The operation of distribution system with the components in deteriorating condition makes the system reliability worsen. It is important to find the solution for balancing failure cost and maintenance benefits such as down-time and reliability. In this paper, time to replace the components in optimum condition based on constant-interval replacement mode is investigated. The optimal replacement time is mainly depended on component’s reliability and the cost ration of preventive replacement and failure replacement. In this paper, equipment inspection method and Weibull Analysis is applied to obtain the accurate reliability estimation. Weibull Analysis is applied with constant-interval replacement model to investigate the optimum replacement time for each component considering the different cost ratios. According to the quantitative results, the determination of the optimal replacement time (OPT) can minimize the total downtime and failure cost. Consequently, the reliability of the system is maximized and estimation also becomes more accurate due to sufficient approach.
Reliability is an important issue for electric power sector. Many electrical components are installed in distribution system and generally the interruption will be occurred if one component is failure. Therefore, reliability of each component should be in the reasonable rage. The failure rate decreases over the working age of the components. The aging components are necessary to replace. On the other hand, the replacement cost also should be considered. A component is required to replace preventively upon reaching age, where the cost of preventive replacement is typically less than that of corrective replacement.
Moreover, reliability distribution is one of the main factors to estimate the optimal replacement time (OPT). Therefore, it is an important issue to forecast reliability accurately. Concerning this issue, the use of the average values is potentially misleading and it has two major drawbacks [
Based on the problems and weak point mentioned above, we proposed how to evaluate the optimum replacement age using equipment inspection method and Weibull distribution method (WDM) considering cost of an unplanned on-line replacement and cost of a planned off-line replacement before failure. In this paper, time varied failure rate is evaluated based on equipment inspection method (EIM) with the condition score of equipment. Then, two parameters of Weibull distribution are figured out by regression method with the input failure data obtained from EIM. Thus, the equipment reliability and optimum replacement age is determined using constant-interval replacement model.
Three conditions of failure rate can be classified as best-condition, average-con- dition and worst-condition. These conditions will be changed accordance with condition and working ages of equipment. The condition of equipment is assigned between 0 and 1 by inspecting with inspection form. Generally, condition score is directly proportional to the working ages of equipment. Condition score zero refers to the best condition or new condition and one refers to the worst condition or damaged condition. The failure rates based on condition score can be calculated as follows [
where, λ is failure rate and x is condition score. The parameters A, B, and Care computed as below.
where,
The failure probability density function f(t), the failure probability distribution function F(t) and the reliability function R(t) can be analyzed using Weibull distribution methods [
(5)
(7)
where, η is a scale parameter and β is a shape parameter. These parameters can be calculated by using probability plot and regression analysis.
The x axis of Weibull plot is natural log value of life time or mean time to failure (MTTF)). The value of the y axis is calculated as shown in (9).
(9)
The proportion of the population that will fail by MTTF can be estimated by using median ranks method.
where, i is the adjusted rank and n is the total number of MTTF tested.
The OPT is to perform preventive replacements at constant intervals of length tp, irrespective of the age of the item. The objective is to determine the optimal interval between preventive replacements to minimize the total expected replacement cost per unit time. Cp is the total cost of a preventive replacement. Cf is the total cost of a failure replacement [
The total expected cost per unit time for preventive replacement at intervals of length tp denoted C(tp) is
where,
(12)
After some differentiating process, we obtain
(13)
In [
According to (8)
(15)
The optimum replacement time and total expected cost per unit time become:
(16)
The test system used in this paper is RBTS Bus 2 system shown in
The test system has four feeders and thirty six feeder sections. Two 33/11 kV transformers for substation and 11/0.4 kV transformers for load points are installed. Breakers also set up in substation and upstream area of every feeder. Disconnecting switches are located in feeder section 7, 21 and 32. Feeder types and lengths are listed in
The λ is the failure rate per year per mile for lines and the failure rate per year for other components. The reliability and system data is shown in
Feeder Type | Length (km) | Feeder section numbers |
---|---|---|
a b c | 0.6 0.75 0.8 | 2, 6, 10, 14, 17, 21, 28, 30, 34 1, 4, 7, 9, 12, 16, 19, 22, 24, 27, 29, 32, 35 3, 5, 8, 11, 13, 15, 18, 20, 23, 26, 31, 33, 36 |
Feeder | Average Load (MW) | Peak Load (MW) | Numbers of customers |
---|---|---|---|
1 2 3 4 | 3.645 2.15 3.106 3.39 | 5.934 3.5 5.057 5.509 | 652 2 632 622 |
Total | 12.291 | 20 | 1908 |
Component | λ(0) | λ(1/2) | λ(1) |
---|---|---|---|
Breakers Disconnecting switch Fuse Transformer 11/0.4 (pole mounted) Power Transformer 11 kv Lines(one mile) | 0.0005 0.002 0.002 0.002 0.0075 0.01 | 0.01 0.014 0.009 0.01 0.04 0.1 | 0.06 0.28 0.06 0.03 0.14 0.06 |
[
The condition score of components to their operating years is described in
Using EIM as mentioned in (1), we can evaluate the different failure rate according to the respective condition score. Firstly, we need to find the parameters A, B and C using three types of failure rate for all components. Then, we can find MTTF for each working age. MTTF based on condition score for transformer is shown in
Variable failure rate (VFR), average failure rate (AFR), worst failure rate and
Transformer | Disconnect switch | Circuit Breaker | Overhead Feeder | ||||
---|---|---|---|---|---|---|---|
Age | Score | Age | Score | Age | Score | Age | Score |
0 - 1 1 - 10 11 - 20 21 - 25 26 - 29 29 - 31 32 - 35 36 - 40 Above 40 | 0.00 0.05 0.10 0.25 0.40 0.50 0.60 0.80 1 | 0 - 1 1 - 5 5 - 10 10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 Above 40 | 0 0.01 0.02 0.03 0.1 0.15 0.2 0.3 0.5 1 | 0 - 1 1 - 3 4 - 5 6 - 8 9 - 10 13 - 15 15 - 20 21 - 25 26 - 30 >30 | 0 0.05 0.07 0.2 0.25 0.3 0.5 0.725 0.75 1 | 0 - 5 6 - 15 16 - 25 26 - 35 35 - 40 40 - 45 45 - 50 51 - 55 56 - 60 65 Above 65 | 0 0.02 0.04 0.06 0.1 0.15 0.2 0.25 0.4 0.5 1 |
Age | Score (x) | λ(x) | MTTF |
---|---|---|---|
0 - 1 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 36 - 40 >40 | 0 0.02 0.05 0.07 0.1 0.25 0.4 0.6 0.8 1 | 0.0075 0.00822 0.009361 0.010167 0.011444 0.019301 0.030307 0.052136 0.086356 0.14 | 132.9680 121.3279 106.5291 98.0925 87.1412 51.6701 32.9051 19.1281 11.5483 7.1233 |
best failure rate condition for circuit breaker can be seen in
For earlier working ages or until 0.5 condition score of circuit breaker, the AFR has over estimation compared with VFR as the value of AFR is higher than VFR. As we can see in figure, the red marker represents the value of VFR at condition score 0.5. The system operators or the utilities will set the maximum condition score and it is not reasonable to use the component until condition score 1. The failure rate at 0.06 failure/year is just for the accidently damaged condition of component. Moreover, they will try to maintain it almost at the best condition or failure rate at 0.005 failure/year. The maximum condition score for CB is set up at 0.75 and the highest failure rate is 0.03 failure/year.
To estimate the reliability function with Weibull distribution, we need to find the shape parameter β, and scale parameter η. Weibull parameters can be easily calculated using regression analysis. MTTF is used as main input parameter for regression analysis. The MTTF is obtained based on VFR by using EIM method.
The result of regression analysis for transformer is shown in
parameters can be estimated from the trend line extracted from regression analysis. The shape parameter β is the slope of the trend line and the scale parameter η is represented to exp (-intercept of y/β). As in
The optimal interval between preventive replacements is determined according to minimizing the total expected replacement cost per unit time. In this case, the ratio of Cp, the total cost of a preventive replacement, and Cf, the total cost of a failure replacement, has influence on tp, the optimum replacement time. The values of Cp and Cf will be changed corresponding to the equipment and the conditions. The cost ratio Cp/Cf, is commonly equal to 1/20. For some expensive system, the failure replacement cost is higher 100 times or more than the preventive replacement cost.
In
The OPT for pole mounted transformer is figured out in
The price of power transformer is relatively high and the Cf cannot be higher
Component | Shape Parameter | Slope Parameter |
---|---|---|
Circuit Breaker Power Transformer Pole-mounted ransformer Disconnecting switch Feeder Type a Feeder Type b Feeder Type c | 1.37 3.7 2.3 1.37 2.23 2.23 2.23 | 286 103 273 286 128 103 96 |
much than Cp as compared with circuit breaker or pole mounted transformer. Therefore, Cf is considered 1.5, 2, 5, and 7.5 times higher than Cp in
The reliability and failure distribution can be also checked at the optimum replacement time. In
cost per time is described as an example. In this example, the ratio of Cp/Cf is 0.05. The reliability probability is 0.8538 and failure probability is 0.1462 at the optimum replacement time of 39.8 years for CB.
This paper proposes how to estimate the optimum replacement time (OPT) by using combination of equipment inspection method, Weibull distribution method and the constant-interval replacement model for preventive maintenance. According to the results, the OPT is mainly depended on reliability probability of equipment and the ration of the total cost of a preventive replacement Cp and the total cost of a failure replacement Cf. Therefore, the estimation of OPT will be accurate if reliability and cost ratio is estimated. In this paper, the reliability estimation is more fact-based using EIM and WDM based on condition score. For the cost ratio, on the contrary, it is difficult to estimate the exact amount. In this paper, the cost ratio is considered in the possible range according to the life expectancy. Estimation of Cp and Cf will be a future consideration. However, in this paper, the OPT is evaluated based on different ratio of Cp and Cf. Consequently, reliability and failure probability can be checked respective to their OPT.
The proposed approach with corresponding computational methods in this paper can assist the system operators to make the economic maintenance and replacement decisions more easily.
This work was supported by Japan International Cooperation Agency.
Lin, O.Z. and Miyauchi, H. (2017) Optimal Replacement Time of Electrical Components Based on Constant-Interval Replacement Model: E- quipment Inspection Method and Weibull Analysis. Energy and Power Engineering, 9, 475-485. https://doi.org/10.4236/epe.2017.94B053