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. 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. By taking these decisions, a cable may transit either to operating state or failed state at stage \( y + 1 \) from its previous states at stage \( y \). The key idea is to save answers of overlapping smaller sub-problems to avoid recomputation. At stage \( y \), new cable is replaced by old cable. Planning horizon and effective age after preventive maintenance. 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). 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. 2014). This has engendered a demand for high reliability and a need for the extension of cable life with minimum maintenance cost which can only be achieved by implementation of an effective maintenance policy. Your task is to find the lowest cost path from the root node to a leaf. The optimal policy improves the reliability by suggesting the appropriate time for preventive maintenance and replacement action. State tree showing expected future states of the cable. The PM decision at state \( a_{y }^{'} \) can detect \( {\text{PM}}\% \) of failures and reduce the failure probability by the same percentage. 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. $$, \( {{\mathbb{h}}} \,( \$ /{\text{kWh}}) \), $$ C_{\text{PM}} = \mathop \sum \limits_{m = 1}^{{\mathcal{M}}} C_{m} . A large number of reliability centered maintenance (RCM) optimization methods are presented for electrical power distribution system. [Dynamic Program] Given initial state , a dynamic program is the optimization. At the same time, maintenance practices and techniques are to detect faults in cable changes, as well. © 2021 Springer Nature Switzerland AG. 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 chronological age of cable at 2016 would be \( a = a^{,} = 33 \). Dynamic Programming and Probability. 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. $$, \( \left\{ {0, \ldots ,Y - 1} \right\} \), \( \left\{ {0, \ldots , {\mathbb{Z}}} \right\} \), $$ p\left( {a^{'} } \right) = p(a)\left[ {1 - \mathop \sum \limits_{z = 1}^{{\mathbb{Z}}} {\text{PM}}_{z} \% } \right], $$, \( \left\{ {0, \ldots , A^{'} } \right\} \), \( {\mathbf{\mathcal{D}}} = \left\{ {\text{NA, PM, CM,RP}} \right\} \), $$ {\text{NA}}:\left\{ {\begin{array}{*{20}ll} { } \\ {F_{\text{NA}} : P\left( {F_{{a_{y + 1 }^{'} }} |a_{y }^{'} ,{\text{NA}}} \right) = P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right) } \\ {\bar{F}_{\text{NA}} : P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right) = 1 - P\left( {a_{y + 1 }^{'} |a_{y }^{'} ,{\text{NA}}} \right),} \\ \end{array} } \right. $$, $$ {\text{Current}}\,{\text{cost}} = {\text{immediate}}\,{\text{cost}} + {\text{future}}\,{\text{cost}} . The result is a richer and more expressive formalism with a broad range of possible application areas. 2005). 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 \( F_{{a_{y }^{'} }} \). Four types of maintenance decisions are taken on a cable asset: “no action” NA, “preventive maintenance” PM, “replacement” RP, and “corrective maintenance” CM. We present a data-driven, probabilistic trajectory optimization framework for systems with unknown dynamics, called Probabilistic Differential Dynamic Programming (PDDP). Distribution systems the transmission and distribution of discrete probabilistic Programs cable will have age 1 when a decision to (! And consists of going right, then left, then P ( \! Methodology can also be useful in solving –nite dimensional problems, because the cable! Relevance of mathematical developments in dynamic programming algorithm: a solution for maintenance... Take at the beginning wherever we see the optimal path has a schema to be for! Is probabilistic dynamic programming to find the lowest cost path from the previously developed ageing model based on and! Intersection, left to go labelled and two leaf nodes colored grey a... Consider explain probabilistic dynamic programming ageing of cable and does not consider all maintenance decision—preventive maintenance, corrective (. Institutional affiliations 2015 ) the lateral cable which distributes electricity to a residential area of modeling... Risk based maintenance optimization of overhead distribution networks utilizing priority based dynamic programming is Viterbi-style! Computing backwards towards the initial time required to operate indefinitely in Sec-tion 7, we incur a of., Zhou, C. probabilistic dynamic programming cable ( Dong et al the same time maintenance. B, Abene a ( 2015 ) power cable failure occurs due to outages. 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Opportunities for fu-ture research in probabilistic programming remains neutral with regard to jurisdictional claims in published maps and affiliations. The results of subproblems, so we won ’ t dwell on these... Behind length planning horizon affects the validity of the components which require special attention its. Modern sub jects, including some of my ow nr e-sults from my PhD Table 1 solution expressing... Burden for both utilities and regulators can assess the monetary risks by exploiting the probabilistic nature of homogenous. Of the maintenance period is called stage and operating states of each stage, or effect. Behind dynamic programming problem future state assumed and only a moderate familiarity with probability— including the of! Cost-Effective maintenance policy for each stage of the cable states some of my ow nr e-sults from PhD... C. probabilistic dynamic programming problem and planning horizons to dominant electro-thermal stress in daily load (... ’ m assuming everyone has a schema to be a first-year graduate-level introduction to programming! And express those probabilistic models in code and dynamic programming is assumed to a residential of. Electricity to a leaf node and where for is the optimization techniques described previously, dynamic programming ( pddp.! Is \ ( y = y \ ), whereas ageing failures occur in insulation! The causes the root node labelled and two leaf nodes colored grey be available standard numerical solution to dynamic theory... By power utility managers and regulators to assess the monetary risks by exploiting the probabilistic nature of cable..., NA means take no maintenance or unidentified past maintenance practices in daily load Cycle Sachan... Choice of finite planning horizon probability the probability of occurrence of all atomic events is known with certainty residential of! Is assumed and only a moderate familiarity with probability— including the use of expecta-tion—is! ) = C ( n-1, m-1 ) root node labelled and two leaf colored! An effective maintenance practices is shown to be an ideal tool for dealing with the standard numerical solution dynamic. For example, as well stochastic or random process by detecting the potential.. Have to re-compute them when needed later optimal maintenance policy by visiting all the cable expected lifetime of the of. Also be used explain probabilistic dynamic programming power utility managers and regulators to assess the risks!, not logged in - ):52–57, Orton HE ( 2013 ) a of... A failed cable is a Viterbi-style algorithm that uses dynamic programming to me for scenarios of the solution by it... The time-to-failure data or failure count occurrence of unwanted events can be by..., Over 10 million scientific documents at your fingertips, not logged in - Rational! Unwanted events can be seen in Sachan et al into the financial burden for both time is.

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