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Effective computing algorithm for maintenance optimization of highly reliable systems
Reliability Engineering and System Safety 109 (2013) 77–85
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Effective computing algorithm for maintenance optimization of highly reliable systems Radim Briˇs a,n, Petr Byczanski b a ˇ Faculty of Electrical Engineering and Computer Science, VSB—Technical University Ostrava, Department of Applied Mathematics 17.listopadu 15, CZ 708 33 Ostrava-Poruba, The Czech Republic b Institute of Geonics AS CR, Department of Applied Mathematics and Computer Science, Studentska 1768, CZ 708 00 Ostrava-Poruba, The Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 7 September 2011 Received in revised form 16 August 2012 Accepted 21 August 2012 Available online 29 August 2012
This paper describes a new iterative numerical algorithm for optimal maintenance strategy respecting a given reliability constraint. It stems from the previous author’s research work which brings a new direct analytical method that enables exact reliability quantifications of highly reliable systems with maintenance (both preventive and corrective), i.e. the instantaneous unavailability function is computed in full machine accuracy. The method takes into account systems with highly reliable and maintained components, including repairable components undergoing to hidden failures. The new numerical algorithm for maintenance optimization introduced in this article fully respects previously developed exact computing methodology to solve a cost optimization problem where decision variable is maintenance. The algorithm, which is based on merits of a high performance language for technical computing MATLAB, results from linear approximation of total system cost that is supposed to be a linear function of frequency of maintenance and from limiting unavailability approximation in each iteration step. The optimization method is demonstrated on two systems from practice—a real power distribution network and high pressure injection system of a nuclear power plant. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Exact computing Maintenance Optimization Unavailability
1. Introduction A reliability engineer’s job is to make systems as reliable as possible. In this paper, we look at how systems can be made as reliable as possible, under the constraints that are imposed. We will consider that reliability optimization will be made by preventive or corrective maintenance. Maintenance can be done to extend the lifetimes of key components. Reliability has become an even greater concern in recent years because high-tech industrial processes, with increasing levels of sophistication, comprise most engineering systems today. Based on enhancing component reliability and providing redundancy, while considering the trade-off between system performance and resources optimal reliability design that aims to determine an optimal system-level configuration has long been an important topic in reliability engineering. Since 1960, many publications have addressed this problem using different system structures, performance measures, optimization techniques, and options for reliability improvement.
n
Corresponding author. Tel.: þ42 59732 3334; fax: þ42 596919597. E-mail addresses: [email protected] (R. Briˇs), [email protected] (P. Byczanski). 0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2012.08.010
Ref. [1] provides good overview of early work in system reliability optimization. It was during the 1970s that various heuristics were developed to solve complex system reliability problems in case that the traditional parametric optimization techniques were insufficient. The authors summarize the developments in optimization techniques, along with recent optimization methods such as metaheuristics. In our previous work [2] we developed a single objective optimization model, with periodically inspected and maintained components, aiming to find out the optimal maintenance policy for each component, by minimizing the cost function and respecting the availability constraint. In order to solve the problem we proposed a genetic algorithm based on simulation method for availability assessment, whose structure includes the first inspection time and the time length between two maintenance interventions for each component. Most of recent work in this area is devoted to multi-state system optimization [3], percentile life as a system performance measure [4], multi-objective optimization [5], active and cold stand-by redundancy [6], optimization techniques [7]. For example, this paper [8] aims to propose a resolution approach for a multi-objective maintenance problem with relation to a system that needs to operate without interruption between two consecutive fixed stops. The maintenance actions must guarantee a reliability level not lower than a fixed value up
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to the next fixed stop, minimizing simultaneously both cost and time of maintenance. We agree with authors that not many papers have explicitly considered the reliability as a basic parameter for the evaluation of system maintenance effectiveness. In other work of the same field [9] the authors propose an exact algorithm to minimize the maintenance cost when a constraint on the reliability is given. The authors show the efficiency of their proposal referring to a real case of a series–parallel system constituted by 200 elements. The paper solves the optimization problem for a system whose major components can be maintained only during a planned system downtime. Measures of system performance are basically of four kinds: reliability, availability, mean time-to-failure, and percentile life. Reliability is widely used and thoroughly studied as primary performance measure for non-maintained systems. For a maintained system that is taken into account in this paper, however, availability (or unavailability) will be considered instead of reliability, because it describes the percentage of time the system really functions. Some important design principles for improving system performance are summarized in [1]. Mostly they are based on reliability-redundancy allocation problems. However, any effort for improvement usually requires resources. Quite often it is hard for a single-objective system to adequately describe a real optimal design problem. For this reason, multi-objective system design problem always deserves a lot of attention. This paper describes a new iterative numerical algorithm for optimization of maintenance cost. It stems from the research presented in [10,11] which brings a direct analytical method that enables exact reliability quantifications of highly reliable systems with maintenance (both preventive and corrective). This computing methodology is also used within the new algorithm for maintenance optimization. General cost-optimization problem based on maintenance may be solved by the algorithm. The method is applicable for systems with highly reliable and maintained components. In this paper the algorithm is demonstrated on two relevant applications from practice—a real power distribution network and high pressure injection system of a nuclear power plant. For a repairable highly reliable system, the question is often when to replace or overhaul an old system. Replacing (or renewing) the system too often is expensive, but an old deteriorating system may fail often requiring many repairs. The objective is to choose the time between renewals to minimize the cost. There are many possible cost models for this scenario, but we will suppose a simple linear cost model.
a discontinuous objective functions that eventually presents local optima. For such problems, it is desired to find global optima not violating any constraint. In addition, requirements such as continuity and differentiability of both objective and constraints functions add yet another conflicting element to the decision problem. Apart from very simplified problems, resolution of such optimization problems requires numerical methods [12–14]. An optimization method based on Genetics Algorithms was successfully investigated in [15] for optimizing surveillance testing and maintenance of components. However, as traditional approaches usually give poor results under these circumstances, new iterative algorithm was investigated in this paper. 2.2. Problem formulation Our optimization problem can be formulated in terms of an objective function f(x) for a given scope, where the optimizer is intended to find the solution constrained by a number of restrictions imposed on the decision variables. The problem in this paper can be defined in terms of the following objective function f(x) to be minimized: f ðxÞ ¼ minC S
ð1Þ
subject to the constraint U S ðxÞ r U 0
ð2Þ
f(x) CS US(x) U0 x¼(x1,y, xn)ARn n
Objective function Total cost of maintenance of a system Maximal system unavailability within a mission time TM A specified limitation of US (maximal permissible value) Decision variable Number of decision variables to be optimized
Both f(x) and US(x) are generally linear or non-linear real-valued functions of the decision variable vector x ¼ ðx1 ,. . .,xn Þ A Rn representing the parameters to be optimized.
3. Unavailability and cost models 3.1. Unavailability model of a highly reliable system
2. Problem formulation 2.1. General overview An optimization problem can be formulated in terms of an objective function, to be either minimized or maximized, under constraints that apply for a given scope (i.e. component, system performance) and the method adopted to perform the parameter optimization (the optimizer). In fact, in optimization, objective functions and constraints cannot be handled independently of the underlying optimizer. In optimizing test intervals based on risk (or unavailability) and cost, like in many engineering optimization problems (i.e. design, reliability etc.), one normally faces multi-modal and non-linear objective functions and a variety of both linear and non-linear constraints. This results in a complex and discrete search space with regions of feasible and unfeasible solutions for
New methodology for exact reliability quantification of highly reliable systems with maintenance was introduced in [10]. It assumes that the system structure is mathematically represented by the use of directed acyclic graph (AG), see more details in [16]. Terminal nodes of the AG that represent system components are established by the definition of deterministic or stochastic process, to which they are subordinate. From them we can compute a function of instantaneous unavailability, of individual terminal nodes. Finally a correspondent instantaneous unavailability function U(x,t) of the highest node (TOP node) which represents reliability behavior of the whole system may be found. For a mission time TM, the following relation must be valid: Uðx,tÞ rU S ðxÞ,t A ½0,T M ;
ð3Þ
Validity of Eq. (3) is clear immediately from definition of US(x)(maximum of unavailability function during TM, for a maintenance strategy represented by decision vector x).
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From optimization point of view it is desired that US(x)rU0 during a mission time TM, i.e. there is defined a maximal value of system unavailability that cannot be overstepped. System unavailability function depends partly on graph structure and partly on component’s unavailability functions. We will assume that the structure of AG, as well as component hazard rate is invariant system characteristics. On the other side, the other component characteristics, as Test Intervals (TIs) or repair rates can be changed within a reasonable range. Just these component characteristics may be used as decision variables because they influence both conflicting functions, i.e. unavailability US(x) and cost CS (herein objective function f(x)). 3.2. Approximation—limiting unavailability Exact unavailability quantification of a complex system within a mission time TM is a hard computing process usually demanding a lot of CPU time. The process was a bit simplified to be integrated within the new iterative algorithm. Because the system unavailability function may be very changeable during TM, the above optimization problem was simplified in the following way. Instead of the maximal system unavailability US(x) within TM, we will consider in the constraint (2) the following system unavailability UL(x): U L ðxÞ ¼
lim U S ðxÞ
TM - þ 1
ð4Þ
On the basis of this definition we can claim that for each time point t the following inequality is valid: U S ðxÞ r U L ðxÞ
ð5Þ
If mission time is increased, then maximal unavailability value during the greater mission time cannot be less than previous maximum. Let us denote m1 ¼US(x), when tA[0, TM1] and m2 ¼US(x), when tA[0, TM2]. If TM1 oTM2, then m1 rm2 , because [0, TM1]C[0, TM2]. If mission time increases from TM1 to TM2, unavailability maximum is searched over increasing length of interval [0, TM2], which contains original interval [0, TM1]. So that inequality (5) must be valid for time going to infinity. In case of a short mission time TM the difference between US(x) and UL(x) may be large and our conclusions will be too conservative. On the other side for long TM the approximation US(x) EUL(x) will be valid and our results and conclusions will be more realistic. This simplification allows the analyst to simplify the computation of cost by the use of a linear approximation, which will be demonstrated in following section.
3.3. Models of highly reliable system components Different unavailability states of system components (sometimes they are named ‘‘basic events’’) are modeled by one of the three following maintenance models. 3.3.1. Model with elements that cannot be repaired Final unavailability function of the component is demonstrated by the distribution function of the time-to-failure: PðtÞ ¼ 1elt ,
ð6Þ
where l is the failure rate. This model has no parameters for optimization. In other words there are no decision variables that may contribute to solve the optimization problem above. Limiting unavailability of the non-repaired
79
ith component is estimated by the following formula: P L ðiÞ ¼ 1eli T M ,
ð7Þ
which may be quantified without loss of accuracy, as is described in [10]. 3.3.2. Model with repairable components Model with repairable components (CM—Corrective Maintenance), i.e. a model which supposes a failure (apparent failure) which is identified at its occurrence and immediately afterwards starts a process leading to its restoration. In this case we can derive a relation on the basis of Laplace’s transformation for a similar unavailability function: m l ðl þ mÞt PðtÞ ¼ 1 þ e lþm lþm i l h ð8Þ 1eðl þ mÞt ,t 4 0 ¼ lþm where m is the repair rate. The function converges for ith component and t-N to the value of P L ðiÞ ¼
li ,t 40 li þ mi
ð9Þ
The only one decision variable that influences unavailability and consequently that might be optimized is repair rate mi. For selected group of industrial applications we suppose that linear approximation of the ith component’s cost C(i) connected with the adequate repair action, contributing to the objective function (1) can be demonstrated as follows: CðiÞ ¼ C R ðiÞ kR ðiÞmi ,
ð10Þ
where kR (i) is positive proportion coefficient between cost and mi. 3.3.3. \Model with periodic test intervals and repairable components Model with periodic test intervals and repairable components, i.e. a model when a possible failure (hidden failure) is identified only at special deterministically predetermined times, appearing with a given period (time-points of periodical inspections). In case of its occurrence at these times an analogical restoration process starts, as in the previous case. If we hereby presume that a time to the end of repair is exponential random variable, it is necessary to derive formula for unavailability function, see [10]: PðtÞ ¼ ð1P C ÞUð1elt Þ m mt lt ðe e Þ , t 40 þP C 1þ ml
ð11Þ
where t is a time which has passed since the last planned inspection, Pc is the probability of a non-functional state of a component at the moment of inspection at the beginning of the interval to the next inspection. Resulting from formula (11) the following maximal limiting unavailability (12) can be derived for the ith component: P L ðiÞ ¼ ð1PL ðiÞÞUð1eli TIðiÞ Þ mi mi TIðiÞ li TIðiÞ e e þ PL ðiÞ 1 þ , mi li
ð12Þ
where TI(i) is the test interval of the ith component. Formula (12) can be used to quantify PL(i) even for a highly reliable component, i.e. without loss of accuracy [11], what is easy to proof. As decision variables in the third component model both mi and TI(i) may be used. Both have direct influence on the
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unavailability function. Cost contribution of the third model to the objective function can be estimated by linear approximation as well: CðiÞ ¼ C R ðiÞ þ C TI ðiÞ kR ðiÞmi þ
kTI ðiÞ , TIðiÞ
ð13Þ
where kR(i), kTI(i) are the positive proportion coefficients between cost and mi , eventually 1/TI(i). In both maintenance models we suppose that cost depends linearly on frequency of maintenance.
3.4. Total cost model and problem reformulation
Execution phase 1 is demonstrated in Fig. 2. A lot of randomly generated points within the projection plane with constant cost, which is demonstrated by dotted line, are investigated to find out the point with minimal unavailability UL(x) and not exceeding the constraint U0, as well. The point is designated as blank point. The starting point of the process is the point found in previous step 1, which is denoted by full black point in the Fig. 2. Execution phase 2 of the step 2 is demonstrated in the following Fig. 3. Starting from the point found in the phase 1, which in Fig. 3 is demonstrated by full black point, a directed line segment is constructed (finishing on surface of the unit cube) in such a direction that is defined by highest drop of cost. This direction is denoted by the arrow in Fig. 3. A lot of randomly generated points within the line are investigated to find out the
Let us denote pj as jth decision variable which can be either repair rate mj or reversed TI(j). Each decision variable can be changed for the optimization purpose within a given allowed range: pj A Dj ,U j The decision variable pj may be transformed to xj by the following way: xj ¼
pj Dj j ¼ 1,. . ., n ðNo: of decision variablesÞ U j Dj
Consequently we can see that xj A ½0,1 So that decision variables are transformed into relative unit space. Further let us denote kj as the cost coefficients kR(j),kTI(j) modified accordingly. Then decision problem (1,2) may be re-formulated as follows: f ðxÞ ¼ minC S
min
U L ðxÞ r U 0
n X
kj xj
Fig. 1. Starting step of the algorithm within unit cube.
ð14Þ
j¼1
In other words the aim of our endeavor is to find optimal vector x in the unit cube:x¼(x1,y, xn) from Rnwhich minimizes the total cost CS and fulfils the constraint condition UL(x) rU0 as well.
x2 1
4. Optimization method 4.1. New iterative algorithm The following new algorithm was generated in the highperformance language MATLAB. The algorithm searches the optimal vector x minimizing total cost in two steps. In the first step a preliminary solution x of the problem given by (14) is found as optimal point on internal diagonal of a unit cube satisfying constraint (2). In the second step the solution x is further rectified within two sub-steps: in the sub-step 1 we try to move to a point with minimal unavailability maintaining constant cost while in the sub-step 2 the cost is effectively dropped maintaining the unavailability constraint U0. Step 2 can be repeated, if necessary. Basic idea is demonstrated as follows using 2D space visualization. In Fig. 1 the starting step 1 of the algorithm is demonstrated. The curve denotes a surface having constant unavailability U0. The hatched area denotes the area in which the constraint fails. First, the internal diagonal of the unit cube in Rn
0
1
x1
Fig. 2. Execution phase 1 in the step 2.
x2 1
½0,0,0,. . .2½1,1,1,. . ., is uniformly apportioned into a lot of points, in which the limiting unavailability function UL(x) is quantified. Then, the admissible point satisfying the constraint is found, denoted by full black point. Generally, the found point is inside the unit cube. Further numerical elaboration of the point is done by applying step 2 which consists of the two execution phases:
0
1 Fig. 3. Execution phase 2 of the step 2.
x1
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one with admissible unavailability UL(x), i.e. not exceeding the limit U0. So that in the Execution phase 1 of the step 2 we try to move to the point with minimal unavailability maintaining constant cost while in the Execution phase 2 the cost is effectively dropped maintaining the unavailability constraint U0. The algorithm may be finished either in phase 1 (Fig. 2), in case that the initial black point minimizes the unavailability within the intended projection plane, or in phase 2 (Fig. 3), in case that the initial black point minimizes cost within intended line. If it has to be to the contrary, step 2 is repeated till such a point in which the difference in comparison with previous point is inconsiderable.
5. Results with tested systems
81
Table 1 Reliability parameters of equipment.
Outlet 22 kV Outlet 110 kV Transformer110/22 kV DTS Disconnector Line 01 Line 02 Line 03 Line 04 Line 05 Line 18 n
l [year 1]
MTTR [h]
PPMn[year]
0.013 0.015 0.059 0.006 0.007 0.0106 0.07 0.028 0.0196 0.0812 0.003
89 48 0.48 4.67 3.84 4.03 4.12 4.12 4.12 4.12 4.22
4 4 2 4 4 1 1 1 1 8
Period of preventive maintenance.
5.1. Tested system from practice The optimization of maintenance policy has been tested on many systems from practice [17]. In this paper it is performed at a real distribution MV (Medium Voltage) network, see Fig. 4. Such system is frequently used in the Czech Republic. Consumption point is a distribution transformer station MV/LV (DTS). This system is supplied from a 110 kV substation that is taken as point with ideal reliability. There is 22 kV substation fed from that 110 kV through transformer 110/22 kV. To improve reliability of 22 kV substation there is a parallel line from 110 kV substation. Network further consists of five outlets, twenty kilometers of lines, and three section disconnecting switches. Their reliability parameters are shown in Table 1 that contains additional data including periods of preventive maintenance. The time to failure of each component is supposed to be exponential, as well as the time to end of repair. Recent data on corrective and preventive maintenance cost of all components included into the network is listed in Table 2. Seeing that there is a power switch only in outlet of substation in the network, any failure in the MV part results in the outage of
Outlet 110 kV
Substation 110 kV
Line 18
Table 2 Corrective and preventive maintenance costs of components included into the network.
DTS Line Transformer(110/22 kV) Disconnecter Outlet n
CMCn[CZK]
PMCnn[CZK]
91,788 2000 25,300 4500 15,000
11,692 5000 45,000 9500 10,000
Corrective maintenance cost–cost of one repair. Preventive maintenance cost–cost of one inspection.
nn
the whole network as far as to this switch. Thus correct function of the system is defined so that the consumption point has to be supplied by the power. In highly reliable applications, when energy supply is crucial for safety reason, the system can be intended as one-out-of-three active parallel system, including three identical branches, as demonstrated in Fig. 4. Responsible workers are sensible to the fact that maintenance is processed unnecessarily frequently without any significant effect. Consequently, cost of the maintenance is too high. We would like to design in this paper such preventive maintenance scheme that meets a prescribed unavailability limit and optimizes cost at the same time.
Transformer 110 / 22 kV
Outlet 22 kV Substation 22kV Outlet 22 kV
Line 01
Disconnector Consumption point
Line 02 Disconnector Line 03
Disconnector
Line 04 Line 05
Fig. 4. Reliability block diagram of the network.
DTS
5.1.1. Unavailability calculations Basic unavailability computation of the highly reliable one-outof-three network system, given by original reliability parameters from Table 1 brings Fig. 5, where the instantaneous unavailability is demonstrated within the mission time of TM ¼10 years. Seeing first 30,000 h of life of the system in Fig. 6 one can observe that the system is very reliable. In Fig. 5 we see that maximal unavailability during the mission time TM ¼10 years is a value close 1.4 10 3. Let us suppose that there is a need to improve the unavailability to the maximal value of U0 ¼2 10 4 (see constraint in (2)) which must be reached with minimal cost at the same time. New optimal maintenance policy saving the constraint U0 has been found by the new iterative algorithm above. The distribution network is mostly composed from repairable components with periodic test intervals. Periods of preventive maintenance TI(i) have been selected as decision variables taking into account the third component model. For simplification, repair rates of the system components mi were supposed to be constants. The optimal case is characterized by the decision vector in Table 3.
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x10-3
U (t)
U (t)
2 x10-4
chronological time [hours]
chronological time [hours]
Fig. 5. Instantaneous unavailability U(t) of the network. Initial case.
log U (t)
log U (t)
Fig. 7. Instantaneous unavailability U(t) of the network. Optimization with constraint U0 ¼2 10 4.
chronological time [hours] Fig. 6. Dependence of log10(unavailability) on time within first 30,000 h of system life.
Table 3 Optimal preventive maintenance for network components. PPMn [h] Outlets Transformer 110/22 kV DTS Disconnector Lines n
8734 24,886 18,350 31,600 7420
Optimal period of preventive maintenance.
Fig. 7 represents the evolution of the instantaneous unavailability of the system for the optimal case when cost CS is minimized, and Fig. 8 first part (first 30,000 h) of the evolution using decimal logarithm of unavailability, for better resolution. If we compare the optimal periods of preventive maintenance in Table 3 with original data in Table 1, we can state that the length of period of transformers increased (from 2 years¼ 17,520 h to 24,886, i.e. growth about 7000 h) while period of DTS and outlets decreased. For DTS the fall from 35,040 h to 18,350 is apparent and for outlets even vigorous fall from 35,040 to 8734 h is apparent. However, this conclusion is not surprising if we take into account the cost in Table 2. Seeing that failure of DTS is very expensive (see cost of one repair) we must perform preventive maintenance more often. The same conclusion holds true for outlets. Contrary conclusion holds true for transformers. Seeing that cost of one preventive inspection is very expensive in comparison with failure (see cost of repair) the preventive maintenance must be performed with longer period.
chronological time [hours] Fig. 8. Dependence of log10(unavailability) on time within first 30,000 h of system life.
Disconnectors seem to be very important components that must be preventively maintained a bit more frequently than before (original period 35,040 and optimal 31,600 h), in spite of the fact that they are highly reliable components. All lines are quite simple components with low-cost maintenance, periods of which are comparable with original data, compare original period 1 year¼8750 with optimal 7420 h.
5.2. Tested system from reference As a second tested system on which the method was applied, the HPIS (High Pressure Injection System) of a nuclear power plant was selected. A simplified HPIS of a Pressurized Water Reactor is shown in Fig. 9, which has been adapted from Ref. [14] and the same system is analyzed also in [18,19]. This system is normally in stand-by and consists of three pumps and seven valves (Model III in Section 3.3) organized as shown in Fig. 9. Under accidental conditions the HPIS can be used to remove heat from the reactor in those events in which steam generators are unavailable. For example, in case of a Small-Break Loss-OfCoolant Accident the HPIS safety function draws water from the Refueling Water Storage Tank (RWST) and must discharge it into the cold legs of the Reactor Cooling System through any of the two injection paths. Normally, pumps discharge into the injection paths A and B through valves 3 and 5, although crossover valves 4, 6 and 7 provide alternative flow paths in case of failure of the normal feed. Table 4 shows typical TI (Test Intervals) requirements included within the HPIS Technical Specifications. In addition, the relevant component reliability and cost data for this system are adopted from [18,19].
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83
Fig. 9. HPIS—a system from Ref. [14].
Table 4 Typical TI for the HPIS. Pumps (P)
Valves (V)
TI [h]
2184
2184
U (t)
Setting
Table 5 Initial values for TI and TP. TI vs TP [h]
T1
PA PB PC V1 V2 V3 V4 V5 V6 V7
TA TB TC TA TC TA
T2
T3
TD TC
Table 5 shows a possible and initial distribution of TI and the time to first test, i.e. TP {TA, TB, TC, TD, TE, TF}, for the components of the HPIS grouped according to the applicable TI {T1, T2, T3}. For example, decision variables associated to pump PA correspond to the set {T1, TA}. In addition, the following relationships also apply, what concerns TI: T1¼2184 [h], T2¼3 T1, T3¼3 T1,and TP [h]: TA¼24, TB¼ 48, TC¼72, TD¼TB, TE ¼TBþT1, TF ¼TBþ2 T1. It follows from Table 5 and the several relationships shown above that decision making process can be based on the following decision vector: X ¼ fTI, TA, TB, TCg The HPIS will be used for the solution of the optimization problem, defined in (1,2) and redefined in (14). Because variables TA, TB, TC are fixed for the general case (Case 1 in [18], which will be used for comparison purpose), the decision on TI is necessary to find. 5.2.1. Assessment of initial conditions First, it is worthy to assess the departure point in order to establish the basis for comparing results after the optimization process. Herein, initial conditions are represented by the particularization of the decision vector for the initial set of TI and TP as follows: x ¼ f2184, 24, 48, 72g
log U (t)
TE TF
Chronological time [hours] Fig. 10. HPIS time-dependent unavailability U(t) and decimal logarithm of the unavailability. Initial case.
Fig. 10 represents the evolution of the time dependent unavailability of the HPIS in the initial case. Basic reliability data are taken from [19], see Table 6. The influence of mean down time due to testing tj on unavailability calculations has been neglected in this paper. Model III does not take this parameter into account in its first version because the algorithm is primarily intended for highly reliable components and systems, where ti o oTI.
5.2.2. Problem resolution In Fig. 10 one can see that maximal unavailability during mission time TM¼50,000 h is a value below and close to 0.03
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Table 6 HPIS-reliability and cost data for Pumps (P) and Valves (V).
P V
Table 7 Comparison of cost.
lj [h 1]
rj
Chc, j [$/h]
Cht, j [$/h]
dj [h]
tj [h]
3.89 10 6 5.83 10 6
5.3 10 4 1.82 10 3
20 20
15 15
24 2.6
4 0.75
lj ¼failure rate of the j-th component.
rj ¼probability of failure on demand of the j-th component. Chc, j ¼costs associated with corrective maintenance of the j-th comp. Cht, j ¼costs associated with preventive testing of the j-th component. dj ¼ mean down time of the j-th component due to corrective maintenance. tj ¼ mean down time of the j-th component due to testing.
CS [$]
initial case
optimal case
8212
13,861
where n is the number of system components and other variables are in context with Table 6. Note that formula (16) is in good accordance with our linear approximation of cost, see formulas (13,14). Using the data from Table 6 total cost CS may be computed for both initial and optimal case. Table 7 demonstrates that fall in US(x) by about 0.02 results in rise in cost to approximately 68%. Similar result was obtained in [18], which is based on decision vector (15), where cost has risen by 62%. We can conclude that despite of using very different methodological approaches to solve the optimization problem demonstrated in Section 2.2, comparable results may be obtained.
log U (t)
6. Conclusions
Chronological time [hours] Fig. 11. HPIS time-dependent unavailability U(t) and decimal logarithm of unavailability. Optimization with constraint 0.01, i.e. log U(t) r 2.
(US(x)¼0.028 in fact). The same optimization problem was solved in [18], wherein US(x) ¼0.056. This difference is caused by neglecting the mean down time of j-th component due to testing. Starting from the constraint U0(x)¼0.03 in Ref. [18], we try improve the unavailability US(x) in a similar way. Consequently, the unavailability constraint condition U0 ¼0.01 has been selected here to find a new optimal maintenance policy. In both cases a new maintenance policy is found resulting in a fall of maximal system unavailability US(x) by about 0.02. In addition, constraints are also imposed to the range of allowed values for decision variables. Thus, the range of allowed values for TP is dependent decision variable as TP depends on TI for the general case. Fig. 11 represents the evolution of the time-dependent unavailability of the HPIS for the optimal case when cost CS is minimized, maintaining US(x)r0.01, (i.e. logUðtÞ r2Þ: The optimal case is characterized by the following decision vector: x ¼ f1275,24,48,72g . We can conclude that the result is comparable with the result of Case 1 in [18]: x ¼ f1128, 24, 48, 72g
ð15Þ
However, this result was achieved by completely different algorithm MOEA (Multiple Objective Evolutionary Algorithm), which uses a Multiple Objective Genetic Algorithm to perform the simultaneous optimization of periodic TI and TP. Finally, the cost of both initial and optimal case has been quantified and compared with the reference. For this reason, basic methodology of the cost estimation has been adopted from [19] n X TM TM CS ¼ C ht,j t j þ C hc,j dj rj þ lj TIj TIj TIj j¼1
ð16Þ
We demonstrated in the paper that the new iterative algorithm to solve a maintenance optimization problem is viable for practical use. The algorithm is based on linear approximation of cost and limiting unavailability approximation. In connection with innovative computing methodology it enables both maintenance optimization and exact unavailability quantification of systems with highly reliable components. Algorithm was demonstrated on two practical systems. On one hand a real distribution MV network from practice has been optimized and on the other hand, HPIS of a Pressurized Water Reactor from references is used. Achieved results of HPIS have been compared with similar calculations from references. It has been shown that in both cases the algorithm is able to solve the given optimization problem. The new computing and optimization method has been numerically realized within the high-performance language MATLAB. All computations above run below 1 s, on Pentium (R) 4 CPU 3.40 GHz, 2.00 GB RAM. Special benefit of the method consists in that it enables to optimize not only preventive maintenance, where test intervals TI(j) are optimized as decision variables, but also corrective maintenance, where repair rates mj may be optimized as decision variables, what was not fully utilized in this paper. Modifications of repair rates were not allowed at both used systems from practice. In the author’s opinion in many practical situations this characteristic has a potential to be modified and improved by applying flexible maintenance team. Very important advantage of the new method for maintenance optimization is that it respects the new computing methodology introduced in [11], which enables the analyst to calculate arbitrary small values of the unavailability function during a mission time exactly, i.e. in full machine accuracy. Such small unavailability values as for example values of order 10 45 are unreachable to be computed by other methods or software. The special computing methodology eliminates errors committed by PC when calculations close to error limit are executed. Computational efficiency was tested and compared with simulation method in [10]. In the future research work a new method will be further tested, compared with other optimization methods and applied to relevant practical applications, taking into account aging components also.
Acknowledgments This article has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070
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supported by Operational Programme ‘Research and Development for Innovations’ funded by Structural Funds of the European Union and state budget of the Czech Republic. References [1] Kuo W, Prasad V. An annotated overview of system reliability optimization. IEEE Transactions on Reliability 2000;49:487–793. [2] Bris R, Chatelet E, Yalaoui F. New method to minimize the preventive maintenance cost of series–parallel systems. Reliability Engineering and System Safety 2003;82:247–55. [3] Lisnianski A, Levitin G, Ben-Haim H. Structure optimization of multi-state system with time redundancy. Reliability Engineering and System Safety 2000;67:103–12. [4] Kim KO, Kuo W. Percentile life and reliability as a performance measures in optima system design. IIE Transactions 2003;35:1133–42. [5] Konak A, Coit DW, Smith AE. Multi-objective optimization using genetic algorithms: a tutorial. Reliability Engineering and System Safety 2006;91: 992–1007. [6] Romera R, Valdes JE, Zequeira RI. Active redundancy allocation in systems. IEEE Transactions on Reliability 2004;53:313–8. [7] Samrout M, Chatelet E, Kouta R, Chebbo N. Optimization of maintenance policy using the proportional hazard model. Reliability Engineering and System Safety 2009;94:44–52. [8] Certa A, Galante G, Lupo T, Passannanti G. Determination of Pareto frontier in multi-objective maintenance optimization. Reliability Engineering and System Safety 2011;96:861–7. [9] Galante G, Passannanti G. An exact algorithm for preventive maintenance planning of series–parallel systems. Reliability Engineering and System Safety 2009;9:1517–25.
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[10] Bris R. Exact reliability quantification of highly reliable systems with maintenance. Reliability Engineering and System Safety 2010;95:1289–92, http://dx.doi.org/10.1016/j.ress.2010.06.004. [11] Bris R, Byczanski P. Direct unavailability computation of a maintained highly reliable system. Proceedings of the IMechE, Part O: Journal of Risk and Reliability 2010;224(3):159–70, http://dx.doi.org/10.1243/1748006XJRR306. [12] Vaurio JK. Optimization of test and maintenance intervals based on risk and cost. Reliability Engineering and System Safety 1995;49:23–36. [13] Fletcher R. Practical Methods of Optimization. New York: Wiley; 1987. [14] Harunuzzaman M, Aldemir T. Optimization of standby safety system maintenance schedules in nuclear power plants. Nuclear Technology 1996;113: 354–67. [15] Munoz A, Martorell S, Serradell V. Genetic algorithms in optimizing surveillance and maintenance of components. Reliability Engineering and System Safety 1997;57:107–20. [16] Bris R. Parallel simulation algorithm for maintenance optimization based on directed Acyclic Graph. Reliability Engineering & System Safety 2008;93: 852–62. ˇ R. Maintenance modeling and optimization applied to [17] Bris R, Rusek S, Gono power distribution networks. In: C. Guedes Soares, editor. Safety and Reliability of Industrial Products, Systems and Structures. CRC Press/Balkema. & Taylor & Francis Group, London, UK; 2010. p. 295–306, ISBN: 978-0-41566392-2. [18] Martorell S, Carlos S, Villanueva JF, Sanchez AI, Galvan B, Salazar D, et al. Use of multiple objective evolutionary algorithms in optimizing surveillance requirements. Reliability Engineering and System Safety 2006;91:1027–38. [19] Podofillini L, Zio E. Designing a risk-informed balanced system by genetic algorithms: comparison of different balancing criteria. Reliability Engineering and System Safety 2008;93:1842–52.
Contents lists available at SciVerse ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Effective computing algorithm for maintenance optimization of highly reliable systems Radim Briˇs a,n, Petr Byczanski b a ˇ Faculty of Electrical Engineering and Computer Science, VSB—Technical University Ostrava, Department of Applied Mathematics 17.listopadu 15, CZ 708 33 Ostrava-Poruba, The Czech Republic b Institute of Geonics AS CR, Department of Applied Mathematics and Computer Science, Studentska 1768, CZ 708 00 Ostrava-Poruba, The Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 7 September 2011 Received in revised form 16 August 2012 Accepted 21 August 2012 Available online 29 August 2012
This paper describes a new iterative numerical algorithm for optimal maintenance strategy respecting a given reliability constraint. It stems from the previous author’s research work which brings a new direct analytical method that enables exact reliability quantifications of highly reliable systems with maintenance (both preventive and corrective), i.e. the instantaneous unavailability function is computed in full machine accuracy. The method takes into account systems with highly reliable and maintained components, including repairable components undergoing to hidden failures. The new numerical algorithm for maintenance optimization introduced in this article fully respects previously developed exact computing methodology to solve a cost optimization problem where decision variable is maintenance. The algorithm, which is based on merits of a high performance language for technical computing MATLAB, results from linear approximation of total system cost that is supposed to be a linear function of frequency of maintenance and from limiting unavailability approximation in each iteration step. The optimization method is demonstrated on two systems from practice—a real power distribution network and high pressure injection system of a nuclear power plant. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Exact computing Maintenance Optimization Unavailability
1. Introduction A reliability engineer’s job is to make systems as reliable as possible. In this paper, we look at how systems can be made as reliable as possible, under the constraints that are imposed. We will consider that reliability optimization will be made by preventive or corrective maintenance. Maintenance can be done to extend the lifetimes of key components. Reliability has become an even greater concern in recent years because high-tech industrial processes, with increasing levels of sophistication, comprise most engineering systems today. Based on enhancing component reliability and providing redundancy, while considering the trade-off between system performance and resources optimal reliability design that aims to determine an optimal system-level configuration has long been an important topic in reliability engineering. Since 1960, many publications have addressed this problem using different system structures, performance measures, optimization techniques, and options for reliability improvement.
n
Corresponding author. Tel.: þ42 59732 3334; fax: þ42 596919597. E-mail addresses: [email protected] (R. Briˇs), [email protected] (P. Byczanski). 0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2012.08.010
Ref. [1] provides good overview of early work in system reliability optimization. It was during the 1970s that various heuristics were developed to solve complex system reliability problems in case that the traditional parametric optimization techniques were insufficient. The authors summarize the developments in optimization techniques, along with recent optimization methods such as metaheuristics. In our previous work [2] we developed a single objective optimization model, with periodically inspected and maintained components, aiming to find out the optimal maintenance policy for each component, by minimizing the cost function and respecting the availability constraint. In order to solve the problem we proposed a genetic algorithm based on simulation method for availability assessment, whose structure includes the first inspection time and the time length between two maintenance interventions for each component. Most of recent work in this area is devoted to multi-state system optimization [3], percentile life as a system performance measure [4], multi-objective optimization [5], active and cold stand-by redundancy [6], optimization techniques [7]. For example, this paper [8] aims to propose a resolution approach for a multi-objective maintenance problem with relation to a system that needs to operate without interruption between two consecutive fixed stops. The maintenance actions must guarantee a reliability level not lower than a fixed value up
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to the next fixed stop, minimizing simultaneously both cost and time of maintenance. We agree with authors that not many papers have explicitly considered the reliability as a basic parameter for the evaluation of system maintenance effectiveness. In other work of the same field [9] the authors propose an exact algorithm to minimize the maintenance cost when a constraint on the reliability is given. The authors show the efficiency of their proposal referring to a real case of a series–parallel system constituted by 200 elements. The paper solves the optimization problem for a system whose major components can be maintained only during a planned system downtime. Measures of system performance are basically of four kinds: reliability, availability, mean time-to-failure, and percentile life. Reliability is widely used and thoroughly studied as primary performance measure for non-maintained systems. For a maintained system that is taken into account in this paper, however, availability (or unavailability) will be considered instead of reliability, because it describes the percentage of time the system really functions. Some important design principles for improving system performance are summarized in [1]. Mostly they are based on reliability-redundancy allocation problems. However, any effort for improvement usually requires resources. Quite often it is hard for a single-objective system to adequately describe a real optimal design problem. For this reason, multi-objective system design problem always deserves a lot of attention. This paper describes a new iterative numerical algorithm for optimization of maintenance cost. It stems from the research presented in [10,11] which brings a direct analytical method that enables exact reliability quantifications of highly reliable systems with maintenance (both preventive and corrective). This computing methodology is also used within the new algorithm for maintenance optimization. General cost-optimization problem based on maintenance may be solved by the algorithm. The method is applicable for systems with highly reliable and maintained components. In this paper the algorithm is demonstrated on two relevant applications from practice—a real power distribution network and high pressure injection system of a nuclear power plant. For a repairable highly reliable system, the question is often when to replace or overhaul an old system. Replacing (or renewing) the system too often is expensive, but an old deteriorating system may fail often requiring many repairs. The objective is to choose the time between renewals to minimize the cost. There are many possible cost models for this scenario, but we will suppose a simple linear cost model.
a discontinuous objective functions that eventually presents local optima. For such problems, it is desired to find global optima not violating any constraint. In addition, requirements such as continuity and differentiability of both objective and constraints functions add yet another conflicting element to the decision problem. Apart from very simplified problems, resolution of such optimization problems requires numerical methods [12–14]. An optimization method based on Genetics Algorithms was successfully investigated in [15] for optimizing surveillance testing and maintenance of components. However, as traditional approaches usually give poor results under these circumstances, new iterative algorithm was investigated in this paper. 2.2. Problem formulation Our optimization problem can be formulated in terms of an objective function f(x) for a given scope, where the optimizer is intended to find the solution constrained by a number of restrictions imposed on the decision variables. The problem in this paper can be defined in terms of the following objective function f(x) to be minimized: f ðxÞ ¼ minC S
ð1Þ
subject to the constraint U S ðxÞ r U 0
ð2Þ
f(x) CS US(x) U0 x¼(x1,y, xn)ARn n
Objective function Total cost of maintenance of a system Maximal system unavailability within a mission time TM A specified limitation of US (maximal permissible value) Decision variable Number of decision variables to be optimized
Both f(x) and US(x) are generally linear or non-linear real-valued functions of the decision variable vector x ¼ ðx1 ,. . .,xn Þ A Rn representing the parameters to be optimized.
3. Unavailability and cost models 3.1. Unavailability model of a highly reliable system
2. Problem formulation 2.1. General overview An optimization problem can be formulated in terms of an objective function, to be either minimized or maximized, under constraints that apply for a given scope (i.e. component, system performance) and the method adopted to perform the parameter optimization (the optimizer). In fact, in optimization, objective functions and constraints cannot be handled independently of the underlying optimizer. In optimizing test intervals based on risk (or unavailability) and cost, like in many engineering optimization problems (i.e. design, reliability etc.), one normally faces multi-modal and non-linear objective functions and a variety of both linear and non-linear constraints. This results in a complex and discrete search space with regions of feasible and unfeasible solutions for
New methodology for exact reliability quantification of highly reliable systems with maintenance was introduced in [10]. It assumes that the system structure is mathematically represented by the use of directed acyclic graph (AG), see more details in [16]. Terminal nodes of the AG that represent system components are established by the definition of deterministic or stochastic process, to which they are subordinate. From them we can compute a function of instantaneous unavailability, of individual terminal nodes. Finally a correspondent instantaneous unavailability function U(x,t) of the highest node (TOP node) which represents reliability behavior of the whole system may be found. For a mission time TM, the following relation must be valid: Uðx,tÞ rU S ðxÞ,t A ½0,T M ;
ð3Þ
Validity of Eq. (3) is clear immediately from definition of US(x)(maximum of unavailability function during TM, for a maintenance strategy represented by decision vector x).
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From optimization point of view it is desired that US(x)rU0 during a mission time TM, i.e. there is defined a maximal value of system unavailability that cannot be overstepped. System unavailability function depends partly on graph structure and partly on component’s unavailability functions. We will assume that the structure of AG, as well as component hazard rate is invariant system characteristics. On the other side, the other component characteristics, as Test Intervals (TIs) or repair rates can be changed within a reasonable range. Just these component characteristics may be used as decision variables because they influence both conflicting functions, i.e. unavailability US(x) and cost CS (herein objective function f(x)). 3.2. Approximation—limiting unavailability Exact unavailability quantification of a complex system within a mission time TM is a hard computing process usually demanding a lot of CPU time. The process was a bit simplified to be integrated within the new iterative algorithm. Because the system unavailability function may be very changeable during TM, the above optimization problem was simplified in the following way. Instead of the maximal system unavailability US(x) within TM, we will consider in the constraint (2) the following system unavailability UL(x): U L ðxÞ ¼
lim U S ðxÞ
TM - þ 1
ð4Þ
On the basis of this definition we can claim that for each time point t the following inequality is valid: U S ðxÞ r U L ðxÞ
ð5Þ
If mission time is increased, then maximal unavailability value during the greater mission time cannot be less than previous maximum. Let us denote m1 ¼US(x), when tA[0, TM1] and m2 ¼US(x), when tA[0, TM2]. If TM1 oTM2, then m1 rm2 , because [0, TM1]C[0, TM2]. If mission time increases from TM1 to TM2, unavailability maximum is searched over increasing length of interval [0, TM2], which contains original interval [0, TM1]. So that inequality (5) must be valid for time going to infinity. In case of a short mission time TM the difference between US(x) and UL(x) may be large and our conclusions will be too conservative. On the other side for long TM the approximation US(x) EUL(x) will be valid and our results and conclusions will be more realistic. This simplification allows the analyst to simplify the computation of cost by the use of a linear approximation, which will be demonstrated in following section.
3.3. Models of highly reliable system components Different unavailability states of system components (sometimes they are named ‘‘basic events’’) are modeled by one of the three following maintenance models. 3.3.1. Model with elements that cannot be repaired Final unavailability function of the component is demonstrated by the distribution function of the time-to-failure: PðtÞ ¼ 1elt ,
ð6Þ
where l is the failure rate. This model has no parameters for optimization. In other words there are no decision variables that may contribute to solve the optimization problem above. Limiting unavailability of the non-repaired
79
ith component is estimated by the following formula: P L ðiÞ ¼ 1eli T M ,
ð7Þ
which may be quantified without loss of accuracy, as is described in [10]. 3.3.2. Model with repairable components Model with repairable components (CM—Corrective Maintenance), i.e. a model which supposes a failure (apparent failure) which is identified at its occurrence and immediately afterwards starts a process leading to its restoration. In this case we can derive a relation on the basis of Laplace’s transformation for a similar unavailability function: m l ðl þ mÞt PðtÞ ¼ 1 þ e lþm lþm i l h ð8Þ 1eðl þ mÞt ,t 4 0 ¼ lþm where m is the repair rate. The function converges for ith component and t-N to the value of P L ðiÞ ¼
li ,t 40 li þ mi
ð9Þ
The only one decision variable that influences unavailability and consequently that might be optimized is repair rate mi. For selected group of industrial applications we suppose that linear approximation of the ith component’s cost C(i) connected with the adequate repair action, contributing to the objective function (1) can be demonstrated as follows: CðiÞ ¼ C R ðiÞ kR ðiÞmi ,
ð10Þ
where kR (i) is positive proportion coefficient between cost and mi. 3.3.3. \Model with periodic test intervals and repairable components Model with periodic test intervals and repairable components, i.e. a model when a possible failure (hidden failure) is identified only at special deterministically predetermined times, appearing with a given period (time-points of periodical inspections). In case of its occurrence at these times an analogical restoration process starts, as in the previous case. If we hereby presume that a time to the end of repair is exponential random variable, it is necessary to derive formula for unavailability function, see [10]: PðtÞ ¼ ð1P C ÞUð1elt Þ m mt lt ðe e Þ , t 40 þP C 1þ ml
ð11Þ
where t is a time which has passed since the last planned inspection, Pc is the probability of a non-functional state of a component at the moment of inspection at the beginning of the interval to the next inspection. Resulting from formula (11) the following maximal limiting unavailability (12) can be derived for the ith component: P L ðiÞ ¼ ð1PL ðiÞÞUð1eli TIðiÞ Þ mi mi TIðiÞ li TIðiÞ e e þ PL ðiÞ 1 þ , mi li
ð12Þ
where TI(i) is the test interval of the ith component. Formula (12) can be used to quantify PL(i) even for a highly reliable component, i.e. without loss of accuracy [11], what is easy to proof. As decision variables in the third component model both mi and TI(i) may be used. Both have direct influence on the
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unavailability function. Cost contribution of the third model to the objective function can be estimated by linear approximation as well: CðiÞ ¼ C R ðiÞ þ C TI ðiÞ kR ðiÞmi þ
kTI ðiÞ , TIðiÞ
ð13Þ
where kR(i), kTI(i) are the positive proportion coefficients between cost and mi , eventually 1/TI(i). In both maintenance models we suppose that cost depends linearly on frequency of maintenance.
3.4. Total cost model and problem reformulation
Execution phase 1 is demonstrated in Fig. 2. A lot of randomly generated points within the projection plane with constant cost, which is demonstrated by dotted line, are investigated to find out the point with minimal unavailability UL(x) and not exceeding the constraint U0, as well. The point is designated as blank point. The starting point of the process is the point found in previous step 1, which is denoted by full black point in the Fig. 2. Execution phase 2 of the step 2 is demonstrated in the following Fig. 3. Starting from the point found in the phase 1, which in Fig. 3 is demonstrated by full black point, a directed line segment is constructed (finishing on surface of the unit cube) in such a direction that is defined by highest drop of cost. This direction is denoted by the arrow in Fig. 3. A lot of randomly generated points within the line are investigated to find out the
Let us denote pj as jth decision variable which can be either repair rate mj or reversed TI(j). Each decision variable can be changed for the optimization purpose within a given allowed range: pj A Dj ,U j The decision variable pj may be transformed to xj by the following way: xj ¼
pj Dj j ¼ 1,. . ., n ðNo: of decision variablesÞ U j Dj
Consequently we can see that xj A ½0,1 So that decision variables are transformed into relative unit space. Further let us denote kj as the cost coefficients kR(j),kTI(j) modified accordingly. Then decision problem (1,2) may be re-formulated as follows: f ðxÞ ¼ minC S
min
U L ðxÞ r U 0
n X
kj xj
Fig. 1. Starting step of the algorithm within unit cube.
ð14Þ
j¼1
In other words the aim of our endeavor is to find optimal vector x in the unit cube:x¼(x1,y, xn) from Rnwhich minimizes the total cost CS and fulfils the constraint condition UL(x) rU0 as well.
x2 1
4. Optimization method 4.1. New iterative algorithm The following new algorithm was generated in the highperformance language MATLAB. The algorithm searches the optimal vector x minimizing total cost in two steps. In the first step a preliminary solution x of the problem given by (14) is found as optimal point on internal diagonal of a unit cube satisfying constraint (2). In the second step the solution x is further rectified within two sub-steps: in the sub-step 1 we try to move to a point with minimal unavailability maintaining constant cost while in the sub-step 2 the cost is effectively dropped maintaining the unavailability constraint U0. Step 2 can be repeated, if necessary. Basic idea is demonstrated as follows using 2D space visualization. In Fig. 1 the starting step 1 of the algorithm is demonstrated. The curve denotes a surface having constant unavailability U0. The hatched area denotes the area in which the constraint fails. First, the internal diagonal of the unit cube in Rn
0
1
x1
Fig. 2. Execution phase 1 in the step 2.
x2 1
½0,0,0,. . .2½1,1,1,. . ., is uniformly apportioned into a lot of points, in which the limiting unavailability function UL(x) is quantified. Then, the admissible point satisfying the constraint is found, denoted by full black point. Generally, the found point is inside the unit cube. Further numerical elaboration of the point is done by applying step 2 which consists of the two execution phases:
0
1 Fig. 3. Execution phase 2 of the step 2.
x1
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one with admissible unavailability UL(x), i.e. not exceeding the limit U0. So that in the Execution phase 1 of the step 2 we try to move to the point with minimal unavailability maintaining constant cost while in the Execution phase 2 the cost is effectively dropped maintaining the unavailability constraint U0. The algorithm may be finished either in phase 1 (Fig. 2), in case that the initial black point minimizes the unavailability within the intended projection plane, or in phase 2 (Fig. 3), in case that the initial black point minimizes cost within intended line. If it has to be to the contrary, step 2 is repeated till such a point in which the difference in comparison with previous point is inconsiderable.
5. Results with tested systems
81
Table 1 Reliability parameters of equipment.
Outlet 22 kV Outlet 110 kV Transformer110/22 kV DTS Disconnector Line 01 Line 02 Line 03 Line 04 Line 05 Line 18 n
l [year 1]
MTTR [h]
PPMn[year]
0.013 0.015 0.059 0.006 0.007 0.0106 0.07 0.028 0.0196 0.0812 0.003
89 48 0.48 4.67 3.84 4.03 4.12 4.12 4.12 4.12 4.22
4 4 2 4 4 1 1 1 1 8
Period of preventive maintenance.
5.1. Tested system from practice The optimization of maintenance policy has been tested on many systems from practice [17]. In this paper it is performed at a real distribution MV (Medium Voltage) network, see Fig. 4. Such system is frequently used in the Czech Republic. Consumption point is a distribution transformer station MV/LV (DTS). This system is supplied from a 110 kV substation that is taken as point with ideal reliability. There is 22 kV substation fed from that 110 kV through transformer 110/22 kV. To improve reliability of 22 kV substation there is a parallel line from 110 kV substation. Network further consists of five outlets, twenty kilometers of lines, and three section disconnecting switches. Their reliability parameters are shown in Table 1 that contains additional data including periods of preventive maintenance. The time to failure of each component is supposed to be exponential, as well as the time to end of repair. Recent data on corrective and preventive maintenance cost of all components included into the network is listed in Table 2. Seeing that there is a power switch only in outlet of substation in the network, any failure in the MV part results in the outage of
Outlet 110 kV
Substation 110 kV
Line 18
Table 2 Corrective and preventive maintenance costs of components included into the network.
DTS Line Transformer(110/22 kV) Disconnecter Outlet n
CMCn[CZK]
PMCnn[CZK]
91,788 2000 25,300 4500 15,000
11,692 5000 45,000 9500 10,000
Corrective maintenance cost–cost of one repair. Preventive maintenance cost–cost of one inspection.
nn
the whole network as far as to this switch. Thus correct function of the system is defined so that the consumption point has to be supplied by the power. In highly reliable applications, when energy supply is crucial for safety reason, the system can be intended as one-out-of-three active parallel system, including three identical branches, as demonstrated in Fig. 4. Responsible workers are sensible to the fact that maintenance is processed unnecessarily frequently without any significant effect. Consequently, cost of the maintenance is too high. We would like to design in this paper such preventive maintenance scheme that meets a prescribed unavailability limit and optimizes cost at the same time.
Transformer 110 / 22 kV
Outlet 22 kV Substation 22kV Outlet 22 kV
Line 01
Disconnector Consumption point
Line 02 Disconnector Line 03
Disconnector
Line 04 Line 05
Fig. 4. Reliability block diagram of the network.
DTS
5.1.1. Unavailability calculations Basic unavailability computation of the highly reliable one-outof-three network system, given by original reliability parameters from Table 1 brings Fig. 5, where the instantaneous unavailability is demonstrated within the mission time of TM ¼10 years. Seeing first 30,000 h of life of the system in Fig. 6 one can observe that the system is very reliable. In Fig. 5 we see that maximal unavailability during the mission time TM ¼10 years is a value close 1.4 10 3. Let us suppose that there is a need to improve the unavailability to the maximal value of U0 ¼2 10 4 (see constraint in (2)) which must be reached with minimal cost at the same time. New optimal maintenance policy saving the constraint U0 has been found by the new iterative algorithm above. The distribution network is mostly composed from repairable components with periodic test intervals. Periods of preventive maintenance TI(i) have been selected as decision variables taking into account the third component model. For simplification, repair rates of the system components mi were supposed to be constants. The optimal case is characterized by the decision vector in Table 3.
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x10-3
U (t)
U (t)
2 x10-4
chronological time [hours]
chronological time [hours]
Fig. 5. Instantaneous unavailability U(t) of the network. Initial case.
log U (t)
log U (t)
Fig. 7. Instantaneous unavailability U(t) of the network. Optimization with constraint U0 ¼2 10 4.
chronological time [hours] Fig. 6. Dependence of log10(unavailability) on time within first 30,000 h of system life.
Table 3 Optimal preventive maintenance for network components. PPMn [h] Outlets Transformer 110/22 kV DTS Disconnector Lines n
8734 24,886 18,350 31,600 7420
Optimal period of preventive maintenance.
Fig. 7 represents the evolution of the instantaneous unavailability of the system for the optimal case when cost CS is minimized, and Fig. 8 first part (first 30,000 h) of the evolution using decimal logarithm of unavailability, for better resolution. If we compare the optimal periods of preventive maintenance in Table 3 with original data in Table 1, we can state that the length of period of transformers increased (from 2 years¼ 17,520 h to 24,886, i.e. growth about 7000 h) while period of DTS and outlets decreased. For DTS the fall from 35,040 h to 18,350 is apparent and for outlets even vigorous fall from 35,040 to 8734 h is apparent. However, this conclusion is not surprising if we take into account the cost in Table 2. Seeing that failure of DTS is very expensive (see cost of one repair) we must perform preventive maintenance more often. The same conclusion holds true for outlets. Contrary conclusion holds true for transformers. Seeing that cost of one preventive inspection is very expensive in comparison with failure (see cost of repair) the preventive maintenance must be performed with longer period.
chronological time [hours] Fig. 8. Dependence of log10(unavailability) on time within first 30,000 h of system life.
Disconnectors seem to be very important components that must be preventively maintained a bit more frequently than before (original period 35,040 and optimal 31,600 h), in spite of the fact that they are highly reliable components. All lines are quite simple components with low-cost maintenance, periods of which are comparable with original data, compare original period 1 year¼8750 with optimal 7420 h.
5.2. Tested system from reference As a second tested system on which the method was applied, the HPIS (High Pressure Injection System) of a nuclear power plant was selected. A simplified HPIS of a Pressurized Water Reactor is shown in Fig. 9, which has been adapted from Ref. [14] and the same system is analyzed also in [18,19]. This system is normally in stand-by and consists of three pumps and seven valves (Model III in Section 3.3) organized as shown in Fig. 9. Under accidental conditions the HPIS can be used to remove heat from the reactor in those events in which steam generators are unavailable. For example, in case of a Small-Break Loss-OfCoolant Accident the HPIS safety function draws water from the Refueling Water Storage Tank (RWST) and must discharge it into the cold legs of the Reactor Cooling System through any of the two injection paths. Normally, pumps discharge into the injection paths A and B through valves 3 and 5, although crossover valves 4, 6 and 7 provide alternative flow paths in case of failure of the normal feed. Table 4 shows typical TI (Test Intervals) requirements included within the HPIS Technical Specifications. In addition, the relevant component reliability and cost data for this system are adopted from [18,19].
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Fig. 9. HPIS—a system from Ref. [14].
Table 4 Typical TI for the HPIS. Pumps (P)
Valves (V)
TI [h]
2184
2184
U (t)
Setting
Table 5 Initial values for TI and TP. TI vs TP [h]
T1
PA PB PC V1 V2 V3 V4 V5 V6 V7
TA TB TC TA TC TA
T2
T3
TD TC
Table 5 shows a possible and initial distribution of TI and the time to first test, i.e. TP {TA, TB, TC, TD, TE, TF}, for the components of the HPIS grouped according to the applicable TI {T1, T2, T3}. For example, decision variables associated to pump PA correspond to the set {T1, TA}. In addition, the following relationships also apply, what concerns TI: T1¼2184 [h], T2¼3 T1, T3¼3 T1,and TP [h]: TA¼24, TB¼ 48, TC¼72, TD¼TB, TE ¼TBþT1, TF ¼TBþ2 T1. It follows from Table 5 and the several relationships shown above that decision making process can be based on the following decision vector: X ¼ fTI, TA, TB, TCg The HPIS will be used for the solution of the optimization problem, defined in (1,2) and redefined in (14). Because variables TA, TB, TC are fixed for the general case (Case 1 in [18], which will be used for comparison purpose), the decision on TI is necessary to find. 5.2.1. Assessment of initial conditions First, it is worthy to assess the departure point in order to establish the basis for comparing results after the optimization process. Herein, initial conditions are represented by the particularization of the decision vector for the initial set of TI and TP as follows: x ¼ f2184, 24, 48, 72g
log U (t)
TE TF
Chronological time [hours] Fig. 10. HPIS time-dependent unavailability U(t) and decimal logarithm of the unavailability. Initial case.
Fig. 10 represents the evolution of the time dependent unavailability of the HPIS in the initial case. Basic reliability data are taken from [19], see Table 6. The influence of mean down time due to testing tj on unavailability calculations has been neglected in this paper. Model III does not take this parameter into account in its first version because the algorithm is primarily intended for highly reliable components and systems, where ti o oTI.
5.2.2. Problem resolution In Fig. 10 one can see that maximal unavailability during mission time TM¼50,000 h is a value below and close to 0.03
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Table 6 HPIS-reliability and cost data for Pumps (P) and Valves (V).
P V
Table 7 Comparison of cost.
lj [h 1]
rj
Chc, j [$/h]
Cht, j [$/h]
dj [h]
tj [h]
3.89 10 6 5.83 10 6
5.3 10 4 1.82 10 3
20 20
15 15
24 2.6
4 0.75
lj ¼failure rate of the j-th component.
rj ¼probability of failure on demand of the j-th component. Chc, j ¼costs associated with corrective maintenance of the j-th comp. Cht, j ¼costs associated with preventive testing of the j-th component. dj ¼ mean down time of the j-th component due to corrective maintenance. tj ¼ mean down time of the j-th component due to testing.
CS [$]
initial case
optimal case
8212
13,861
where n is the number of system components and other variables are in context with Table 6. Note that formula (16) is in good accordance with our linear approximation of cost, see formulas (13,14). Using the data from Table 6 total cost CS may be computed for both initial and optimal case. Table 7 demonstrates that fall in US(x) by about 0.02 results in rise in cost to approximately 68%. Similar result was obtained in [18], which is based on decision vector (15), where cost has risen by 62%. We can conclude that despite of using very different methodological approaches to solve the optimization problem demonstrated in Section 2.2, comparable results may be obtained.
log U (t)
6. Conclusions
Chronological time [hours] Fig. 11. HPIS time-dependent unavailability U(t) and decimal logarithm of unavailability. Optimization with constraint 0.01, i.e. log U(t) r 2.
(US(x)¼0.028 in fact). The same optimization problem was solved in [18], wherein US(x) ¼0.056. This difference is caused by neglecting the mean down time of j-th component due to testing. Starting from the constraint U0(x)¼0.03 in Ref. [18], we try improve the unavailability US(x) in a similar way. Consequently, the unavailability constraint condition U0 ¼0.01 has been selected here to find a new optimal maintenance policy. In both cases a new maintenance policy is found resulting in a fall of maximal system unavailability US(x) by about 0.02. In addition, constraints are also imposed to the range of allowed values for decision variables. Thus, the range of allowed values for TP is dependent decision variable as TP depends on TI for the general case. Fig. 11 represents the evolution of the time-dependent unavailability of the HPIS for the optimal case when cost CS is minimized, maintaining US(x)r0.01, (i.e. logUðtÞ r2Þ: The optimal case is characterized by the following decision vector: x ¼ f1275,24,48,72g . We can conclude that the result is comparable with the result of Case 1 in [18]: x ¼ f1128, 24, 48, 72g
ð15Þ
However, this result was achieved by completely different algorithm MOEA (Multiple Objective Evolutionary Algorithm), which uses a Multiple Objective Genetic Algorithm to perform the simultaneous optimization of periodic TI and TP. Finally, the cost of both initial and optimal case has been quantified and compared with the reference. For this reason, basic methodology of the cost estimation has been adopted from [19] n X TM TM CS ¼ C ht,j t j þ C hc,j dj rj þ lj TIj TIj TIj j¼1
ð16Þ
We demonstrated in the paper that the new iterative algorithm to solve a maintenance optimization problem is viable for practical use. The algorithm is based on linear approximation of cost and limiting unavailability approximation. In connection with innovative computing methodology it enables both maintenance optimization and exact unavailability quantification of systems with highly reliable components. Algorithm was demonstrated on two practical systems. On one hand a real distribution MV network from practice has been optimized and on the other hand, HPIS of a Pressurized Water Reactor from references is used. Achieved results of HPIS have been compared with similar calculations from references. It has been shown that in both cases the algorithm is able to solve the given optimization problem. The new computing and optimization method has been numerically realized within the high-performance language MATLAB. All computations above run below 1 s, on Pentium (R) 4 CPU 3.40 GHz, 2.00 GB RAM. Special benefit of the method consists in that it enables to optimize not only preventive maintenance, where test intervals TI(j) are optimized as decision variables, but also corrective maintenance, where repair rates mj may be optimized as decision variables, what was not fully utilized in this paper. Modifications of repair rates were not allowed at both used systems from practice. In the author’s opinion in many practical situations this characteristic has a potential to be modified and improved by applying flexible maintenance team. Very important advantage of the new method for maintenance optimization is that it respects the new computing methodology introduced in [11], which enables the analyst to calculate arbitrary small values of the unavailability function during a mission time exactly, i.e. in full machine accuracy. Such small unavailability values as for example values of order 10 45 are unreachable to be computed by other methods or software. The special computing methodology eliminates errors committed by PC when calculations close to error limit are executed. Computational efficiency was tested and compared with simulation method in [10]. In the future research work a new method will be further tested, compared with other optimization methods and applied to relevant practical applications, taking into account aging components also.
Acknowledgments This article has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070
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