# A Maintenance Cost Study of Transformers Based on Markov Model Utilizing Frequency of Transition Approach

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## Abstract

**:**

## 1. Introduction

## 2. Health Index of Transformer

_{j}is the assigned rating given for each factor, HIF

_{j}is the assigned score for each factor and j is the number of condition data. According to [9,10], the constant values of A and B are 60% and 40%, respectively, which correspond to parameters of transformers and tap changers. In this study, the constants, A and B, were not considered since only transformer condition data are available for analysis. Furthermore, due to the limitation of data in the database, only oil quality, dissolved gases and furanic compounds were considered and the updated formulation of HI can be seen in Equation (2):

## 3. Markov Model Structure with Health Index

#### 3.1. Markov Chain Concepts

_{ij}[32,33]. Transition matrix,

**P**, represents the transition probabilities. The corresponding transition matrix,

**P**, for 5 states of MM used in this study is shown in Equation (3) as following:

_{ij}(t) was considered in a particular year. In addition, each state probabilities should be equal to one, for example, P

_{55}+ P

_{54}= 1, P

_{44}+ P

_{43}= 1, P

_{33}+ P

_{32}= 1 and P

_{22}+ P

_{21}= 1. The final state was set as 1, assuming that all transformers would end up in the poorest condition and remained at the last state. This concept of the MM is known as an absorbing model. Once the transition matrix has been developed, the future states of the HI of transformers can be determined based on the current state shown in Equation (4):

**H**

_{n}

_{+1}is the next state in a particular interval and

**H**

_{n}is the current state. The future state of the HI can also be determined by the MM based on the following equation:

**H**

_{0}is the initial state and

**H**

_{t}is the future deterioration state in t.

#### 3.2. Transition Probability Derivation

_{ij}is the transition probability for equipment remaining at the existing state and 1 − P

_{ij}is the probability for equipment moving to the next state. Both probabilities depend on the present state j. In order to determine the future state for the next interval, t + 1, the transition probabilities were calculated, where each iteration fulfilled the Chapman-Kolmogorov rules and can be seen in Equations (8)–(12):

#### Frequency of Transition

_{ij}is the number of transformers transit from a state i to j in one year period and ${n}_{i}=\sum _{j}{n}_{ij}$ is the number of transformers in state i before the transition.

**H**

_{0}equal to [1 0 0 0 0].

## 4. Maintenance Policy and Cost

**R**, is given by:

**PR**[27]. These Markov chain models were solved to find future-state probabilities at a certain year, n, by iterating the initial-state distribution matrix with the combined effect of deterioration and maintenance policy matrices, as shown in Equation (15):

**C**, and hence, the expected maintenance cost of each future state for each year, n, can be determined based on Equation (16):

## 5. Application of Markov Model for 33/11 kV Distribution Transformers

^{2}value of the line fitting was 0.8826. It is found that the data exceeded the confidence interval, but were still within the prediction interval.

^{2}values were 0.8864 and 0.8619 for the year 2014 and 2015, respectively.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

CA | Condition Assessment |

CBM | Condition Based Monitoring |

CM | Corrective Maintenance |

DGA | Dissolved Gas Analysis |

FA | Furfural Analysis |

HI | Health Index |

HIF | Health Index Factor |

MM | Markov Model |

OEM | Original Engineering Manufacturer |

OQA | Oil Quality Analysis |

RM | Ringgit Malaysia |

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**Figure 4.**State probabilities in the year 2015 based on transformer transition states for the year 2013/2014.

**Figure 5.**Predicted versus actual numbers of transformer in the year 2015 based on transformer transition states for the year 2013/2014.

**Figure 6.**Predicted versus actual numbers of transformer in the year 2014 based on transformer transition states for the year 2012/2013.

**Figure 7.**Predicted versus actual numbers of transformer in the year 2015 based on transformer transition states for the year 2012/2013.

**Figure 8.**Predicted state probabilities for State 1 in the year 2015 based on transformer transition states for the year 2013/2014 and 2012/2013.

**Figure 9.**Actual and predicted transformer populations in the year 2015 based on transformer transition states for the year 2013/2014 and 2012/2013.

**Figure 10.**Maintenance cost for transformers in each state for transition states for the year 2013/2014.

**Figure 11.**Maintenance cost for transformers in each state for transition states for the year 2012/2013.

**Figure 12.**Total estimated maintenance cost for transformers in each year based on transformer transition states for the year 2012/2013 and 2013/2014.

Condition | Health Index | Description |
---|---|---|

Very Good | 85–100% | Some aging or minor deterioration of a limited number of components. |

Good | 70–84% | Significant deterioration of some components. |

Fair | 50–69% | Widespread significant or serious deterioration of specific components. |

Poor | 30–49% | Widespread serious deterioration. |

Very Poor | 0–29% | Extensive serious deterioration. |

**Table 2.**Maintenance action, intervention state, maintenance activity and cost estimation breakdown.

Maintenance Action | Intervention State | Activity | Estimated Cost (RM) |
---|---|---|---|

Condition assessment | 5, 4, 3, 2, 1 | Annual routine monitoring ^{1} | 49,500.00 |

Minor work | 2 | Oil regeneration + major parts restoration ^{2} | 30% from new unit |

Major work | 1 | Replacement | 1,700,000.00 |

^{1}Annual routine monitoring by OEM will consider physical inspection, dissolved gases, oil quality and furanic compounds.

^{2}It may include the tap changers, bushing, labor, oil drying, tank and sealing, wiring and control systems. Rewinding and core replacement are not included [36].

Markov Model State | Corresponding HI Range |
---|---|

State 5 | 85–100% |

State 4 | 70–84% |

State 3 | 50–69% |

State 2 | 30–49% |

State 1 | 0–29% |

State | Year 2012 | Year 2013 | Year 2014 | Year 2015 |
---|---|---|---|---|

State 5 | 23 | 19 | 17 | 7 |

State 4 | 25 | 23 | 15 | 12 |

State 3 | 46 | 46 | 42 | 42 |

State 2 | 21 | 26 | 39 | 44 |

State 1 | 5 | 6 | 7 | 15 |

To From | State 5 | State 4 | State 3 | State 2 | State 1 |
---|---|---|---|---|---|

State 5 | 17 | 2 | 0 | 0 | 0 |

State 4 | 0 | 13 | 10 | 0 | 0 |

State 3 | 0 | 0 | 34 | 12 | 0 |

State 2 | 0 | 0 | 0 | 25 | 1 |

State 1 | 0 | 0 | 0 | 0 | 1 |

To From | State 5 | State 4 | State 3 | State 2 | State 1 |
---|---|---|---|---|---|

State 5 | 19 | 4 | 0 | 0 | 0 |

State 4 | 0 | 19 | 6 | 0 | 0 |

State 3 | 0 | 0 | 40 | 6 | 0 |

State 2 | 0 | 0 | 0 | 20 | 1 |

State 1 | 0 | 0 | 0 | 0 | 1 |

**Table 7.**Numbers of transformers in each of the states based on transition states for the year 2013/2014 with the maintenance policy model.

Age | State 5 | State 4 | State 3 | State 2 | State 1 |
---|---|---|---|---|---|

0 | 120 | 0 | 0 | 0 | 0 |

1 | 107 | 13 | 0 | 0 | 0 |

2 | 96 | 18 | 6 | 0 | 0 |

3 | 86 | 21 | 12 | 1 | 0 |

4 | 77 | 21 | 18 | 4 | 0 |

5 | 70 | 20 | 22 | 8 | 0 |

6 | 64 | 19 | 25 | 12 | 0 |

7 | 59 | 17 | 27 | 16 | 1 |

8 | 55 | 16 | 27 | 21 | 1 |

9 | 52 | 15 | 27 | 24 | 2 |

10 | 50 | 14 | 26 | 28 | 2 |

11 | 49 | 13 | 26 | 29 | 3 |

12 | 47 | 13 | 25 | 31 | 4 |

13 | 47 | 12 | 24 | 32 | 5 |

14 | 46 | 12 | 23 | 34 | 5 |

15 | 46 | 11 | 22 | 35 | 6 |

16 | 46 | 11 | 21 | 35 | 7 |

17 | 46 | 11 | 21 | 35 | 7 |

18 | 46 | 11 | 20 | 35 | 8 |

19 | 46 | 11 | 20 | 35 | 8 |

20 | 46 | 11 | 19 | 35 | 9 |

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**MDPI and ACS Style**

Yahaya, M.S.; Azis, N.; Mohd Selva, A.; Ab Kadir, M.Z.A.; Jasni, J.; Kadim, E.J.; Hairi, M.H.; Yang Ghazali, Y.Z.
A Maintenance Cost Study of Transformers Based on Markov Model Utilizing Frequency of Transition Approach. *Energies* **2018**, *11*, 2006.
https://doi.org/10.3390/en11082006

**AMA Style**

Yahaya MS, Azis N, Mohd Selva A, Ab Kadir MZA, Jasni J, Kadim EJ, Hairi MH, Yang Ghazali YZ.
A Maintenance Cost Study of Transformers Based on Markov Model Utilizing Frequency of Transition Approach. *Energies*. 2018; 11(8):2006.
https://doi.org/10.3390/en11082006

**Chicago/Turabian Style**

Yahaya, Muhammad Sharil, Norhafiz Azis, Amran Mohd Selva, Mohd Zainal Abidin Ab Kadir, Jasronita Jasni, Emran Jawad Kadim, Mohd Hendra Hairi, and Young Zaidey Yang Ghazali.
2018. "A Maintenance Cost Study of Transformers Based on Markov Model Utilizing Frequency of Transition Approach" *Energies* 11, no. 8: 2006.
https://doi.org/10.3390/en11082006