4, the failure probability of cables was obtained by NHPP. However, we are interested in one approach where the problem is solved backwards, through a sequence of smaller sub-problems. 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. It finds the minimum cost for \( y = Y \), then \( y = Y - 1, \) then \( y = Y - 2 \) and so on. The probabilistic case, where there is a probability dis- tribution for what the next state will be, is discussed in the next section. The optimisation model considers the probabilistic nature of cables failures. 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. A new cable section has chronological age 1 when a decision to replace (RP) cable is taken. $$, \( ({\text{stage}}:y = 0\,{\text{to}}\,14) \), \( ({\text{stage}}:y = 0\,{\text{to}}\,39). The cost of detecting the exact fault location in an underground cable is much higher than overhead cable. 2. Table 2 shows the impact of maintenance by effective age. 2014; Orton 2013). In The First Gene: The Birth of Programming, Messaging and Formal Control, Abel, D. L., Ed. Maintenance planning starts from \( y = 0 \); at this stage, the chronological age of cable is \( a \). 2015a). 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. (2015a). IEEE Trans Power Deliv 22(4):2000–2009, Moghaddam KS, Usher JS (2011) Preventive maintenance and replacement scheduling for repairable and maintainable systems using dynamic programming. Let’s consider the problem a little more generally in the next figure. The first part of the algorithm is utilized to obtain all possible states which a cable might visit in future by quantifying the effect of maintenance actions on cable state. 1. This is called the Plant Equation. The probability of failure of cables under no maintenance or unidentified past maintenance practices is shown in Fig. The time-to-failure data can be modeled by the Weibull distribution. 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. Recursively define the value of the solution by expressing it in terms of optimal solutions for smaller sub-problems. (4) \( \bar{F}_{\text{PM}} \) = 0.60 \( \times \) 0.98 and \( F_{\text{PM}} \) = 0.40 \( + \) 0.60 \( \times \) 0.02]. volume 8, pages117–127(2019)Cite this article. 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 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). Abbasi E, Firuzabad MF, Jahromi AA (2009) Risk based maintenance optimization of overhead distribution networks utilizing priority based dynamic programming. 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. optimal objective) for when the summation is started from , rather than . $$, $$ {\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.} Therefore, there are two possible kinds of repair. We briefly explain the principles behind dynamic programming and then give its definition. 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. (16) and (17). p(j \i,a,t)the probability that the next period’s state will be j, given that the current (stage t) state is iand action a First criteria are focused on the decline in the performance of cable insulation and second criteria are focused on the loss of ability to resist fire (Yang 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. 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. 2016). Here. Probabilistic programming allows rapid prototyping of complexly structured probabilistic models without requiring the design of model-specific inference algorithms. The first is a Viterbi-style algorithm that uses dynamic programming to find the single most likely parse for a given text. They adopted a risk management approach to consider the actual condition of the electrical components and expected financial risk in the model. 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. Degradation of cable insulation with respect to service life. The model represents life-cycle cost approach and it can provide an appropriate time to utilize diagnostic test information in a cost-effective manner. Only the decision of replacement \( ({\text{RP}}) \) is taken at \( y = Y \) if the cable has reached the end of the life time \( (a^{'} = A^{'} ) \) and has failed \( (F) \). The PM repair cost depends on the type of preventive maintenance action taken on the detected potential failure location. The input maintenance and failure cost are shown in Table 3. 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? (2005), Lassila et al. Thus, it considers the fact that cable is a repairable component (Sachan et al. 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. 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. The corrective maintenance restores the cable back to its operational state after the occurrence of a failure by cutting and splicing in a new cable section. 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. $$, \( \left\{ {{\text{NA}}, {\text{PM}}, {\text{RP, and CM}}} \right\}.\), $$ C_{\text{RP}} = \left( {C_{\text{cable}} + C_{\text{inst}} } \right) l. $$, $$ C_{F} = \mathop \sum \limits_{{\mathbb{h}} \in H} \left( {d_{{{\mathbb{h}}}} + b_{{{{\mathbb{h}}}\text{ }}} t_{\text{r}} } \right)L_{{\mathbb{h}}} . A random failure can occur due to degradation in a small section of a cable circuit such as poor workmanship, a manufacturing defect, or sudden mechanical (Sachan et al. where is the minimum cost from to a leaf node and where for is the node to the lefthand-side or righthand-side of . A large number of reliability centered maintenance (RCM) optimization methods are presented for electrical power distribution system. For each edge, there is a cost. [Dynamic Program] Given initial state , a dynamic program is the optimization. (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 \( {\text{km}} \). $$, \( \left( {C_{{{\text{RE}}_{\text{PM}} }} } \right) \), $$ {\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}} }} . Moreover, Dynamic Programming algorithm solves each sub-problem just once and then saves its answer in a table, thereby avoiding the work of re-computing the answer every time. The key idea is to save answers of overlapping smaller sub-problems to avoid recomputation. But as we will see, dynamic programming can also be useful in solving –nite dimensional problems, because of its recursive structure. I've been staring at this problem for hours and I'm still as lost as I was at the beginning. 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. 2015b), whereas ageing failures occur in cable insulation due to dominant electro-thermal stress in daily load cycle (Sachan et al. 3. \\ \end{array} } \right. We report on a probabilistic dynamic programming formulation that was designed specifically for scenarios of the type described. The probability of failure and XLPE insulation degradation level is shown in Fig. Contribution of PM methods towards the reduction in failure probability of cable can be obtained by Eq. CM would restore cable to an operating state with “good as new”, “bad as old”, “worse than before”, and failed conditions. 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. A computer programming method the reliability of power cable is replaced by cable. Substantially towards the initial stage \ ( y \ ) inventory modeling are presented for power... Model-Specific inference algorithms within a few years of time frame ( Orton 2013, 2015.. An event a is a general method aimed at solving multistage optimization problems difficult to establish a for. State depends on the type described a = a^ {, } = 33 ). 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