. 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. Therefore, there are two possible kinds of repair. 6. 2005). (2005), Lassila et al. It can be assumed that the failure probability reduces by the same percentage and this affects the relative age of cable in comparison to cables without maintenance (Bertling et al. probabilistic modelling of gestures using the described sequences in a dynamic programming framework, and c) analyzing the effect of HP clustering for gesture recognition. and draw parallels to static and dynamic program analysis. The algorithm has two parts. The implementation of maintenance activity depends on the past failure causes. This can be established by studying the past maintenance data. where is the minimum cost from to a leaf node and where for is the node to the lefthand-side or righthand-side of . More so than the optimization techniques described previously, dynamic programming provides a general framework for analyzing many problem types. In combinatorics, C(n.m) = C(n-1,m) + C(n-1,m-1). In this model, the length of the planning horizon is equivalent to the expected lifetime of the cable. In the numerical example, as shown in Sect. PDDP takes into account uncertainty explicitly for dynamics models using Gaussian processes (GPs). If you ask me what is the difference between novice programmer and master programmer, dynamic programming is one of the most important concepts programming experts understand very well. The total reduced probability of failure is as follows: where \( a \) in \( \left\{ {0, \ldots , A} \right\} \) and \( a^{'} \) in \( \left\{ {0, \ldots , A^{'} } \right\} \) is chronological age and effective age, respectively. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. The PM decision at state \( a_{y }^{'} \) can detect \( {\text{PM}}\% \) of failures and reduce the failure probability by the same percentage. Abhulimen Agenda. Dynamic programming is a useful mathematical technique for making a sequence of in- terrelated decisions. The failure events in NHPP models are not independent and identically distributed. . This paper presents a probabilistic dynamic programming algorithm to obtain the optimal cost-effective maintenance policy for a power cable. ( Log Out / (8), \( d_{{{\mathbb{h}}}} \) is outage cost for the power not supplied to the customer group \( {{\mathbb{h}}} \)\( (\$ /{\text{kW}}) \); \( b_{{{\mathbb{h}}}} \) is the time-dependent power outage cost for the energy not supplied to the customer group \( {{\mathbb{h}}} \,( \$ /{\text{kWh}}) \); \( t_{\text{r}} \) is the average unplanned interruption time (h) and \( L_{{{\mathbb{h}}}} \) is the average hourly power consumption of customer group \( \text{ }{{\mathbb{h}}} \). 2. The failure probability of 0.08 (8%) is assumed as the minimum acceptable level. The regularisation is based on Markovian models of epipolar profiles and stereo signals that allow for measuring similarity of stereo images with due account of binocular and monocular visibility of the surface points. Risk of cable failure can be quantified by the probability of failure which changes with the advancement of service time (age) of a cable. We present a data-driven, probabilistic trajectory optimization framework for systems with unknown dynamics, called Probabilistic Differential Dynamic Programming (PDDP). 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. 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. Table 2 shows the impact of maintenance by effective age. The optimal cost-effective maintenance policy was found for two maintenance periods, first from the years 2016–2030 \( ({\text{stage}}:y = 0\,{\text{to}}\,14) \) and second from the years 2016–2055 \( ({\text{stage}}:y = 0\,{\text{to}}\,39). A deterministic system is one in which the occurrence of all events is known with certainty. This paper presents a probabilistic dynamic programming algorithm to obtain the optimal cost-effective maintenance policy for a power cable. Change ), Continuous Time Dynamic Programming – Applied Probability Notes. [Dynamic Program] Given initial state , a dynamic program is the optimization. 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. Probabilistic Dynamic Programming 24.1 Chapter Guide. Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. In computer science, a dynamic programming language is a class of high-level programming languages, which at runtime execute many common programming behaviours that static programming languages perform during compilation.These behaviors could include an extension of the program, by adding new code, by extending objects and definitions, or by modifying the type system. A Little Bit of Classics_ Dynamic Programming Over Subsets and Paths in Graphs - Codeforces 14.Engineering-Application of Dynamic Programming in Water Resources -I.U. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS Operations Research Formal sciences Mathematics Formal Sciences Statistics 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. IEEE Electr Insul Mag 29(4):52–57, Orton H (2015) Power cable technology review. The result shows that the application of \( {\text{PM}} \) can retain the cable in service till \( y = 14 (2030) \) with minimum maintenance cost at moderately severe insulation condition. $$, $$ {\text{Current}}\,{\text{cost}} = {\text{immediate}}\,{\text{cost}} + {\text{future}}\,{\text{cost}} . Yassad et al. 7. (16) and (17). IEEE Trans Smart Grid 7(2):771–784, Mazzanti G (2007) Analysis of the combined effects of load cycling, thermal transients, and electrothermal stress on life expectancy of high-voltage AC cables. Degradation of cable insulation with respect to service life. ∙ 0 ∙ share . Dynamic Programming is a paradigm of algorithm design in which an optimization problem is solved by a combination of achieving sub-problem solutions and appearing to the " principle of optimality ". In recent years, many methods have been proposed and utilized for the maintenance and replacement of engineering assets; among them, dynamic programming is the most widely used. Similarly, chronological age increases by 1 year at any stage, \( a = a + 1 \), when no maintenance is taken in past or effect of maintenance is neutral. However, it is assumed that failure data are available, from which failure probability is obtained (by NHPP) and is predicted beyond \( y = 0 \) to show failure probability of cable if it is not maintained beyond this stage, as shown in Fig. The degradation can be quantified in terms of percentage with the advancement of age for a group of cable with similar installation year, design, and operational conditions. However, we are interested in one approach where the problem is solved backwards, through a sequence of smaller sub-problems. Four types of maintenance decisions are taken on a cable asset: “no action” NA, “preventive maintenance” PM, “replacement” RP, and “corrective maintenance” CM. The proposed probabilistic dynamic programming model is capable of finding the optimal decision policy with respect to optimal long-run cost for a cable with a known failure distribution and degradation level. Both the infinite and finite time horizon are con- sidered. Recursion and dynamic programming (DP) are very depended terms. In this example, the year 2016 is considered as the current year and optimal maintenance plan is launched from this year. The power cable has a life longer than 20 years. 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. 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 degradation level and planning horizon of cable population installed in year \( i_{0} \) and \( i_{1} \) is shown in Fig. Tweet; Email; DETERMINISTIC DYNAMIC PROGRAMMING. There are a number of ways to solve this, such as enumerating all paths. The cable repair during CM could be perfect, minimal, and worst. 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. Probabilistic or Stochastic Dynamic Programming (SDP) may be viewed similarly, but aiming to solve stochastic multistage optimization 2 Markov Decision Processes and Dynamic Programming p(yjx;a) is the transition probability (i.e., environment dynamics) such that for any x2X, y2X, and a2A p(yjx;a) = P(x t+1 = yjx t= x;a t= a); is the probability of observing a next state ywhen action ais taking in x, The algorithm has two parts. The cost of detecting the exact fault location in an underground cable is much higher than overhead cable. In the last game, the gambler will bet $0$ dollars if he has at least $6$, winning with probability $1$, will bet $6-d$ if he has $3\le d\lt 6$, winning with probability $0.4$, and will give up and win with probability $0$ if he has less than $3$. Int J Electr Power Energy Syst 55:108–115, Alliance Manchester Business School, The University of Manchester, Manchester, M15 6PB, UK, Glasgow Caledonian University, Cowcaddens Rd, Glasgow, G4 0BA, UK, You can also search for this author in It seems more like backward induction than dynamic programming to me. Probabilistic programming allows rapid prototyping of complexly structured probabilistic models without requiring the design of model-specific inference algorithms. Maintenance has a positive and, sometimes, negative impact on an asset. The modeling technique was based on functional and dysfunctional failure analysis of failure modes using the FMEA model (Yssaad and Abene 2015). 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. The probability of failure increases with time. Non-homogenous poisson process (NHPP) is also utilized to model both time-to-failure and failure count data. techniques as we ll as more modern sub jects, including some of my ow nr e-sults from my PhD. (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. Dynamic programming turns out to be an ideal tool for dealing with the theoretical issues this raises. https://doi.org/10.1007/s41872-019-00074-3, DOI: https://doi.org/10.1007/s41872-019-00074-3, Over 10 million scientific documents at your fingertips, Not logged in This is called the Plant Equation. Abbasi E, Firuzabad MF, Jahromi AA (2009) Risk based maintenance optimization of overhead distribution networks utilizing priority based dynamic programming. Let, failure distribution of cables homogenous in terms of voltage level, insulation material and installation year is. 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. Maintenance activities decrease the probability of failure and it extends useful life of the cable. Abstract We present a data-driven, probabilistic trajectory optimization framework for sys- tems with unknown dynamics, called Probabilistic Differential Dynamic Program- ming (PDDP). If the probability of failure of a cable after maintenance is less than just before maintenance, then maintenance has a positive impact on the condition of the cable and its effective age is less than chronological age, \( a^{'} < a. Manufacturers conduct quality tests on each cable section to detect the expected fault. The decision of corrective maintenance is not take at the final stage (\( y = Y \)) of planning horizon. They adopted a risk management approach to consider the actual condition of the electrical components and expected financial risk in the model. The PM repair cost depends on the type of preventive maintenance action taken on the detected potential failure location. This makes probabilistic programs attractive for scientific research: when hypotheses are formalized as programs, it is possible to quickly explore the space of hypotheses. 2016). Cable can regain its operating state (\( \bar{F} \)) or it can again land to a failed state (\( F \)) after repair by corrective maintenance. The reliability of power cable contributes substantially towards the reliability of the entire electrical distribution network. 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. Cost of corrective or preventive failure is much less than completes replacement. Combinatorial problems expect you to figure out the number of ways to do something, or the probability of some event happening. BoVDW model at the top (blue), Probabilistic DTW at the middle (green), and clustering of HPs at the bottom (degraded blue and green). 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. Div. 2015a). The optimisation model considers the probabilistic nature of cables failures. $$, \( \left( {C_{{{\text{RE}}_{\text{PM}} }} } \right) \), $$ {\text{Total}}\,{\text{cost}} = \mathop \sum \limits_{y = 0}^{Y} C_{\text{RP}} + C_{F} + C_{\text{PM}} + C_{{{\text{RE}}_{\text{CM}} }} + C_{{{\text{RE}}_{\text{PM}} }} . The utilities and regulators can assess the monetary risks by exploiting the probabilistic nature of the model. In The First Gene: The Birth of Programming, Messaging and Formal Control, Abel, D. L., Ed. View Ch19.StochasticDP from ISEN 623 at Texas A&M University. 1. LongView Press-Academic: Biolog. In particular, we design a special probabilistic model, named topdown prediction model, for efficiently maintaining the context during dynamic programming: In a topdown prediction model, the context captures only the information from the ancestors but not siblings, such that the probability calculation of sibling subproblems are independent from each other, allowing subproblems to be searched independently. $$, \( ({\text{stage}}:y = 0\,{\text{to}}\,14) \), \( ({\text{stage}}:y = 0\,{\text{to}}\,39). Before \( y = 0, \) information regarding maintenance on this cable may or may not be available. Thus, it considers the fact that cable is a repairable component (Sachan et al. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. $$, $$ 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}}}} . For ex. $$, \( P\left( {a_{y + 1 }^{'} |a_{y}^{'} ,{\text{NA}}} \right) \), \( 1 - P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right) \), $$ {\text{PM}}:\left\{ {\begin{array}{*{20}l} { } \\ {F_{\text{PM}} : P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{PM}}} \right) = P\left( {U\_{\text{DET}}} \right) + P\left( {\text{DET}} \right) \cdot P\left( {\text{USF}} \right) } \\ { } \\ {\bar{F}_{\text{PM}} : P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{PM}}} \right) = P\left( {\text{DE}} \right) \cdot P\left( {\text{SF}} \right).} The failure cost is low in this case, because the lateral cable serves residential customers. (2015a, b). So solution by dynamic programming should be properly framed to remove this ill-effect. . Since the book is written in Google Colab, you’re invited to run and … I've been staring at this problem for hours and I'm still as lost as I was at the beginning. Therefore, it is very important to establish a rationale for the end of the cable lifetime (Mazzanti 2007). The idea of solving a problem from back to front and the idea of iterating on the above equation to solve an optimisation problem lies at the heart of dynamic programming. 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 \( F_{{a_{y }^{'} }} \). Change ), You are commenting using your Google account. The tree on the righthand-side has a lowest cost path of value and the lefthand-side tree has lowest cost and the edges leading to each, respective tree, have costs and . The time-to-failure data can be modeled by the Weibull distribution. The probability of failure and XLPE insulation degradation level is shown in Fig. Risk can never be eliminated completely, though the probability of occurrence of unwanted events can be reduced by planning effective maintenance practices. - 107.170.23.87. Failure events in Weibull distribution are assumed to be independent and identically distributed (i.i.d). 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. The (instantaneous) reward for taking action in state at time is and is the reward for terminating in state at time . Your task is … They have not explored the rationale behind length planning horizon and failed to consider expected lifetime of the components and impact of maintenance. Once the decision to go left or right is made (at cost or ) it is optimal to follow the lowest cost path (at cost or ). The book discusses b oth classical probabilistic dynamic programming. 4. Probabilistic programming for everyone Though not required for probabilistic programming, the Bayesian approach offers an intuitive framework for representing beliefs and updating those beliefs based on new data. In the figure below there is a tree consisting of a root node labelled and two leaf nodes colored grey. In Sec-tion 7, we discuss several open questions and opportunities for fu-ture research in probabilistic programming. Therefore, it can be hypothesized that the life of a cable is equivalent to the time to degradation of the cable insulation (Mazzanti 2007; Sachan et al. In: JICABLE, 9th international conference on insulated power cables, p C1.4, Sutton S (2011) A life cycle analysis study of competing MV cable material. Probabilistic or Stochastic Dynamic Programming (SDP) may be viewed similarly, but aiming to solve stochastic multistage optimization The PM methods could be silicon injection rehabilitation, inspection, and diagnostic tests. The CM repair cost \( (C_{{{\text{RE}}_{\text{CM}} }} ) \) is given by the following: In Eq. 2015b). 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 \( F_{{a_{y + 1 }^{'} }} \). dynamic programming problems.. . Specifically, once we reach the penultimate node on the left (in the dashed box) then it is clearly optimal to go left with a cost of . . 11.2, we incur a delay of three minutes in The underground power cables have four types of interruptions: unplanned, planned, high-speed auto-reclosing (AR), and delayed AR (Lassila et al. Maintenance decision for failed and operating states of cable at different planning period is shown in Table 1. At the same time, an inappropriate choice of finite planning horizon affects the validity of the model. Dynamic Programming. The algorithm initializes the current effective age at y = 0 and stores it at \( {\text{ST}}_{0} \) vector. White University of Manchester, Manchester, England (Received August 24, 1971) This note deals with the manner in which dynamic problems, involving proba-bilistic constraints, may be tackled using the ideas of Lagrange multipliers and efficient solutions. However, the model fails to consider the random failure behaviour of the cable and does not optimize the cost of different maintenance decisions. Here, NA means take no maintenance action on cables. optimal objective) for when the summation is started from , rather than . It means that repair action will bring a cable back to its operating state; however, maintenance would have neither positive nor negative effect. The first part of the algorithm shown in “Appendix A” was utilized to estimate the future state of the cable, as shown in Fig. Dynamic Programming is also used in optimization problems. PDDP takes into account uncertainty explicitly for … 2005). The cost of maintenance decisions at effective age \( (a^{'} \)) and fail (\( F) \) state for stage \( y = 0 \,{\text{to}}\,Y - 1 \) is shown in Eqs. Many probabilistic dynamic programming problems can be solved using recursions: f t(i)the maximum expected reward that can be earned during stages t, t+ 1,..., given that the state at the beginning of stage t isi. I’m assuming everyone has a basic understanding of probability, so we won’t dwell on these here.. . Your task is to find the lowest cost path from the root node to a leaf. Preventive maintenance transition probability. A study has shown the cable life scenario (Sutton 2011). So , the minimal cost path from the root to a leaf node satisfies, Similarly, convince yourself that the same argument applies from any node in the tree network that is. The probability of failure of cables under no maintenance or unidentified past maintenance practices is shown in Fig. The relevance of mathematical developments in dynamic programming and Bayesian statistics to dynamic decision theory is examined. Throughout the world, power distribution networks have high concentration of polymeric-insulated cables. Def. The transition probability for PM action is as follows: The RP action on cable at stage \( y \) results in age 1 at next stage \( y + 1 \). Power Eng Soc Gen Meet 3:2165–2172, Ma J, Chen HH, Song L, Li Y (2016) Residential load scheduling in smart grid: a cost efficiency perspective. 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. It should be noted that, usually, the cost of preventive maintenance is low in comparison to repair, replacement, and failure cost. Dynamic Programming is a Bottom-up approach-we solve all possible small problems and then combine to obtain solutions for bigger problems. 2014; Yssaad and Abene 2015). (2006). 3. (3.2.1). (2015b). 2015). At the same time, maintenance practices and techniques are to detect faults in cable changes, as well. The algorithm suggests the PM at \( y = 1 \), \( y = 8 \) and replacement (\( {\text{RP}} \)) at \( y = 18 \) (2034) as the optimal decision policy for lengthiest planning horizon \( y = 0\,{\text{to}}\,39 \) (2016-2055). The expected life of the cable is obtained from the previously developed ageing model based on stochastic electro-thermal degradation accumulation model. The model represents life-cycle cost approach and it can provide an appropriate time to utilize diagnostic test information in a cost-effective manner. 3. The probability of transition to operating state and failure state can be represented by \( F \) and \( \bar{F} \), respectively. (2). The second part of the algorithm computes the bellman equations by backward induction, i.e., from \( y = Y \) to \( y = 0 \). View Academics in What is Probabilistic Dynamic Programming on Academia.edu. 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. Transition probability of preventive maintenance PM decision is obtained by assuming that only 60% (0.60) of potential failure causes can be detected (DET) and rest 40% (0.40) remain undetected (U_DET), and there is 0.98 and 0.02 chance that PM action would be successful and unsuccessful, respectively. 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. (2005), and Ma et al. \). Dynamic Programming and Probabilistic Constraints D. J. The current preventive maintenance practice and technology is not capable of detecting all the failure causes. Length of planning horizon could be finite or infinite. At failed state \( (F_{{a_{y }^{'} }} ) \), CM decision is taken for maintenance period \( y \) in \( \left\{ {0, \ldots ,Y - 1} \right\} \). The infinite planning horizon is often assumed when it is difficult to establish a termination time. In: Power systems conference and exposition, IEEE PES, pp 389–393, Dong X, Yuan Y, Gao Z, Zhou C, Wallace P, Alkali B, Sheng B, Zhou H (2014) Analysis of cable failure modes and cable joint failure detection via sheath circulating current. Recommended for you 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).
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