 and 4 show the repair and replacement characteristics of circuit-breakers and transformers with a series of increments. Since the ageing leads to the degradation of equipment, the number of repair and replacement for circuit-breakers and transformers begins to grow rapidly. This means that the preventive replacement of circuit-breakers and transformers should be planned at 20 and 34 operation years, respectively. It is obvious that transformers have more numbers of repair and replacement than circuit-breakers. The
0.1
number of repair and corrective replacement
checking whether the component has reached its economic life. The economic life is also referred to as the minimum cost life or optimal maintenance interval. The optimal maintenance interval is the solution that minimizes the sum of these maintenance cost rates:
0.05
0
˛
E
M
ϑ
11.6 8.3
2.2E0 2.0E0
6.8M0 7.2M0
40 ◦ C 40 ◦ C
0
10
20
year Fig. 3. Calculated expected number of repair and corrective replacement (dotted) for circuit-breakers according to strategy 2.
X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
1
number of repair and replacement
number of repair and corrective replacement
0.4
0.2
0.5
0 0
1713
0
7
14
21
28
0
10
year
20
30
35 Fig. 5. Calculated expected number of repair (below-dotted), corrective and preventive (above-dotted) replacements for circuit-breakers according to strategy 1.
year Fig. 4. Calculated expected number of repair (dotted) and corrective replacement for transformers according to strategy 2.
number of occasions depends on the failure rate of the equipment. When the failure rate increases, the number and cost rate of corrective and consequential maintenance (including repair) associated with failures increase as the operation time increases. On the other hand, the number and cost rate of preventive maintenance generally decreases under the age-dependent strategy 1. In the case of either strategy 1 or strategy 2, those failed circuit breakers and transformers have more repairs than corrective replacements. The probability of the equipment degradation increases, so we would perform more corrective maintenance. Taking into account this dependence it is clear that the contributions due to preventive maintenance should decrease, because corrective maintenances have opposite effect. The total cost rate, the sum of these three individual cost rates, exhibits a minimum, and so an “optimum” or target level of reliability is achieved. It will be useful to compare strategy 1 (Figs. 5 and 6) with strategy 2 for circuit breakers and transformers. In accordance with strategy 1 or strategy 2, circuit breakers have different optimal preventive intervals of 28 or 20 years with minimum cost rate, respectively. Strategy 2 is more wasteful since more preventive replacements occur and more unfailed circuit-breakers are removed than those under a similar policy based on age. As might be suspected however, the cost rate of transformers will be more under strategy 1. This is because stochastic failures cause more expensive costs of corrective and consequential maintenance. As utilizing strategy 1, the preventive replacement cannot be fully scheduled, and the policy may therefore be complex to administer. The final result in the economic evaluation is to estimate the annual cost of maintenance according to different maintenance strategies of strategy 1 and strategy 2. Table 4 shows the annual maintenance cost and optimal maintenance interval according to
Table 4 Annual maintenance cost (D /year) and optimal maintenance interval (year) according to different maintenance strategies. Strategy
Statistic 1 2
Circuit-breaker
Transformer
Cost rate
Interval
Cost rate
Interval
260 100 158
10 28 20
3535 266 250
2 26 34
number of repair and replacement
1
0.5
0
0
10
year
20
30
Fig. 6. Calculated expected number of repair (below-dotted), corrective and preventive (above-dotted) replacements for transformers according to strategy 1.)
different maintenance strategies. From the failure statistic [1], circuit breakers and transformers have the actual cost rate of 260 D /year and 3535 D /year for maintenance when a preventive maintenance is executed every 10 and 2 years. However, different optimal intervals and annual costs of maintenance for circuit breakers and transformers can be obtained according to strategy 1 or strategy 2. By minimizing the cost rate of 100 and 158 D /year, the interval of preventive maintenance is optimized by 28 or 20 years if strategy 1 or strategy 2 is caught into effect for circuit-breakers, respectively. Under these circumstances of strategies 1 and 2, transformers have cost rate of 266 or 250 D /year with the interval of 26 and 34 years, respectively. Comparing the actual cost rate of 260 and 3535 D /year, circuit breakers and transformers have a significant decrease in cost rate by the implement of effective maintenance strategies. 5. Conclusion Assuming that all the activities have the same period of operation, the block based maintenance yields a fewer failure and a higher cost than the age based maintenance. In this case the minimum cost rate with an optimized interval can be obtained since the contribution from preventive maintenance is compensated by
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X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
the contributions from repair and corrective maintenance. The only way to find the minimum risk is to optimize the maintenance process to obtain the appropriate value of the operation period and the maintenance duration for a set of fixed reliability parameters. The optimization process also impacts the cost, because risks not only strongly depend on the failure rate of the equipment in the operation activity but also on the cost of the implementing maintenance actions. The results show that it is not difficult to conclude which strategy is better in a general sense. For circuit-breakers, the better strategy is strategy 1 and the worse strategy is to perform strategy 2 in view of cost rate. To the contrary, transformers have the minimum cost rate with strategy 2. References [1] U. Zickler, A. Machkine, M. Schwan, A. Schnettler, X. Zhang, E. Gockenbach, Asset management in distribution systems considering new knowledge on component reliability and damage costs, in: 15th Power Systems Computation Conference, Session 6, Belgium, 2005. [2] J. Endrenyi, S. Aboresheid, R.N. Allan, G.J. Anders, S. Asgarpoor, R. Billinton, N. Chowdhury, E.N. Dialynas, M. Fipper, R.H. Fletcher, C. Grigg, J. McCalley, S. Meliopoulos, T.C. Mielnik, P. Nitu, N. Rau, N.D. Reppen, L. Salvaderi, A. Schneider, Ch. Singh, The present status of maintenance strategies and the impact of maintenance on reliability, IEEE Transactions on Power Systems 16 (November (4)) (2001) 638–646. [3] X. Zhang, E. Gockenbach, Asset management of transformers based on condition monitoring and standard diagnosis, IEEE Electrical Insulation Magazine 4 (2008) 26–40. [4] C. Neumann, R. Huber, D. Meurer, R. Plath, U. Schichler, S. Tenbohlen, K.H. Weck, The impact of insulation monitoring and diagnostics on reliability and exploitation of service life, in: GIGRE, C4-201, Paris, 2006. [5] M. Bengtsson, E. Olsson, P. Funk, M. Jackson, Technical design of condition based maintenance system – a case study using sound analysis and case-based reasoning, in: Proceedings of the 8th Maintenance and Reliability Conference, Knoxville, USA, 2004. [6] Z.Y. Wang, Y.L. Liu, P.J. Griffin, Neutral net and expert system diagnose transformer faults, IEEE Computer Applications in Power 13 (1) (2000) 50–55. [7] J. Schlabbach, Reliability centered maintenance of MV circuit-breakers, in: IEEE Porto Power Tech. Conference, Portugal, September, 2001.
[8] G. Balzer, Condition assessment and reliability centered maintenance of high voltage equipment, in: Proceeding of International Symposium on Electrical Insulating Materials, Kitakyushu, Japan, June, 2005, pp. 259–264. [9] L. Bertling, A reliability-centered asset maintenance method for assessing the impact of maintenance in power distribution systems, IEEE Transactions on Power Systems 20 (February (1)) (2005) 75–82. [10] Y. Jiang, Z. Zhang, T. Van Voorhis, J. McCalley, Risk-based maintenance optimization for transmission equipment, in: Proceeding of 35th North American Power Symposium, Rolla, USA, October, 2003, pp. 416–423. [11] S. Martorell, A. Munoz, V. Serradell, Age-dependent models for evaluating risk and costs of surveillance and maintenance of components, IEEE Transactions on Reliability 45 (1996) 433–442. [12] P. Jirutitijaroen, C. Singh, The effect of transformer maintenance parameters on reliability and cost: a probabilistic model, Electric Power Systems Research 72 (3) (2004) 213–224. [13] X. Zhang, E. Gockenbach, Assessment of the actual condition of the electrical components in the medium-voltage networks, IEEE Transaction on Reliability 55 (2) (2006) 361–368. [14] X. Zhang, E. Gockenbach, Modeling the component reliability of distribution systems based on the evaluation of failure statistic, IEEE Transaction on Dielectrics and Electrical Insulation 14 (5) (2007) 1183–1191. [15] T. Nakagawa, Maintenance Theory of Reliability, Springer, 2005. [16] H.Z. Wang, H. Pham, Reliability and Optimal Maintenance, Springer, 2006.
Xiang Zhang received the B.Sc., M.Sc. and Ph.D. degrees in Electrical Engineering from Xi’an Jiaotong University, Xi’an, China, in 1989, 1992, and from the Aachen University of Technology, Aachen, Germany, in 2002, respectively. From 1992 to 1997 she was a research engineer at Xi’an High Voltage Apparatus Research Institute, Xi’an, China. Currently she is a research fellow on asset management of electrical equipment and networks of the Schering-Institute of High Voltage Technology at the University of Hanover, Germany. Her main areas of interest include high voltage apparatus, gas discharge, arc modeling, and asset management.
Ernst Gockenbach (M’83-SM’88-F’01) received the M.Sc. and Ph.D. degrees in Electrical Engineering from the Technical University of Darmstadt, Darmstadt, Germany, in 1974 and 1979, respectively. From 1979 to 1982, he worked at Siemens AG, Berlin, Germany. From 1982 to 1990, he worked with E. Haefely AG, Basel, Switzerland. Since 1990, he has been professor and director of the Schering-Institute of High Voltage Technology at the University of Hanover, Germany. He is member of VDE and CIGRE, chairman of GIGRE Study Committee D1 Materials and Emerging Technologies for Electro-technology, and a member of national and international Working Groups (IEC, IEEE) for Standardization of High Voltage Test and Measuring Procedures.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Age-dependent maintenance strategies of medium-voltage circuit-breakers and transformers夽 Xiang Zhang ∗ , Ernst Gockenbach Institute of Electric Power Systems, Division of High Voltage Engineering, Leibniz University of Hannover, Callinstrasse 25A, Hanover 30167, Germany
a r t i c l e
i n f o
Article history: Received 14 October 2008 Received in revised form 13 February 2011 Accepted 10 March 2011 Available online 7 May 2011 Keywords: Age Circuit-breaker Cost Failure rate Maintenance strategy Transformer
a b s t r a c t The general life and reliability models of electrical equipment are essential to evaluate their actual conditions because of the degradation of equipment. To optimize the maintenance strategies for maximal reliability and minimal cost in a quantitative way, available maintenance models of ageing equipment shall be found to describe actual maintenance actions in time-series processes. In particular, a functional relationship between failure rate and maintenance measures is to be developed for electrical equipment. This paper demonstrates some actual examples by applying these models and the results show the value of using a systematic quantitative approach to investigate the effect of different maintenance strategies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Deregulation of the power system market has forced electric utilities to scrutinize investment and maintenance expenditures much more rigorously than in the past. Many focus their investments on maintaining the competitive position of power generating assets, while trying to squeeze more performance from ageing power delivery assets with less expenditure. However, in the power transmission and distribution systems, the annual expenditures for maintenance and replacement average only 1% or less (equal to 20 billion Euro for distribution systems in Germany and 400 billion Dollar for transmission systems in USA), which corresponds to an expected lifetime for more than 100 years of electrical equipment [1]. Owing to the limited reinvestments, the age of electrical equipment will increase so that the utilities have to face various market requirements. On the one hand, customers are paying for a service and the authorities are imposed regulation, supervision, and compensation depending on the degree to which contracts and other obligations are fulfilled. On the other hand, utilities must ensure that their expenditure is cost-effective.
夽 This work was supported by the National Research Council of Germany under the Contract SPP 1101. ∗ Corresponding author. E-mail address: [email protected] (X. Zhang). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.03.018
The easiest way and the most widely used strategy today is the time-based maintenance (TBM) [2]. There are fixed time intervals for inspections and for certain maintenance works. However, it seems that the time intervals chosen are far from the safe side and the total replacement expenditure is extremely expensive, as there are many inspections revealing no problems at all. So, the time intervals obviously can be extended – the question is from which point of time the occurrence of failures will increase significantly. In order to obtain useful information about the actual conditions of equipment, condition monitoring technique and conditionbased maintenance (CBM) has been well developed [3–5]. Condition monitoring technique means mainly sensor development, data acquisition, data analysis, and development of methods for determination of equipment condition and early fault recognition. The importance of monitoring methods can recognize which measured parameter affects the ageing of equipment to a greater or lesser degree. It should support the introduction of conditionbased maintenance and help to avoid unexpected outages. The cost of sufficient instrumentation can often be quite large and off-line measurement even causes more significant outage. In the majority of cases, there are neither communication links nor suitable sensors available for monitoring from remote. In terms of numerical protection devices and control systems, those engineers are suffering from data overload. Also, from the technical side, it is not always as simple as possible. Even if some types of equipment can easily be observed by measuring simple values as temperature or
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X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
pressure, it is not trivial to turn this measured data into actionable knowledge about health of the equipment. In recent years, the reliability centered maintenance (RCM) has been established [6–8]. RCM is an industrial improvement approach focused on identifying and establishing the operational, maintenance, and capital improvement policies that will manage the risks (combination of severity and frequency) of equipment failure most effectively. As an economic and reliable life cycle management, the condition and the importance of the equipment must be combined and evaluated in order to create a cost-effective maintenance strategy, addressing dominant modes and causes of equipment failure. The artificial neutral network which exhibits the non-linear input–output relationships between equipment reliability and system maintenance is integrated into maintenance management system (expert system) via a commercial software tool. The reliability centered maintenance based on the analysis of “condition-importance” and the use of “weighting factors”, depends heavily on practical experiences in diagnostic analysis and maintenance measures. That has followed so-called reliability centered asset management (RCAM) since the last years [9–12]. The research approach deals with the life assessment (e.g. failure rate) and the maintenance tasks (e.g. inspection rate) based on several discrete exponential distributions. The assumption of exponential distribution spent in each stage implies constant failure/maintenance rate which is assigning to each operation state. In that way, the deterioration processes of equipment consist of a chain of discrete Markov models. Therefore, this parameter evaluated by using time between failures as input data for special network facility, is of the most concern in the analysis. These chosen parameters represent past experience by single facts or numbers, without assigning any degree of likelihood to future expectation. Therefore the approach does not provide enduring analytical models of the deterioration processes, by which the consequences of each failure can be effectively used on a predictive basis for the future. On the basis of the exponential distribution, the fitted curves are only a mathematical simulation that is not concerned with any information about the technical parameters and the operating conditions of electrical equipment, thus they can not give a complete understanding of the physical implications in failures. Up to now, the impacts of maintenance on reliability and cost can be analyzed only for those selected strategies. The reason is a lack of an available reliability model for the deteriorating electrical equipment and a theoretical maintenance model put into practice for power systems. In fact, only a continuous quantitative description (mathematical models) can lead to an optimal solution of the systematic maintenance strategies. Therefore, this broad perspective of asset management is recognized as “relating maintenance effort to system availability and total cost, with the aim to reach an optimal maintenance management”. This new research topic is world-wide challenge for all researchers on asset management of the power transmission and distribution systems.
2. Age-dependent reliability model The failure causes may identify the origin of detrimental effects, contributing to the failure of electrical equipment. The influence factors producing an ageing change of equipment are divided in different stresses such as electrical, thermal, mechanical and ambient stresses. The stresses of electric ageing are partial discharges, tracking, treeing, electrolysis and space charges. The intensity and progress of electrical ageing depends on the electric field strength in the insulation. The consequences of thermal ageing comprise chemical, physical and thermal–dynamic changes by chemical degradation,
polymerization, depolymerization of organic materials, diffusion and thermal–mechanical effects as a result of thermal extension or contraction. A mechanical ageing of equipment can only occur by solid material as a result of mechanical tractive and shear forces during transportation, installation and operation. The substantial reasons of mechanical ageing are electrodynamic, electromagnetic and thermal forces. Ageing caused by ambient conditions comes from the reaction of material to humidity, air, chemical, biological substances, weathering, pollution and radiation. But more or less the electrical equipment is simultaneously stressed by various influence factors. A thorough electro-thermo-mechanical life model of the electrical component can be established according to the Inverse Power Law and the Arrhenius Model. This can simply be done by assuming that ageing rate under these combined stresses is the product of ageing rates under each single stress [13,14]: L = L0
E −(n−bT ) M −m ·
E0
M0
· e−BT ,
T=
1 1 − ϑ0 ϑ
(1)
where E, M, T and L are the electrical, mechanical, thermal stresses and lifetime, respectively. E0 and M0 are the scale-parameters for the lower limit of electrical and mechanical stresses respectively (below which the ageing can be neglected) and L0 is the corresponding lifetime. n, m and B are the voltage-endurance coefficient, the mechanical stress-endurance coefficient and the activation energy of thermal degradation reaction, respectively. b is the correct coefficient which takes into account the reaction of materials due to combined stress application. ϑ and ϑ0 are the absolute temperature and the reference temperature. The failure of an electrical component may occur if over-voltage or mechanical stress is applied, or if the electrical component is aged by temperature or time. A criterion for the ageing of electrical components is consistent with the electrical, thermal or mechanical stress. Thus the relationship between lifetime and failure probability of electrical equipment may be determined by:
˛(n−bT ) m˛ ˛ E M L
Pr(L) = 1 − exp −
·
E0
M0
·
L0
·e
˛BT
(2)
Therefore, the lifetime L of an ageing component as a random variable t has a cumulative probability distribution F(t) ≡ Pr{L ≤ t} with right continuous, and a probability density function f(t) = dF(t)/dt
˛[n−b((1/ϑ )−(1/ϑ))] m˛ 0 E M
F(t) ≡ Pr{L ≤ t} = 1 − exp −
×
t ˛ L0
· e˛B((1/ϑ0 )−(1/ϑ))
E0
·
M0 (3)
Ageing is usually measured based on the term of a failure rate function. Failure rate is the most important quantity in maintenance theory. The instant failure rate function h(t) is defined as h(t) =
f (t) 1 dF(t) = · ¯ ¯ dt F(t) F(t)
(4)
The failure rate is the most important reliability items for electrical equipment. The failure rate allows electrical equipment in different asset classes to be compared with each other, and to make reference to several criteria like age, number of maintenance, time between events, etc. Whatever criteria for assessment are chosen, proper maintenance activities can be performed.
X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
3. Modeling risk and costs
=
cr
¯ ¯ S(x)h(x) dx − S() + cc S() + cp S()
0
3.1. Defining strategies and their contributions
0
During its life, the equipment might undergo several maintenance activities in order to improve its reliability, which is achieved by knowing its operability or by reducing its age. For circuitbreakers or transformers, all the typical activities are grouped in two strategies:
Strategy 2.
Maintenance with periodic preventive maintenance
The component is operated under a block-based periodic replacement. Imperfect repair between successive replacements is made after each failure so that the failure rate remains below a certain threshold level. Assumption.
1 S()
E{Z |L < } =
Notation.
Imperfect repair
When using a replacement, the system is “as good as new” after the repair action is completed. When the minimal repair is used, the component is “as bad as old” after the repair action. The replacement and minimal repair may thus be considered as two extreme cases, and components subject to imperfect repair will be somewhere between these two extremes [15]. Hence, an imperfect repair model is suggested for a component: the repair is a perfect repair with a certain failure probability Pr(t); the repair is a minimal one with 1 − Pr(t). In the case, the successive perfect repair times form a renewal process with interarrival time distribution
S(t) = 1 − exp
−
t
Pr(x)h(x) dx
(5)
y
¯ Pr(x)h(x) dx dS(y) 0
(8)
0
or the expected number of repairs when L ≥ [16]
¯ Pr(x)h(x) dx
E{Z |L ≥ } =
(9)
0
For Strategy 2. By definition the total expected cost C() of one cycle is given by the sum of preventive replacement cost, corrective replacement cost and repair cost within a preventive replacement:
C() =
All maintenance time is negligible.
(7)
¯ S(x) dx
Proof 1. Assume that Z represents the number of minimal repairs during the time interval (0, min {, L}), we have the expected number of repairs when L < [16]
Strategy 1. Maintenance with sequential preventive maintenance Consider a process where the component is subject to age-based sequential replacement. Imperfect repair between successive replacements is made after each failure so that the failure rate remains below a certain threshold level.
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[cr · (expected number of repairs during a preventive replacement) + cc · (expected number of corrective replacements during a preventive replacement) + cp ] one cycle of a preventive replacment
(10)
The total expected maintenance cost rate C() when using the interval time is equal to C() =
cr Nr () + cc Nc () + cp
(11)
Proof 2. If N() represents the number of failures during (0, ], N() is obviously the renewal function for the renewal process with the interarrival time distribution S(t) and can be determined by the solution method to the renewal function in renewal theory [16]. Therefore, the expected number of corrective replacements can be gotten.
Nc () = S() +
Nc ( − x) dS(x)
(12)
0
0
and the corresponding failure rate Pr(t)·h(t).
and the expected number of repairs
3.2. Modeling strategies
Nr () = expected number of repairs during a corrective replacement · cumulative probability of a corrective replacement
Within each strategy there are several component states and each one contributes to component risk. To calculate the various contributions to the total cost for each strategy, the renewal theory is used in time-series processes. In our maintenance model, two maintenance strategies are performed by the different combinations of preventive maintenance, repair or corrective replacement. For Strategy 1. Let C denotes the total expected maintenance cost per unit of time over an indefinitely long period. By definition the total expected cost C() of one cycle is given by the sum of cost rates of preventive maintenance and repair or corrective replacement and repair:
C() =
+ expected number of repairs during a preventive replacement · cumulative probability of a preventive replacement
¯ Nr ( − x) dS(x) = E{Z |L < }S() + E{Z |L ≥ }S()
+ 0
+
0
¯ S(x) h(x) dx − S() +
Nr ( − x) dS(x) = 0
Nr ( − x) dS(x) 0
(13)
{[cr · (expected number of repairs during a corrective replacement) + cc ] · cumulative probability of a corrective replacement + [cr · (expected number of repairs during a preventive replacement) + cp ] · cumulative probability of a preventive replacement} one cycle of a replacement (6)
where cc , cp and cr are the costs of the corrective- and consequential replacement, preventive replacement and imperfect repair respectively. cc includes unplanned replacement cost and unplanned unavailability cost. Thus, the total expected maintenance cost rate C() is equal to C() =
¯ {[cr E{Z |L < } + cc ] · S() + [cr E{Z |L ≥ } + cp ] · S()}
0
¯ S(x) dx
3.3. Optimizing strategies It is expected to decide at which intervals to maintain the component and whether to keep the component in an old state or replace the component with an identical component. These decisions are made by comparing the annual cost of the component and
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X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714 Table 3 Costs of different components (D ).
total cost rate
Component
cost rate
Optimum frequency preventive maintenance corrective-&consequential maintenance
Transformer Circuit-breaker
cc
3535 260
4771 6270
cr 172 55
0.3
maintenance frequency Fig. 1. Optimization of maintenance frequency.
Table 1 The parameters of Eq. (3). n
m
b (K)
B (K)
7.0
2.3
6000
17,000
L0 (year)
ϑ0 (K)
E0 (kV/mm)
M0 (N/mm2 )
4.5 × 10
298
5.0
2.4 × 10−4
failure rate (1/year)
repair
4
cp
0.2
0.1
0
0
7
14
21
28
35
year
dC() =0 d
(14)
Fig. 1 shows that the cost rate of preventive maintenance generally increases with higher maintenance frequency. On the other hand, the cost rates of corrective and consequential maintenance (including repair) associated with failures decrease as the maintenance frequency increases. The total cost rate, the sum of these three individual cost rates, exhibits a minimum, and so an “optimum” or target level of reliability is achieved. Thus the integrated cost benefit analyses involving customers and their decisions, in an asset management’s view, could be a more rewarding and applicable approach. 4. Simulation results from application studies The significant requirement in applying reliability models is a choice of the model parameters. This research work is initiated to collect nearly 120,000 failure data of historical events of electrical equipment in the special failure statistic from the year 1920 to 2005 [1]. These failure record data is dependent more on the component age than the calendar year. Furthermore, certain information on specific damage data, especially without disturbance of network operation, has been acquired. Therefore, the detailed failure statistic can provide information about the actual conditions and economic assessment of these increasingly old components as shown in Tables 1–3. In light of this goal, this work attempts to integrate so sufficient failure data into the reliability models that failure rate of electrical equipment can be modeled as a function of operating Table 2 The parameters of Eq. (3). Component
Transformer Circuit-breaker
Fig. 2. Calculated failure rates for circuit-breakers and transformers (dotted).
history, operational stresses and component type (Fig. 2). The failure rates increase with operation year and their increases become larger after a certain operation year. When a circuit-breaker is overhauled, the value of failure rate decreases to the value of the more new equipment. However, as wear parts without exchange may remain, the value of failure rate will increase like the shift curve. Under the maintenance policy of strategy 2, repair and corrective replacement are likely to be performed several times during the lifetime of equipment. Figs. 3 and 4 show the repair and replacement characteristics of circuit-breakers and transformers with a series of increments. Since the ageing leads to the degradation of equipment, the number of repair and replacement for circuit-breakers and transformers begins to grow rapidly. This means that the preventive replacement of circuit-breakers and transformers should be planned at 20 and 34 operation years, respectively. It is obvious that transformers have more numbers of repair and replacement than circuit-breakers. The
0.1
number of repair and corrective replacement
checking whether the component has reached its economic life. The economic life is also referred to as the minimum cost life or optimal maintenance interval. The optimal maintenance interval is the solution that minimizes the sum of these maintenance cost rates:
0.05
0
˛
E
M
ϑ
11.6 8.3
2.2E0 2.0E0
6.8M0 7.2M0
40 ◦ C 40 ◦ C
0
10
20
year Fig. 3. Calculated expected number of repair and corrective replacement (dotted) for circuit-breakers according to strategy 2.
X. Zhang, E. Gockenbach / Electric Power Systems Research 81 (2011) 1709–1714
1
number of repair and replacement
number of repair and corrective replacement
0.4
0.2
0.5
0 0
1713
0
7
14
21
28
0
10
year
20
30
35 Fig. 5. Calculated expected number of repair (below-dotted), corrective and preventive (above-dotted) replacements for circuit-breakers according to strategy 1.
year Fig. 4. Calculated expected number of repair (dotted) and corrective replacement for transformers according to strategy 2.
number of occasions depends on the failure rate of the equipment. When the failure rate increases, the number and cost rate of corrective and consequential maintenance (including repair) associated with failures increase as the operation time increases. On the other hand, the number and cost rate of preventive maintenance generally decreases under the age-dependent strategy 1. In the case of either strategy 1 or strategy 2, those failed circuit breakers and transformers have more repairs than corrective replacements. The probability of the equipment degradation increases, so we would perform more corrective maintenance. Taking into account this dependence it is clear that the contributions due to preventive maintenance should decrease, because corrective maintenances have opposite effect. The total cost rate, the sum of these three individual cost rates, exhibits a minimum, and so an “optimum” or target level of reliability is achieved. It will be useful to compare strategy 1 (Figs. 5 and 6) with strategy 2 for circuit breakers and transformers. In accordance with strategy 1 or strategy 2, circuit breakers have different optimal preventive intervals of 28 or 20 years with minimum cost rate, respectively. Strategy 2 is more wasteful since more preventive replacements occur and more unfailed circuit-breakers are removed than those under a similar policy based on age. As might be suspected however, the cost rate of transformers will be more under strategy 1. This is because stochastic failures cause more expensive costs of corrective and consequential maintenance. As utilizing strategy 1, the preventive replacement cannot be fully scheduled, and the policy may therefore be complex to administer. The final result in the economic evaluation is to estimate the annual cost of maintenance according to different maintenance strategies of strategy 1 and strategy 2. Table 4 shows the annual maintenance cost and optimal maintenance interval according to
Table 4 Annual maintenance cost (D /year) and optimal maintenance interval (year) according to different maintenance strategies. Strategy
Statistic 1 2
Circuit-breaker
Transformer
Cost rate
Interval
Cost rate
Interval
260 100 158
10 28 20
3535 266 250
2 26 34
number of repair and replacement
1
0.5
0
0
10
year
20
30
Fig. 6. Calculated expected number of repair (below-dotted), corrective and preventive (above-dotted) replacements for transformers according to strategy 1.)
different maintenance strategies. From the failure statistic [1], circuit breakers and transformers have the actual cost rate of 260 D /year and 3535 D /year for maintenance when a preventive maintenance is executed every 10 and 2 years. However, different optimal intervals and annual costs of maintenance for circuit breakers and transformers can be obtained according to strategy 1 or strategy 2. By minimizing the cost rate of 100 and 158 D /year, the interval of preventive maintenance is optimized by 28 or 20 years if strategy 1 or strategy 2 is caught into effect for circuit-breakers, respectively. Under these circumstances of strategies 1 and 2, transformers have cost rate of 266 or 250 D /year with the interval of 26 and 34 years, respectively. Comparing the actual cost rate of 260 and 3535 D /year, circuit breakers and transformers have a significant decrease in cost rate by the implement of effective maintenance strategies. 5. Conclusion Assuming that all the activities have the same period of operation, the block based maintenance yields a fewer failure and a higher cost than the age based maintenance. In this case the minimum cost rate with an optimized interval can be obtained since the contribution from preventive maintenance is compensated by
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the contributions from repair and corrective maintenance. The only way to find the minimum risk is to optimize the maintenance process to obtain the appropriate value of the operation period and the maintenance duration for a set of fixed reliability parameters. The optimization process also impacts the cost, because risks not only strongly depend on the failure rate of the equipment in the operation activity but also on the cost of the implementing maintenance actions. The results show that it is not difficult to conclude which strategy is better in a general sense. For circuit-breakers, the better strategy is strategy 1 and the worse strategy is to perform strategy 2 in view of cost rate. To the contrary, transformers have the minimum cost rate with strategy 2. References [1] U. Zickler, A. Machkine, M. Schwan, A. Schnettler, X. Zhang, E. Gockenbach, Asset management in distribution systems considering new knowledge on component reliability and damage costs, in: 15th Power Systems Computation Conference, Session 6, Belgium, 2005. [2] J. Endrenyi, S. Aboresheid, R.N. Allan, G.J. Anders, S. Asgarpoor, R. Billinton, N. Chowdhury, E.N. Dialynas, M. Fipper, R.H. Fletcher, C. Grigg, J. McCalley, S. Meliopoulos, T.C. Mielnik, P. Nitu, N. Rau, N.D. Reppen, L. Salvaderi, A. Schneider, Ch. Singh, The present status of maintenance strategies and the impact of maintenance on reliability, IEEE Transactions on Power Systems 16 (November (4)) (2001) 638–646. [3] X. Zhang, E. Gockenbach, Asset management of transformers based on condition monitoring and standard diagnosis, IEEE Electrical Insulation Magazine 4 (2008) 26–40. [4] C. Neumann, R. Huber, D. Meurer, R. Plath, U. Schichler, S. Tenbohlen, K.H. Weck, The impact of insulation monitoring and diagnostics on reliability and exploitation of service life, in: GIGRE, C4-201, Paris, 2006. [5] M. Bengtsson, E. Olsson, P. Funk, M. Jackson, Technical design of condition based maintenance system – a case study using sound analysis and case-based reasoning, in: Proceedings of the 8th Maintenance and Reliability Conference, Knoxville, USA, 2004. [6] Z.Y. Wang, Y.L. Liu, P.J. Griffin, Neutral net and expert system diagnose transformer faults, IEEE Computer Applications in Power 13 (1) (2000) 50–55. [7] J. Schlabbach, Reliability centered maintenance of MV circuit-breakers, in: IEEE Porto Power Tech. Conference, Portugal, September, 2001.
[8] G. Balzer, Condition assessment and reliability centered maintenance of high voltage equipment, in: Proceeding of International Symposium on Electrical Insulating Materials, Kitakyushu, Japan, June, 2005, pp. 259–264. [9] L. Bertling, A reliability-centered asset maintenance method for assessing the impact of maintenance in power distribution systems, IEEE Transactions on Power Systems 20 (February (1)) (2005) 75–82. [10] Y. Jiang, Z. Zhang, T. Van Voorhis, J. McCalley, Risk-based maintenance optimization for transmission equipment, in: Proceeding of 35th North American Power Symposium, Rolla, USA, October, 2003, pp. 416–423. [11] S. Martorell, A. Munoz, V. Serradell, Age-dependent models for evaluating risk and costs of surveillance and maintenance of components, IEEE Transactions on Reliability 45 (1996) 433–442. [12] P. Jirutitijaroen, C. Singh, The effect of transformer maintenance parameters on reliability and cost: a probabilistic model, Electric Power Systems Research 72 (3) (2004) 213–224. [13] X. Zhang, E. Gockenbach, Assessment of the actual condition of the electrical components in the medium-voltage networks, IEEE Transaction on Reliability 55 (2) (2006) 361–368. [14] X. Zhang, E. Gockenbach, Modeling the component reliability of distribution systems based on the evaluation of failure statistic, IEEE Transaction on Dielectrics and Electrical Insulation 14 (5) (2007) 1183–1191. [15] T. Nakagawa, Maintenance Theory of Reliability, Springer, 2005. [16] H.Z. Wang, H. Pham, Reliability and Optimal Maintenance, Springer, 2006.
Xiang Zhang received the B.Sc., M.Sc. and Ph.D. degrees in Electrical Engineering from Xi’an Jiaotong University, Xi’an, China, in 1989, 1992, and from the Aachen University of Technology, Aachen, Germany, in 2002, respectively. From 1992 to 1997 she was a research engineer at Xi’an High Voltage Apparatus Research Institute, Xi’an, China. Currently she is a research fellow on asset management of electrical equipment and networks of the Schering-Institute of High Voltage Technology at the University of Hanover, Germany. Her main areas of interest include high voltage apparatus, gas discharge, arc modeling, and asset management.
Ernst Gockenbach (M’83-SM’88-F’01) received the M.Sc. and Ph.D. degrees in Electrical Engineering from the Technical University of Darmstadt, Darmstadt, Germany, in 1974 and 1979, respectively. From 1979 to 1982, he worked at Siemens AG, Berlin, Germany. From 1982 to 1990, he worked with E. Haefely AG, Basel, Switzerland. Since 1990, he has been professor and director of the Schering-Institute of High Voltage Technology at the University of Hanover, Germany. He is member of VDE and CIGRE, chairman of GIGRE Study Committee D1 Materials and Emerging Technologies for Electro-technology, and a member of national and international Working Groups (IEC, IEEE) for Standardization of High Voltage Test and Measuring Procedures.