# Probabilistic dynamic programming algorithm: a solution for optimal maintenance policy for power cables

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## Abstract

This paper presents a probabilistic dynamic programming algorithm to obtain the optimal cost-effective maintenance policy for a power cable. 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. The optimisation model considers the probabilistic nature of cables failures. 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. The model can be used by power utility managers and regulators to assess the financial risk and schedule maintenance.

## Keywords

Probability Optimization Maintenance Algorithm Failure## 1 Introduction

Power cables play an integral part in the transmission and distribution of electricity. The reliability of power cable contributes substantially towards the reliability of the entire electrical distribution network. 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. 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.

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 dynamic programming approach can provide the optimal cost-effective and reliability-centered maintenance policy for the assets which are required to operate indefinitely. 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. 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. In another paper, Korpijärvi and Kortelainen (2009) showed the application of dynamic programming for the maintenance of electric distribution system. Abbasi et al. (2009) developed a priority-based dynamic programming model to schedule the maintenance of the overhead distributed network. They adopted a risk management approach to consider the actual condition of the electrical components and expected financial risk in the model. An application of dynamic programming for maintenance of power cable was presented by Bloom et al. (2006). The model represents life-cycle cost approach and it can provide an appropriate time to utilize diagnostic test information in a cost-effective manner. However, the model fails to consider the random failure behaviour of the cable and does not optimize the cost of different maintenance decisions.

A large number of reliability centered maintenance (RCM) optimization methods are presented for electrical power distribution system. 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. (2016). This method optimizes only PM cost and reliability index does not consider the ageing of cable insulation. Yassad et al. presented two system-level RCM optimization methods (Yssaad et al. 2014; Yssaad and Abene 2015). Both methods identify the components which require special attention and its goal is to minimize the corrective and preventive maintenance cost by maximizing reliability. The modeling technique was based on functional and dysfunctional failure analysis of failure modes using the FMEA model (Yssaad and Abene 2015). These methods do not consider all maintenance decision—preventive maintenance, corrective maintenance, and replacement. They have not explored the rationale behind length planning horizon and failed to consider expected lifetime of the components and impact of maintenance.

In this paper, probabilistic dynamic programming algorithm is proposed to obtain optimal cost-effective maintenance policy for power cables in each stage (or year) of the planning period. In this model, the length of the planning horizon is equivalent to the expected lifetime of the cable. The expected life of the cable is obtained from the previously developed ageing model based on stochastic electro-thermal degradation accumulation model. The maintenance policy in this model includes preventive maintenance, corrective maintenance, replacement, and do nothing as a set of decisions. The algorithm first finds the future state of the cable by qualifying the effect of each maintenance decision, and then, it uses backward induction method to solve the dynamic programming recursive equations consisting of future state transition probability. The random failure behaviour of the power cable is included in the model by considering it as a stochastic or random process. This work specifies the process of applying the failure data, maintenance data, and diagnostic test data in the decision-making process. The proposed methodology can also be used in the maintenance of other electrical components, as well.

## 2 Proposed methodology

## 3 Probabilistic dynamic programming algorithm

### 3.1 PART 1: estimation of future state of the cable

#### 3.1.1 Length of maintenance period

Length of planning horizon could be finite or infinite. The infinite planning horizon is often assumed when it is difficult to establish a termination time. At the same time, an inappropriate choice of finite planning horizon affects the validity of the model. The power cables can operate a certain number of years before they become completely obsolete. A cable has two types of failure criteria. 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. 2016). A cable has a finite lifetime. Therefore, it is very important to establish a rationale for the end of the cable lifetime (Mazzanti 2007).

The lifetime of the cables is usually obtained by modeling the historical failure data which have high fluctuations due to the presence of both random and ageing failures (Sachan et al. 2015a, b). The fluctuating data source should not be utilized to develop the long-term maintenance policy which includes proactive replacement as one of the high investment maintenance decisions. Cost of corrective or preventive failure is much less than completes replacement. 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. The preventive maintenance improves the reliability by detecting the potential failures.

Power cable failure occurs due to random, ageing, or cumulated effect of both the causes. 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. 2015b), whereas ageing failures occur in cable insulation due to dominant electro-thermal stress in daily load cycle (Sachan et al. 2015a, b). Most common mode of insulation failure is electrical breakdown of insulation, breakdown at the electrical interface, and insulation thermal breakdown (Dong et al. 2014; Orton 2013). The insulation is the weakest link of a power cable in terms of degradation or failure. Completely degraded insulation leads to unrecoverable failure; after this type of failure event, any kind of maintenance action is ineffective. 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. 2015a). Throughout the world, power distribution networks have high concentration of polymeric-insulated cables. 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. 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. 2015a).

#### 3.1.2 Set of states and maintenance decisions for each stage

Four types of maintenance decisions are taken on a cable asset: “no action” NA, “preventive maintenance” PM, “replacement” RP, and “corrective maintenance” CM. Here, NA means take no maintenance action on cables. The preventive maintenance is taken to reduce potential failures in near future. The corrective maintenance is only carried out on cables in a failed state. The replacement action renews an old cable with a new cable.

*CM*decision is taken for maintenance period \( y \) in \( \left\{ {0, \ldots ,Y - 1} \right\} \). The decision of corrective maintenance is not take at the final stage (\( y = Y \)) of planning horizon. 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. Maintenance decision for failed and operating states of cable at different planning period is shown in Table 1.

Maintenance decision for all states

State | |||
---|---|---|---|

Operating state \( (a_{y }^{'} ) \)\( y \) in \( \left\{ {0, \ldots ,Y} \right\} \). | Fail state \( (F_{{a_{y }^{'} }} ) \)\( y \) in \( \left\{ {0, \ldots ,Y - 1} \right\} \) | Fail \( (F_{{a_{Y }^{'} }} ) \) or operating state \( (a_{y }^{'} ) \) \( y = Y \) | |

Maintenance decisions \( ({\mathbf{\mathcal{D}}}) \) | \( {\text{NA, PM, RP}} \) | \( {\text{CM}} \) | \( {\text{RP}} \) |

#### 3.1.3 Effect of maintenance

Maintenance has a positive and, sometimes, negative impact on an asset. The risk of failure of an important asset like cable can translate into the financial burden for both utilities and customers. Risk can never be eliminated completely, though the probability of occurrence of unwanted events can be reduced by planning effective maintenance practices. Effect of maintenance on the cable must be quantified appropriately to make an effective maintenance plan.

Risk of cable failure can be quantified by the probability of failure which changes with the advancement of service time (age) of a cable. The probability of failure is estimated from either time-to-failure data or failure count. The time-to-failure data can be modeled by the Weibull distribution. Failure events in Weibull distribution are assumed to be independent and identically distributed (i.i.d). It treats cable as a non-repairable component or it was not maintained in the past (Tang et al. 2015). Non-homogenous poisson process (NHPP) is also utilized to model both time-to-failure and failure count data. The failure events in NHPP models are not independent and identically distributed. Thus, it considers the fact that cable is a repairable component (Sachan et al. 2015b). In this research, cable is assumed to a repairable component. In the numerical example, as shown in Sect. 4, the failure probability of cables was obtained by NHPP. A detailed application of NHPP on power cable can be seen in Sachan et al. (2015b).

Effective age

Effect of maintenance | Effective age |
---|---|

Positive effect | \( a^{'} < a \) |

Neutral effect | \( a^{'} = a \) |

Negative effect | \( a^{'} > a \) |

### 3.2 Part 2: determination of optimal maintenance policy

#### 3.2.1 Transition probability

A maintenance decision \( ({\mathbf{\mathcal{D}}}) \) on a cable at any stage \( y \) transforms it to another state at next stage \( y + 1 \). Transition probability depends on current state and maintenance decisions \( {\mathbf{\mathcal{D}}} = \left\{ {\text{NA, PM, CM,RP}} \right\} \). 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 probability of transition to operating state and failure state can be represented by \( F \) and \( \bar{F} \), respectively. Transition property represents Markov property. According to Markov property, future state depends on the current state.

*y*+ 1 by taking NA, PM, and RP decisions on cable operating at state \( a_{y }^{'} \) is as follows:

- (A)
No action (NA)

- (B)
Preventive maintenance (PM)

- (C)
Replacement (RP)

The *RP* action on cable at stage \( y \) results in age 1 at next stage \( y + 1 \). The power cable has a life longer than 20 years. A study has shown the cable life scenario (Sutton 2011). According to this study, XLPE, TR-XLPE, and EPR cables have a lifespan of 30, 50, and 45 years, respectively. Manufacturing techniques, material, design, and installation method improve within a few years of time frame (Orton 2013, 2015). At the same time, maintenance practices and techniques are to detect faults in cable changes, as well.

- (D)
Corrective maintenance

#### 3.2.2 Maintenance cost

- A.
Replacement cost

- B.
Failure and unplanned interruption cost

- C.
Maintenance cost

- D.
Repair cost

In Eq. (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. The PM repair cost \( \left( {C_{{{\text{RE}}_{\text{PM}} }} } \right) \) is usually less than CM repair cost \( (C_{{{\text{RE}}_{\text{CM}} }} ) \), because CM repair action is taken after the occurrence of the failure which includes a high cost for detection and repair of a failed section of the cable. The PM repair cost depends on the type of preventive maintenance action taken on the detected potential failure location. 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. 2016).

#### 3.2.3 Objective and recursion equation formulation

*Objective*Obtain optimal maintenance policy that minimizes the total maintenance cost over a finite planning horizon \( 0 < y < Y \). The total cost of no action on a cable, replacement, preventive maintenance, and corrective maintenance decision is given by the following equation:

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. Minimum maintenance cost at stage \( y \) of planning horizon is \( V_{y} \) and expected future cost of maintenance at stage \( y + 1 \) is \( V_{y + 1} \). Minimum maintenance cost incurs due to decisions at the end of planning horizon at stage \( y = Y \) for state \( a^{'} \) (effective age) is zero, \( F \) (fail) is replacement cost, and \( a^{'} = A^{'} \) is both failure and replacement cost shown in Eq. (13) to (15). 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 \( (F) \).

\( {\text{for}}\,y = Y \),

## 4 Numerical example

In this example, the year 2016 is considered as the current year and optimal maintenance plan is launched from this year. The chronological age of cable at 2016 would be \( a = a^{,} = 33 \). At the initial stage \( y = 0 \), the effective age is equal to chronological age. It was estimated that, by the years 2030 and 2055, the entire insulation of the cable is expected to reach 75% of moderately severe and 99.8% of severe of insulation degrade level, respectively (can be seen in Fig. 6). 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 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 \( ({\text{stage}}:y = 0\,{\text{to}}\,39). \).

*U*_DET), and there is 0.98 and 0.02 chance that PM action would be successful and unsuccessful, respectively. The transition probability of PM action is \( \bar{F}_{\text{PM}} \) = 0.59 and \( F_{\text{PM}} \) = 0.41 [from Eq. (4) \( \bar{F}_{\text{PM}} \) = 0.60 \( \times \) 0.98 and \( F_{\text{PM}} \) = 0.40 \( + \) 0.60 \( \times \) 0.02]. The failure probability of 0.08 (8%) is assumed as the minimum acceptable level. The \( {\text{PM }} \) and \( {\text{RP}} \) decisions are not taken below this level.

Maintenance and failure cost

Cost | Value |
---|---|

1. Failure or unplanned interruption cost | |

Number of residential households | 34 |

Average annual residential load | 6000 \( {\text{kWh}} \) |

Average hourly power consumption \( (L_{{\mathbb{h}}} ) \) | 23.28 \( {\text{kW}} \) \( \left( { = \frac{{6000\,{\text{kW}}\,{\text{h}} \times 34}}{{365\,{\text{days}} \times 24\,{\text{h}}}}} \right) \) |

Average unplanned interruption time in hours \( (t_{\text{r}} ) \) | 2.5 h |

Power outage cost \( (d_{{\mathbb{h}}} ) \) | 1.84 $/\( {\text{kW}} \) |

Time-dependent power outage cost \( (b_{{{\mathbb{h}}\text{ }}} ) \) | 6.7 \( \$ /{\text{kWh}} \) |

Average failure cost \( (C_{\text{F}} ) \) | $ 411.59 |

2. Maintenance cost: the average cost of diagnostic tests and inspection is negligible in compared to repair and replacement cost Average maintenance cost for 500 \( m \) cable \( (C_{\text{M}} ) \) | $ 300.0 (620.00 $/km) |

3. Repair cost: the average repair cost of single failure is | |

Average CM repair \( (C_{{R_{\text{CM}} }} ) \) | $ 5100.00 |

Average PM repair cost \( (C_{{R_{\text{PM}} }} ) \) | $ 320.00 |

4. Cost of replacement | |

Cost of new XLPE cable ( | 6.6 $/m |

Cost of installation ( | 108.8 $/m |

Replacement cost of 520 m cable ( | $60,008.0 |

## 5 Conclusion

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. The optimal policy improves the reliability by suggesting the appropriate time for preventive maintenance and replacement action. The utilities and regulators can assess the monetary risks by exploiting the probabilistic nature of the model.

## Notes

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