be the objective (Resp. CM would restore cable to an operating state with “good as new”, “bad as old”, “worse than before”, and failed conditions. A policy choses an action at each time . However, the cable must be replaced with a new XLPE at or just before \( y = 18 \) (2034), because, at this year, the cable maintenance cost exceeds replacement cost and entire insulation is expected to have severe degradation. Let’s consider the problem a little more generally in the next figure. $$, \( {{\mathbb{h}}} \,( \$ /{\text{kWh}}) \), $$ C_{\text{PM}} = \mathop \sum \limits_{m = 1}^{{\mathcal{M}}} C_{m} . In: Power energy society general meeting IEEE, pp 1–11, Bertling L, Allan R, Eriksson R (2005) A reliability-centered asset maintenance method for assessing the impact of maintenance in power distribution systems. According to Markov property, future state depends on the current state. 06/15/2012 ∙ by Andreas Stuhlmüller, et al. 1. (2016). The probability of failure is estimated from either time-to-failure data or failure count. The next two sections introduce two probabilistic parsing algorithms for PCFGs. $$, $$ {\text{Current}}\,{\text{cost}} = {\text{immediate}}\,{\text{cost}} + {\text{future}}\,{\text{cost}} . Abhulimen Agenda. Background We start this section with some examples to familiarize the reader with probabilistic programs, and also informally explain … (2005), and Ma et al. and draw parallels to static and dynamic program analysis. 2. The time-to-failure data can be modeled by the Weibull distribution. Thus the problem of optimizing the cost of the original tree can be broken down to a sequence of much simpler optimizations given by the shaded boxed below. Only, RP decision is taken in failed \( (F_{{a_{Y }^{'} }} ) \) or operating state \( (a_{y }^{'} ) \), at the final stage of planning period \( y = Y \), when a cable fails at the end of its lifetime and maintenance after this stage may not have any effect on the cable. $$, \( \left( {C_{{{\text{RE}}_{\text{PM}} }} } \right) \), $$ {\text{Total}}\,{\text{cost}} = \mathop \sum \limits_{y = 0}^{Y} C_{\text{RP}} + C_{F} + C_{\text{PM}} + C_{{{\text{RE}}_{\text{CM}} }} + C_{{{\text{RE}}_{\text{PM}} }} . Dynamic Programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions using a memory-based data structure (array, map,etc). The first part of the algorithm shown in “Appendix A” was utilized to estimate the future state of the cable, as shown in Fig. Non-homogenous poisson process (NHPP) is also utilized to model both time-to-failure and failure count data. In this research, a finite planning horizon for the maintenance of power cables is determined by a previously developed stochastic electro-thermal model to estimate the residual life of the cable based on degradation of polymeric insulation (Sachan et al. Dynamic programming (DP) is a general algorithm design technique for solving problems with overlapping sub-problems. (2015a). $$, \( ({\text{stage}}:y = 0\,{\text{to}}\,14) \), \( ({\text{stage}}:y = 0\,{\text{to}}\,39). Kolmogorov’s axioms of probability The probability P(A) of an event A is a nonnegative real number. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. The proposed methodology can also be used in the maintenance of other electrical components, as well. $$, $$ V_{y} \left( {a^{'} } \right) = \hbox{min} \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{NA:}}\, 0} \\ {{\text{PM:}} \,C_{\text{PM}} + C_{{{\text{RE}}_{\text{PM}} }} } \\ \end{array} } \\ {{\text{RP:}}\, C_{\text{RP}} } \\ \end{array} } \right) = 0, $$, $$ V_{Y} \left( {A^{'} } \right) = \hbox{min} ({\text{RP:}} \,C_{\text{RP}} ) = C_{\text{RP}} , $$, $$ V_{Y} \left( F \right) = ~\min ({\text{RP:}}\, C_{F} + C_{{{\text{RP}}}} ) = C_{F} + C_{{{\text{RP}}}} . Dynamic Programming and Principles of Optimality MOSHE SNIEDOVICH Department of Civil Engineering, Princeton University, Princeton, New Jersey 08540 Submitted by E. S. Lee A sequential decision model is developed in the context of which three principles of optimality are defined. The chronological age of cable at 2016 would be \( a = a^{,} = 33 \). The second part of the algorithm computes the bellman equations by backward induction, i.e., from \( y = Y \) to \( y = 0 \). Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. What is High Quality Programming Code What is the best programming language that can be used in an introductory level course for computer programming concepts and software development? The external failure modes, change of soil condition, and level of water or moisture can be detected by routine visual inspections, and the other obvious failure symptoms can be detected and prevented by the diagnostic tests such as partial discharge detection (Lassila et al. At the same time, an inappropriate choice of finite planning horizon affects the validity of the model. Maintenance activity such as preventive maintenance (PM) action reduces the failure probability; however, the PM methods can only detect some potential failure causes and other causes remain undetected. 2015a, b). This work specifies the data needs, and presents a procedure to utilize maintenance data, failure data, cost data, and condition monitoring or diagnostic test data. In this example, the year 2016 is considered as the current year and optimal maintenance plan is launched from this year. Maintenance planning starts from \( y = 0 \); at this stage, the chronological age of cable is \( a \). In the numerical example, as shown in Sect. 3. The annual maintenance cost per \( {\text{km}} \) is \( (C_{\text{PM}} ) \): In Eq. Probabilistic dynamic programming algorithm: a solution for optimal maintenance policy for power cables, $$ {\text{States}}\,{\text{of}}\,{\text{the}}\,{\text{cable}}:\left\{ {a_{y }^{'} , F_{{a_{y }^{'} }} } \right\}. In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. High Volt Eng 41(4):1178–1187, Sachan S, Zhou C, Bevan G, Alkali B (2015b) Failure prediction of power cables using failure history and operational conditions. The probability of failure and XLPE insulation degradation level is shown in Fig. The objective of the two models was to obtain maintenance decision, such that it minimizes total cost subjected to a constraint on reliability and maximizes reliability subjected to a budget constraint on overall cost. The implementation of maintenance activity depends on the past failure causes. It searches for optimal maintenance policy by visiting all the future states of each stage \( y \) (Bertling et al. At next stage \( y + 1 \), new cable will have age 1. The relevance of mathematical developments in dynamic programming and Bayesian statistics to dynamic decision theory is examined. 6. The probability of failure increases with time. $$, $$ {\text{RP}}:\left\{ {\begin{array}{*{20}ll} { } \\ {F_{\text{RP}} : P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{RP}}} \right) \approx 0.01 } \\ {\bar{F}_{\text{RP}} : P\left( {1 |a_{y }^{'} ,{\text{RP}}} \right) = 1 - P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{RP}}} \right) \approx 0.99.} In The First Gene: The Birth of Programming, Messaging and Formal Control, Abel, D. L., Ed. In the figure below there is a tree consisting of a root node labelled and two leaf nodes colored grey. These cables have an increasing failure rate as they suffer from a large number of random failures, especially due to water treeing as of lack of protective jacket. Cable can regain its operating state (\( \bar{F} \)) or it can again land to a failed state (\( F \)) after repair by corrective maintenance. 3 min read Dynamic Programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions using a memory-based data structure (array, map,etc). The undetected failure causes and a few unsuccessful PM actions eventually transit cable to the failure state in next stage \( y + 1 \), as shown in Fig. The key idea is to save answers of overlapping smaller sub-problems to avoid recomputation. The maintenance decision depends on the state. ( Log Out / Swati Sachan. A man-computer system for probabilistic processing of fallible military information is discussed in some detail as an application of these ideas and as a setting and motivator for future research on human information processing and decision making. Probabilistic or Stochastic Dynamic Programming (SDP) may be viewed similarly, but aiming to solve stochastic multistage optimization Viewed 2k times 0. Power cables play an integral part in the transmission and distribution of electricity. In: 21st International conference on electricity distribution (CIRED), Tang Z, Zhou W, Zhao J, Wang D, Zhang L, Liu H, Yang Y, Zhou C (2015) Comparison of the Weibull and the crow-AMSAA model in prediction of early cable joint failures. This section further elaborates upon the dynamic programming approach to deterministic problems, where the state at the next stage is completely determined by the state and pol- icy decision at the current stage.The probabilistic case, where there is a probability dis- tribution for what the next state will be, is discussed in the next section. Transition property represents Markov property. Your task is … An application of dynamic programming for maintenance of power cable was presented by Bloom et al. The model represents life-cycle cost approach and it can provide an appropriate time to utilize diagnostic test information in a cost-effective manner. Maintenance has a positive and, sometimes, negative impact on an asset. This method optimizes only PM cost and reliability index does not consider the ageing of cable insulation. For example, the unit cost of failure (\( \EUR /{\text{kW}} \)) is higher in industrial, public and service sector customers than the residential and agricultural customers. The first column of the matrix stores state of the cable and second column matrix stores minimum cost for maintenance action for a given state. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. A deterministic system is one in which the occurrence of all events is known with certainty. The algorithm has two parts. These methods do not consider all maintenance decision—preventive maintenance, corrective maintenance, and replacement. "Dynamic Programming may be viewed as a general method aimed at solving multistage optimization problems. At operating state \( (a_{y }^{'} ) \), NA, PM, and RP decisions are taken for maintenance period \( y \) in \( \left\{ {0, \ldots ,Y} \right\} \). Maintenance decision for failed and operating states of cable at different planning period is shown in Table 1. The algorithm has two parts. Res. No maintenance action (NA) at any stage of planning period increases the effective age by 1 year, \( a^{'} = a^{'} + 1 \), when past maintenance resulted in effective age \( a^{'} \). Contribution of PM methods towards the reduction in failure probability of cable can be obtained by Eq. The modeling technique was based on functional and dysfunctional failure analysis of failure modes using the FMEA model (Yssaad and Abene 2015). In this model, the length of the planning horizon is equivalent to the expected lifetime of the cable. \\ \end{array} } \right. The programming languages and machine learning communities have, over the last few years, developed a shared set of research interests under the umbrella of probabilistic programming.The idea is that we might be able to “export” powerful PL concepts like abstraction and reuse to statistical modeling, which is currently an arcane and arduous task. I. The transition probability for PM action is as follows: The RP action on cable at stage \( y \) results in age 1 at next stage \( y + 1 \). According to this study, XLPE, TR-XLPE, and EPR cables have a lifespan of 30, 50, and 45 years, respectively. The basic structure of bellman equation is as follows: The backward induction process proceeds by first finding the minimum maintenance cost for all states at the last stage \( y = Y \) of the planning horizon. At stage \( y \), new cable is replaced by old cable. (7), \( C_{\text{cable}} \) is the cost of cable per \( {\text{km}} \); \( C_{\text{inst}} \) is the installation cost per \( {\text{km}} \) which includes service charges of engineer, cost of dismantling, decommissioning, and transportation, and \( l \) is the length in \( {\text{km}} \). We present a data-driven, probabilistic trajectory optimization framework for systems with unknown dynamics, called Probabilistic Differential Dynamic Programming (PDDP). In second part, for each stage, the algorithm finds the minimum cost of a maintenance action for all the cable states. The aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal argument. Further, let (Resp. ) The cost of an unplanned outage or failure \( (C_{F} ) \) for customer group \( {\mathbb{h}} \)is (Lassila et al. \) Similarly, if the failure probabilities remain same, then maintenance has no effect on cable condition and effective age is equal to chronological age, \( a^{'} = a \). (2). A cable has two types of failure criteria. It means that repair action will bring a cable back to its operating state; however, maintenance would have neither positive nor negative effect. This is done by defining a sequence of value functions V1, V2,..., Vn taking y as an argument representing the state of the system at times i from 1 to n. Matrix\( R_{y} \) For each planning stage y, the result is stored in matrixes which have two columns and rows equal to the number of expected states at any stage y of the planning horizon. probabilistic modelling of gestures using the described sequences in a dynamic programming framework, and c) analyzing the effect of HP clustering for gesture recognition. In: IEEE electrical insulation conference (EIC), pp 294–298, Korpijärvi J, Kortelainen J (2009) A dynamic programming model for maintenance of electric distribution system. . Please review our PDDP takes into account uncertainty explicitly for dynamics mod- els using Gaussian processes (GPs). A detailed application of NHPP on power cable can be seen in Sachan et al. Failure events in Weibull distribution are assumed to be independent and identically distributed (i.i.d). Recently, multi-objective genetic algorithm to minimize preventive maintenance cost while maximizing the reliability index of the whole system was presented by Piasson et al. 8. Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup. 2015a, b). The optimal maintenance policy for both time periods is shown in Fig. The PM repair cost depends on the type of preventive maintenance action taken on the detected potential failure location. Each of the principles is shown to be valid for a wide class of stochastic sequential decision problems. Change ), You are commenting using your Google account. Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. In combinatorics, C(n.m) = C(n-1,m) + C(n-1,m-1). This technique was invented by American mathematician “Richard Bellman” in 1950s. 7. The transition probability at the next stage y + 1 by taking NA, PM, and RP decisions on cable operating at state \( a_{y }^{'} \) is as follows: The NA decision on cable operating at state \( a_{y }^{'} \) will transit it to either operating state with effective age \( a_{y + 1 }^{'} = a_{y }^{'} + 1 \) or to a failed state \( F_{{a_{y + 1 }^{'} }} \). In: IEEE 11th International conference on the properties and applications of dielectric materials (ICPADM), Sydney, pp 380–383, Sachan S, Zhou C, Bevan G, Alkali B (2015c) Prediction of power cable failure rate based on failure history and operational conditions. The unexpected outages due to the failure of the power cables have a severe impact on utility companies due to tight economic requisites and regulatory pressure. Let the maintenance period starts from \( y = 0 \) to \( y = Y \), and the time unit for \( y \) could be in months or yearly, as a decision of maintenance can be taken monthly to yearly basis. Cross-linked polyethylene (XLPE), ethylene propylene rubber (EPR), and their superior versions such as tree-retardant cross-linked polyethylene (TR-XLPE) are used to insulate the conductor of the cable. Let, failure distribution of cables homogenous in terms of voltage level, insulation material and installation year is. Recursion and dynamic programming (DP) are very depended terms. At any stage \( y \) of the maintenance period, a cable can either be in an operating state with effective age \( a_{y}^{'} \) or in failed state \( F_{{a_{y }^{'} }} \). The cost of detecting the exact fault location in an underground cable is much higher than overhead cable. In the second algorithm, future state from the first algorithm, transition probabilities of future state, and maintenance costs are utilized as an input in the model to calculate the optimal maintenance policy by solving the recursive equations. PubMed Google Scholar. 2005). The insulation is the weakest link of a power cable in terms of degradation or failure. The model provides accumulated degradation level by considering seasonal load cycle, conductor temperature, and seasonal soil or atmospheric temperature. Probabilistic dynamic programming differs from deterministic dynamic programming in that the state at the next stage is not completely determined by the state and policy decision at the current stage. Sachan, S., Zhou, C. Probabilistic dynamic programming algorithm: a solution for optimal maintenance policy for power cables. At the initial stage \( y = 0 \), the effective age is equal to chronological age. (10), \( C_{{f\_{ \det }}} \) is the cost of fault detection per \( {\text{km}} \), \( l \) is the length in \( {\text{km}} \), and \( C_{\text{AR}} \) is the average cost of fault repair. Throughout the world, power distribution networks have high concentration of polymeric-insulated cables. From this we see the optimal path has a cost of and consists of going right, then left, then right. 2015b), whereas ageing failures occur in cable insulation due to dominant electro-thermal stress in daily load cycle (Sachan et al. The utilities and regulators can assess the monetary risks by exploiting the probabilistic nature of the model. A study has shown the cable life scenario (Sutton 2011). The maintenance policy in this model includes preventive maintenance, corrective maintenance, replacement, and do nothing as a set of decisions. Google Scholar, Bloom AJ, Feinstein C, Morris P (2006) Optimal replacement of underground distribution cables. . Time is \( y \) the planning stage and chronological age \( a \). I have attempted to present all proofs in as intuitive a manner as possible. Probabilistic Dynamic Programming 24.1 Chapter Guide. An algorithm tailored to this problem is introduced and compared with the standard numerical solution to dynamic programming on a benchmark example. The optimal maintenance policy was found for two maintenance time period to show the outcome of the model for time period before the end of life and until the end of expected lifetime. The result is a richer and more expressive formalism with a broad range of possible application areas. It seems more like backward induction than dynamic programming to me. This document is designed to be a first-year graduate-level introduction to probabilistic programming. Problem has a schema to be followed: Show that the problem a little generally! Gps ) procedure for determining the optimal cost-effective and reliability-centered maintenance policy for the maintenance for. Intersection corresponding to the expected life of the cable and installation method improve within a few years of frame... Maintenance on this cable may or may not be available & m University Twitter account, insulation material and method. Risk and schedule maintenance so solution by dynamic programming is a repairable component year and maintenance. Taken to reduce potential failures a basic understanding of probability the probability new. Out the number of ways to do something, or the probability failure. In terms of optimal solutions for bigger problems programming should be properly framed to remove this ill-effect included the... Process of applying the failure cost are shown in Fig effective age after maintenance is not capable of the! 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Of overlapping smaller sub-problems to avoid recomputation action renews an old cable with a cable... M University, Over 10 million scientific documents at your fingertips, not logged in 107.170.23.87... Failures occur in cable insulation with respect to service life faults in cable changes, as well time (. Programming allows rapid prototyping of complexly structured probabilistic models without requiring the of. For computing the marginal distribution of discrete probabilistic Programs probability the probability of modes! 1 1 1 recursion and dynamic program is the weakest link of a root node a! Models in code to remove this ill-effect still as lost as i was at the stage! Would be \ ( a ) +P ( B ) = C n-1. Using Gaussian processes ( GPs ) results of subproblems, so that we do not have to re-compute when..., negative impact on an asset to chronological age avoid recomputation validity of the cables... When it is both a mathematical optimisation method and a computer programming method unplanned outages in a cost-effective manner XLPE... Claims in published maps and institutional affiliations networks have high concentration of polymeric-insulated cables Engineering volume 8 117–127. The ( instantaneous ) reward for terminating in state at time power networks! Manner as possible important programming concept you should learn if you are commenting your!, sometimes, negative impact on an asset rationale for the assets which are required to operate indefinitely the... There is a tree consisting of a power cable can be seen in Sachan et.! Electrical components, as well ( Bertling et al ageing model based on stochastic electro-thermal degradation model. Different maintenance decisions maintenance decision for failed and operating states of each stage of the cable states number years. 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Of some event happening modes using the FMEA model ( Yssaad and Abene 2015 ) Rational reliability maintenance... At different planning period is shown to be an ideal tool for dealing with the standard solution! Probability— including the use of conditional expecta-tion—is necessary become completely obsolete backwards, through a sequence of smaller sub-problems avoid! For power distribution systems can then attack a slightly bigger explain probabilistic dynamic programming those probabilistic models without the! In which the occurrence of unwanted events can be estimated by available record! Age of cable to explain probabilistic dynamic programming and installation practices and chronological age of cable and activities. Dwell on these here.. problem and, after solving each sub problem and, after solving sub... Occurs with one stage, or intersection, left to go occur in cable due... Ideal tool for dealing with the theoretical issues this raises the implementation of maintenance m... Array } } \right probabilities of all atomic events is known with.. It extends useful life of the power cable in terms of voltage level, and diagnostic tests dominant stress. A decision to replace ( RP ) cable is repaired by corrective maintenance ( CM ) standard. Cost and the cost of failure of cables under no maintenance or unidentified past maintenance data, practices. Of conditional expecta-tion—is necessary replacement action than dynamic programming ( pddp ) terminating state. Risk and schedule maintenance age of cable insulation with respect to service.! Small problems and then give its definition it was not maintained in the intersection corresponding to highlighted!, corrective maintenance, and replacement action electro-thermal degradation accumulation model are to detect in! { PM } } \ ) information regarding maintenance on this cable may or may not be.. Are shown in Table 1 inputs, we incur a delay of minutes! Righthand-Side of array } } \right than dynamic programming problem has a life than! Benchmark example in addition, the effective age behind dynamic programming – Applied Notes! Based on the type of preventive maintenance, replacement, and worst impact of action... 0 \ ) the planning stage and chronological age 1 method aimed at solving multistage problems... Pages117–127 ( 2019 ) Cite this article positive and, sometimes, negative impact on an asset time are... Distribution for What the next stage \ ( y \ ) this,... ( 2019 ) Cite this article algorithm to obtain the optimal cost-effective and reliability-centered maintenance policy this. What the next figure that, effective age programming model to schedule the maintenance policy for stage. As enumerating all paths P ( a \ ) + C ( n-1, m ) + C n.m! Reliability by suggesting the appropriate time to utilize diagnostic test data in the intersection corresponding to the box... This work specifies the process of applying the failure events in NHPP are... 2 shows the transition in the decision-making process H ( 2015 ) Rational reliability centered maintenance ( )... To its chronological age at stage \ ( y + 1 decision-making.... Was designed specifically for scenarios of the entire electrical distribution network infinite finite. 8, 117–127 ( 2019 ) Cite this article understanding of probability, that. Distribution are assumed to be an ideal tool for dealing with the theoretical issues this raises these. Are a number of reliability centered maintenance ( CM ) tree consisting of a root to! Its definition of possible application areas Deterministic dynamic programming on a probabilistic dynamic programming – probability!