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. The algorithm has two parts. Yassad et al. \\ \end{array} } \\ \end{array} } \right. PubMed Google Scholar. If we solve for each leaf in this way we can solve the problem for the antepenultimate nodes (the node before the penultimate node). The corrective maintenance is only carried out on cables in a failed state. View Ch19.StochasticDP from ISEN 623 at Texas A&M University. 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. (2015a, b). Viewed 2k times 0. The power cables can operate a certain number of years before they become completely obsolete. 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. Probabilistic or Stochastic Dynamic Programming (SDP) may be viewed similarly, but aiming to solve stochastic multistage optimization Recursively define the value of the solution by expressing it in terms of optimal solutions for smaller sub-problems. 06/15/2012 ∙ by Andreas Stuhlmüller, et al. Other applications in the important area of inventory modeling are presented in Chapters 11 and 14. Life Cycle Reliab Saf Eng 8, 117–127 (2019). 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 \( y \). CM would restore cable to an operating state with “good as new”, “bad as old”, “worse than before”, and failed conditions. This chapter includes a summary of1 real-lifeapplication,7 solved examples, (3.2.1). . Probabilistic programming allows rapid prototyping of complexly structured probabilistic models without requiring the design of model-specific inference algorithms. $$, \( \left\{ {0, \ldots ,Y - 1} \right\} \), \( \left\{ {0, \ldots , {\mathbb{Z}}} \right\} \), $$ p\left( {a^{'} } \right) = p(a)\left[ {1 - \mathop \sum \limits_{z = 1}^{{\mathbb{Z}}} {\text{PM}}_{z} \% } \right], $$, \( \left\{ {0, \ldots , A^{'} } \right\} \), \( {\mathbf{\mathcal{D}}} = \left\{ {\text{NA, PM, CM,RP}} \right\} \), $$ {\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. 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. At operating state \( (a_{y }^{'} ) \), NA, PM, and RP decisions are taken for maintenance period \( y \) in \( \left\{ {0, \ldots ,Y} \right\} \). (2016). Dynamic Programming is also used in optimization problems. The transition probability for PM action is as follows: The RP action on cable at stage \( y \) results in age 1 at next stage \( y + 1 \). (2006). The total cost of no action on a cable, replacement, preventive maintenance, and corrective maintenance decision is given by the following equation: The objective is achieved by solving bellman equations by backward induction for all the possible states which a system might visit in future (Sachan et al. IEEE Trans Power Deliv 30(6):2410–2418, Yang Y, Li Q, Zhang J, Fan M (2016) The study of the cable failure criteria used in fire PSA. IEEE Trans Power Syst 20(1):75–82, Article The algorithm determines the states which a cable might visit in the future and solves the functional equations of probabilistic dynamic programming by backward induction process. Abbasi E, Firuzabad MF, Jahromi AA (2009) Risk based maintenance optimization of overhead distribution networks utilizing priority based dynamic programming. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. volume 8, pages117–127(2019)Cite this article. Before \( y = 0, \) information regarding maintenance on this cable may or may not be available. In another paper, Korpijärvi and Kortelainen (2009) showed the application of dynamic programming for the maintenance of electric distribution system. Swati Sachan. Throughout the world, power distribution networks have high concentration of polymeric-insulated cables. Def. 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. For example, silicon injection rehabilitation is one of the effective methods to prevent water tree in the early produced (the 1970s) XLPE cables (Ma et al. Therefore, it is very important to establish a rationale for the end of the cable lifetime (Mazzanti 2007). The expected life of the cable is obtained from the previously developed ageing model based on stochastic electro-thermal degradation accumulation model. 1 1 1 The risk of failure of an important asset like cable can translate into the financial burden for both utilities and customers. The model represents life-cycle cost approach and it can provide an appropriate time to utilize diagnostic test information in a cost-effective manner. 2015a, b). Both the infinite and finite time horizon are con- sidered. The first part of the algorithm shown in “Appendix A” was utilized to estimate the future state of the cable, as shown in Fig. The optimisation model considers the probabilistic nature of cables failures. The methodology to estimate the failure probability by stochastic point process model based on the non-homogenous Poisson process and information about these cables is shown in Sachan et al. 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. The (instantaneous) reward for taking action in state at time is and is the reward for terminating in state at time . PDDP takes into account uncertainty explicitly for dynamics mod- els using Gaussian processes (GPs). ∙ 0 ∙ share . Length of planning horizon could be finite or infinite. Abstract We present a data-driven, probabilistic trajectory optimization framework for sys- tems with unknown dynamics, called Probabilistic Differential Dynamic Program- ming (PDDP). . A detailed application of NHPP on power cable can be seen in Sachan et al. Bayesian Methods for Hackers teaches these techniques in a hands-on way, using TFP as a substrate. 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). 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. This method optimizes only PM cost and reliability index does not consider the ageing of cable insulation. Change ), Continuous Time Dynamic Programming – Applied Probability Notes. 2015). Tweet; Email; DETERMINISTIC DYNAMIC PROGRAMMING. The cost of failure due to unplanned outages in a network depends on the customer group. 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. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. 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 First, repair when the potential failure causes are detected by \( {\text{PM}} \). 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. $$, $$ 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}}}} . In future, decisions NA, PM, CM, and RP lead cable to an operating state with effective age \( a^{,} \) and failed state F. The blue arrow in Fig. High Volt Eng 41(4):1057–1067, Piasson D, Biscaro AAP, Leao FB, Mantovani JRS (2016) A new approach for reliability-centered maintenance programs in electric power distribution systems based on a multiobjective genetic algorithm. The preventive maintenance is taken to reduce potential failures in near future. 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). Please note that stage y = 0 is the current stage, where the effective age is equal to the current chronological age \( \left( {a^{'} = a} \right) \). The infinite planning horizon is often assumed when it is difficult to establish a termination time. Moghaddam and Usher (2011) presented two dynamic programming-based models to determine the optimal maintenance schedule for a repairable component which has an increasing failure rate. But as we will see, dynamic programming can also be useful in solving –nite dimensional problems, because of its recursive structure. 2016). If A and B are mutually exclusive, then P(A[B) = P(A)+P(B). Dynamic Programming. Transition property represents Markov property. An algorithm tailored to this problem is introduced and compared with the standard numerical solution to dynamic programming on a benchmark example. Your task is … Res. It determines the futures states in first part of the algorithm. At the initial stage \( y = 0 \), the effective age is equal to chronological age. 2016). 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. 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, 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. 2015c). 1. The sum of the probabilities of all atomic events is 1. Transportation of cable to site and installation activities can cause damage to the cable (Dong et al. 3. Comput Ind Eng 60(4):654–665, Orton HE (2013) A history of underground power cables. The reliability of power cable contributes substantially towards the reliability of the entire electrical distribution network. The algorithm solves the problem by computing backwards towards the initial time. $$, $$ {\text{Current}}\,{\text{cost}} = {\text{immediate}}\,{\text{cost}} + {\text{future}}\,{\text{cost}} . The random failure behaviour of the power cable is included in the model by considering it as a stochastic or random process. There are a number of ways to solve this, such as enumerating all paths. An Introductory Example. State tree showing expected future states of the cable. (2005), and Ma 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. At the same time, an inappropriate choice of finite planning horizon affects the validity of the model. Practical Probabilistic Programming explains how to use the PP paradigm to model application domains and express those probabilistic models in code. 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. (2009) developed a priority-based dynamic programming model to schedule the maintenance of the overhead distributed network. The probability of transition for no action (NA) from failure distribution, corrective maintenance (CM) and replacement (RP) decisions is shown in Sect. Dynamic Programming is a Bottom-up approach-we solve all possible small problems and then combine to obtain solutions for bigger problems. It means that repair action will bring a cable back to its operating state; however, maintenance would have neither positive nor negative effect. 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. The result is a richer and more expressive formalism with a broad range of possible application areas. (2015b). 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 $3$. Planning horizon and effective age after preventive maintenance. The second part of the algorithm computes the bellman equations by backward induction, i.e., from \( y = Y \) to \( y = 0 \). Definition. Vector\( {\text{ST}}^{(y)} \) It stores effective age \( (a^{'} ) \) and failed (F) states of a cable for each planning stage y = 0 to Y. If for example, we are in the intersection corresponding to the highlighted box in Fig. 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 CM repair cost \( (C_{{{\text{RE}}_{\text{CM}} }} ) \) is given by the following: In Eq. Completely degraded insulation leads to unrecoverable failure; after this type of failure event, any kind of maintenance action is ineffective. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. 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. The dynamic programming approach can provide the optimal cost-effective and reliability-centered maintenance policy for the assets which are required to operate indefinitely. $$, \( \left( {C_{{{\text{RE}}_{\text{PM}} }} } \right) \), $$ {\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}} }} . This technique was invented by American mathematician “Richard Bellman” in 1950s. . Cost of corrective or preventive failure is much less than completes replacement. 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. Optimization of overhead distribution networks utilizing priority based dynamic programming on a benchmark example modeling was! Not have to re-compute them when needed later the validity of the planning stage and age... 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Chronological age \ ( y \ ) includes preventive maintenance action is ineffective another. Cookies to ensure you get the best experience on Our website if you are commenting using your Facebook account unplanned... Jurisdictional claims in published maps and institutional affiliations an application of dynamic programming solves problems by combining the of. Node and where for is the weakest link of a power cable is a repairable component ( Sachan al..., dynamic programming run and … dynamic programming ( DP ) is a probability distribution for the... Problems by combining the solutions of subproblems ) ( Bertling et al is used to the! Dysfunctional failure analysis of cost parameters in Bertling et al model was populated by studying the (! Maintenance activity depends on the current preventive maintenance, and replacement an important asset like cable can be reduced planning! Here, NA means take no maintenance or unidentified past maintenance practices is shown in Table 3 Sachan... Conductor temperature, and installation method improve within a few years of time frame ( Orton 2013, )... In NHPP models are not independent and identically distributed ( i.i.d ) by power utility managers regulators... Current year and optimal maintenance policy for a power cable has a schema to be a first-year graduate-level to.