Physical life of transformer generally refers to the life determined by the physical wear and tear, which is also the life of the transformer from the time put into service until retired due to physical wear and tear. Physical life of transformer is mainly determined by tangible wear also related to self-quality status, the use of technology and maintenance quality. But, technical life of transformer commonly means that with the development of science and technology, existing transformer will be replaced by more advanced transformer before its physical or economic life is ended. Therefore, technical life of transformer mainly indicates the time from transformer investment to its loss of use value due to technological progress. Because of the impact by the intangible wear, technical life of transformer is generally shorter than the natural life. Based on the above, transformer physical life and technical life are defined from the use of the transformer itself.

Economic life of transformer generally refers to the life determined by operation and maintenance costs of transformer. Because of transformer worn out in its late use, operation and maintenance costs of transformer is increasing. Under that circumstances, there will have a service cycle which is economic life of transformer when the average running cost is below a certain limit. Economic life is often used to analyze and determine the best life and the best update time of the transformer [11]. Therefore, we can determine the best life and the best update time of the transformer when the economic life is certain. Based on the above, there will be a year that the average running cost is minimum and the year can be considered the economic life of transformer.

According to the investment structure of transformer in the actual operation of power enterprises, the cost of transformer based on life cycle theory contains initial investment cost, operation cost, maintenance cost, fault cost and equipment residual value. The life cycle cost of transformer can also be divided into variable cost (including maintenance cost, fault cost) that vary with the failure rate and fixed cost (including initial investment cost, operating cost and equipment residual value) that do not vary with the failure rate.

1) Initial investment cost

The initial investment cost includes all cost before the transformer is utilized, mainly including purchase, mounting cost, test cost and so on.

where is purchase cost, is installation and mounting cost.

2) Operation cost

The operation cost mainly consists of the loss of transformer in operating.

where is the equivalent no-load loss, is the equivalent load loss, is load fluctuation loss factor, is the load factor which means the transformer loading over its time, is run time, is the purchasing cost of electricity, is the loss rate of load per year,

where is the empirical coefficient, represents the number of hours in maximum load per year.

3) Maintenance cost and fault cost

Maintenance cost and fault cost vary with the failure rate. Therefore, different transformer failure time cause different economic losses. In order to quantify the economic loss of the transformer, this paper classifies transformer failures into general, severe, and catastrophic failures [12].

where is maintenance cost, is fault cost, refers to general, severe, and catastrophic failures, is type failure of transformer, is the maintenance cost caused by type failure, is the comprehensive loss caused by type failure, is a correcting parameters.

4) equipment residual value

Equipment residual value refers to the residual value of recycling when the equipment retired.

where is equipment residual value, is constant coefficient for Simplified representation.

In general, the Weibull distribution can be used to describe the failure rate of electrical equipment with time. Equipment cumulative failure probability refers to probability of failure in the specified conditions and time, also known as unreliable function or life distribution function in the failure rate model [13].

where is reliable function.

Therefore, assume that the service life of electrical equipment distribution in line with the three-parameters Weibull distribution of cumulative failure probability function:

where is form parameter, , is scale parameter, , is location parameter, and equipment will not usually fail before, refers to the operating time of equipment.

The failure rate of equipment in the life cycle can be defined as the failure rate per hour after time when equipment has not failed.

where is the failure rate of equipment in the life cycle.

Usually in the process of equipment operation, its failure rate goes through the following three stages: early failure rate of the characteristics is decreasing, known as practiced period; the middle of the failure rate is almost constant, known as using period; lately failure rate of the characteristics is increasing, known as wear-out period. As the Figure 1 shows:

The curve above can be well fitted by the Weibull distribution. The failure rate function can be described as:

where three stages of the curve respectively correspond to the form parameter 0 < m < 1, m = 1, m > 1 three situations.

The failure rate curve can be fitted piecewise through the analysis of the history statistics from the same type electrical equipment based on the Weibull distribution. Then, form parameter, scale parameter, location parameter can be estimated according to the fitted curve. Finally, the failure rate function is determined.

Frist, assuming that location parameter is known, form parameter and scale parameter can be estimated by Levenberg―Marquardt algorithm. Then, location parameter is determined by the nonlinear fitting method. Specific steps are as follows:

1) Assuming is known as, according to the failure rate function:

2) Based on the initial value of the parameters, obtained by the above formula, two data points located on the Weibull distribution are selected to construct the equations:

3) According to initial value of the parameters, , and N sets of data, we can use Matlab for iterative calculation based on Levenberg-Mar- quardt algorithm.

4) Parameter and scale parameter can be determined form the calculation in step3.

5), which is the cut-off point of the failure rate stages, can be calculated by the part of n sets of data based on non-linear fitting,

6) The failure rate interval can be determined by the fitted curve and error which are calculated based on Matlab:

where refers to the prediction of equipment failure rate on year.

The historical data of transformer SFPSZ-120000/220 can be found in reference [15]. From the Figure 2, the demarcation point between using period and wear-out period is about 12 years.

Frist, the location parameter of Weibull contribution can be assumed as 12, , which means the transformer goes into wear-out period on 12 years. Next, the failure rate of transformer can be fitted based on Levenberg-Marquardt algorithm in order to get form parameter, scale parameter . Then, location parameter can be determined as based on nonlinear Fitting. Finally, the failure rate curve can be fitted as Figure 3. The failure rate function can be described as:

The failure rate can be interval:

The upper and lower limit of failure rate is represented in Figure 3.

The annual average cost intervals of transformer can be calculated based on the failure rate fitted above. Life cycle parameter of transformer is presented as Table 1:

According to the data in the table and the failure rate calculated above, the upper and lower limits of the average annual cost in different operating years is presented in Figure 4:

In Figure 4, the annual average cost of LCC reaches a minimum of 2519.802 thousand ¥ in 33 years when the parameters of the transformer take the upper

limit; the annual average cost of LCC reaches a minimum of 2438.77 thousand ¥ in 31 years when the parameters of the transformer take the lower limit. Therefore, the optimal economic life interval of transformer is 31 - 33 years. When the transformer reaches the optimal economic life interval between 31 to 33 years, the power enterprise can consider the overall transformer repair or replacement.