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We study a multistate model for an aging piece of equipment under condition-based maintenance and apply an expectation maximization algorithm to obtain maximum likelihood estimates of the model parameters. Because of the monitoring discontinuity, we cannot observe any state's duration. The observation consists of the equipment's state at an inspection or right after a repair. Based on a proper construction of stochastic processes involved in the model, calculation of some probabilities and expectations becomes tractable. Using these probabilities and expectations, we can apply an expectation maximization algorithm to estimate the parameters in the model. We carry out simulation studies to test the accuracy and the efficiency of the algorithm.

A multistate model may provide more flexibility than a traditional binary state model for modeling equipment conditions. In a multistate model, a piece of equipment is allowed to experience more than two possible states, for example, completely working, partially working, partially failed, and completely failed. Even if every state has an exponential duration distribution, the equipment has a nonexponentially lifetime distribution. Therefore, many authors utilize multistate models for equipment in many disciplines. Examples arise from electric power systems [

On the other hand, maintenance is an indispensable action to keep the reliability level of industrial equipment. To reduce the total maintenance costs, condition-based maintenance is usually implemented. References [

We assume that every state of the equipment has an exponential duration distribution specified by its parameter. In the paper, we investigate the parameter estimation problem, based on the data obtained from routine inspections and repairs.

Such a parameter estimation problem for a multistate model plays an important role in many applications [

There is an alternative method to obtain an MLE. To better illustrate the idea, we start from a complete data set of every state’s durations through continuous monitoring. As it has an exponential duration distribution, the reciprocal of the average of durations of a state is the MLE of its parameter. This simple example gives a clue to the case of condition-based maintenance. Then, given the incomplete data from routine inspections and repairs, we can calculate conditional expectations of every state’s duration, and we may deduce an MLE. The theoretical foundation of the above discussion is the expectation maximization (EM) algorithms. The EM algorithms are first introduced in [

Recently, a few researchers have tried to use EM algorithms for problems similar to ours. For example, based on a complicated discussion, [

In this paper, we first propose a proper mathematical framework and a special technique of distributions to simplify the computation of conditional expectations. Then we apply an EM algorithm to obtain MLEs of the model parameters, with the observation of routine inspections and repairs. The paper is organized into seven sections. Section

In this section, we propose the mathematical framework for our problem. In the following Model

In the multistate model, a piece of equipment with

In Model

To introduce some derivational concepts to describe the behavior of the equipment clearly, we introduce several families of auxiliary random variables. These random variables,

For

For

Moreover,

We now apply the EM algorithm to estimate the parameter

The EM algorithm, introduced in [

As the pdf of the repairs

Based on the theory of EM algorithms, (see, [

Let

Set

Stop and output

Set

To calculate the conditional expectations involved in (

Assume that random variables

The proof of Lemma

Under the assumption and notations in Lemma

Let

Let

Suppose that a random variable

By computing the definite integral of the pdf of

It follows from the independence among

Then we can calculate several auxiliary important conditional expectations.

Under the assumption and notations of Lemma

In the following, we calculate three expectations respectively.

We have

We have

When

Now we turn to the following Lemma

For random variables involved in Model

Finally, we arrive at conditional expectations in (

We assume that there is a guess

Except some trivial cases, this result follows from Theorem

In this section, we test the efficiency and accuracy of the EM algorithm, based on simulation data.

At first, we investigate the necessity of the condition “

Now we turn to the assumption of Lemma

Let

Similar discussions can be applied to other theorems and lemmas in Section

Secondly, we present a method to simulate the behavior of a 4-state model for an aging piece of equipment under condition-based maintenance, based on some preset parameters. The simulation can provide us a test data set.

Without maintenance, the equipment would run through states 1, 2, 3 orderly to a failure state 4. With maintenance, the straight path to a failure is regularly deflected by inspections and maintenance. In an inspection, if we find that the current state is 1 or 2, no repair is applied. If the state is 3, an appropriate repair is carried out. In this case, the state of the equipment right after the repair is a random variable, which is 1 with probability 0.1, is 2 with probability 0.3, and is 3 with probability 0.6. If the state of the equipment is 4, we must replace it with a new piece of equipment, and the state right after this replacement is 1. The duration of states 1, 2, and 3 have exponential distributions specified by parameters 0.3, 0.29, and 0.5. These parameters are suggested by [

Now we turn to five experiments in which we test the efficiency and accuracy of the EM algorithm, based on simulation data provided by the above method.

In the first experiment, we study the relationship between convergence error and the number of iterations. As EM algorithms are recursive, this experiment will provide us with an intuitive stop criterion for Algorithm

Relationship between convergence error and the number of iterations.

In the second experiment, we run the EM algorithm for different initial values. Figure

Estimated parameters for different initial values.

In the third experiment, we run the EM algorithm for different sample sizes. Given the number of inspections

Estimated parameters with different numbers of inspections.

20 | 0.4089 | 0.2700 | 0.7639 |

40 | 0.3754 | 0.3123 | 0.6581 |

100 | 0.3342 | 0.2778 | 0.5887 |

200 | 0.3271 | 0.3040 | 0.5414 |

400 | 0.3215 | 0.2831 | 0.5354 |

1000 | 0.3075 | 0.2983 | 0.5256 |

In the fourth experiment, we repeat the third experiment 100 times to obtain the standard errors of each estimated parameter. Table

Standard errors of estimated parameters.

20 | 0.0902 | 0.0632 | 0.2337 |

40 | 0.0462 | 0.0582 | 0.1681 |

100 | 0.0354 | 0.0280 | 0.0963 |

200 | 0.0265 | 0.0263 | 0.0740 |

400 | 0.0178 | 0.0164 | 0.0532 |

1000 | 0.0129 | 0.0114 | 0.0406 |

Finally, in the fifth experiment, we compare our method with another method.

As indicated in Section

On the other hand, although we assume that, in our method, the benchmark interval is 1 for simplicity of argument, it is not too difficult to implement our method for other benchmark intervals. We run the simulating equipment for 400 time units and obtain inspection data sets for different benchmark intervals, respectively.

Figure

Estimated parameters of two methods for different benchmark intervals.

In this section, we apply the EM algorithm to a real-world dataset. The dataset is a 25-year record of inspections and repairs of a power transformer substation under condition-based maintenance. The power transformer substation consists of two power transformers, and it is a cool backup system. Once the primary transformer incurs faults, the secondary transformer replaces it automatically, and it can generally serve the entire load. The benchmark interval of inspections is three months. During an inspection, a transformer is repaired if it incurs faults, and the repaired primary transformer is put into operation. The repair is perfect because we can replace a transformer if necessary.

The behavior of the power transformer substation can be described by a 3-state model with a warning value

Inspections of a power transformer substation.

Inspection no. | 0 | 1–10 | 11 | 12–37 | 38 | 39–55 | 56 | 57–73 | 74 | 75–80 | 81 | 82-83 | 84 | 85–98 | 99 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

— | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 3 | 1 | 2 | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | — |

Based on the dataset given in Table

We use the total time on test (TTT) plot to check the efficiency of the result. The TTT plot is obtained by plotting

against

The TTT plots for real-world dataset and the simulated dataset.

In this paper, we have addressed the parameter estimation problem for the multistate model for aging equipment under condition-based maintenance. Based on the memoryless property of exponential distributions, we proposed a convenient mathematical framework for the problem. In this framework, the calculation of involved conditional expectations became tractable. Then we applied an EM algorithm to obtain MLE of parameters. A sequence of simulation experiments shows that the estimated parameters converge to the preset value of the parameters, even for a moderate number of inspections. Moreover, the EM algorithm is stable for different length of benchmark intervals. Hence, the algorithm can be recommended for practical applications. It is convenient to extend the algorithm to the situation with random benchmark intervals.

In this paper, we assume that the state of the equipment can be directly observed at an inspection. However, it is a difficult task for many types of industrial equipment. In these cases, some other variables may be used to obtain estimates of the states. To establish a full model consisting of equipment, inspections, indirect variables, and investigating parameter estimations, it is challenging and valuable for practical situations.

We proceed by an induction on

For

Assume that for

It is obvious that

So the result is true for

For a time instance

Let

It is obvious that

Then, we have the following:

The authors acknowledge the funding of the National Natural Science Foundation of China (Grant Number 50977073). Qihong Duan acknowledges the funding of the National Natural Science Foundation of China (Gant Number 70971109).