DECLARATION 2- PUBLICATIONS
3.5 Results and discussion
3.5.3 Comparison of PDF and CDF for reactors
Figure 3-15: Cumulative benefits: (a) end-life and (b) mid-life
Figure 3-16: Comparison of Normal and Weibull PDFs for 500kV reactors
Figure 3-17: Comparison of Normal and Weibull CDFs for 500kV reactors
20 22 24 26 28 30 32 34 36 38 40
0 0.02 0.04 0.06 0.08 0.1
0.12 Plots of normal and Weibull PDFs:=29.727,=3.909,=31.361,=9.407
Years
Probability density function
Normal distribution Weibull distribution
20 22 24 26 28 30 32 34 36 38 40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 Plots of normal and Weibull CDFs:=29.727,=3.909,=31.361,=9.407
Years
Cumulative density function
Normal distribution Weibull distribution
Figures 3-18 and 3-19 show how the risk trending applies to the data for the reactors. The figures show that for the reactors, the risk in the first twenty years is very low. This is due to the high reliability of reactors, in general.
Figure 3-18: Application of the risk trending model to reactors (end-life renewal)
Figure 3-19: Applying the risk trending model to reactors (mid-life renewal)
0 2 4 6 8 10 12
-0.5 0 0.5 1 1.5 2
2.5 Risk and risk reduction levels (major end-life renewal)
Age group,
Risk factor
Risk-business as usual Mid-life renewal End-life renewal Mid-life risk reduction End-life risk reduction
0 2 4 6 8 10 12
-0.5 0 0.5 1 1.5 2
2.5 Risk and risk reduction levels (major mid-life renewal)
Age group,
Risk factor
Risk(Business as usual) Risk(Mid-life renewal) Risk(End-life renewal) Mid-life risk reduction End-life risk reduction
As stated earlier on, the reactor curves presented in Figures 3-16 to 3-19 serve the purpose of demonstrating how the model can be used to model the risk of various types of physical assets if the failure functions or the life modelling parameters are known (determined). Just like power transmission transformers, failure of the reactors is age related. They do not portray any infantile mortality failure.
In this chapter, the failure risk of physical assets was modelled. Inferences from system dynamics, the Markov analysis, the Weibull distribution and the bathtub curve analysis were assembled to come up with the model. The power grid AM system was represented by causal loop diagrams from systems thinking. A subsystem with components operating in a high-load regime was expressed as a system in state-space transitions so as to enable the application of the Markov processes. System dynamics principles were used to adapt the inferences into a linear algebraic form of a risk trending model containing a failure risk function. The failure risk function was expressed in terms of parameters of the Weibull distribution. The parameters were estimated using the MLE. Only failure data was used to estimate the parameters according to methods in [35], [81].
The sensitivities of the risk factor to variations in the number of components renewed and the number admitted to or relieved from a high-operating-load regime during the lifespan were simulated. The focus of the sensitivity analysis was on the impact of major mid-life and end-life renewal strategies on the risk factor. The sensitivity analysis was applied to a set of AM systems made up of substation transmission transformers. This was able to track the magnitudes of risk amplifications and attenuations caused by the application of renewal strategies.
The use of the Weibull distribution made it possible to model with a few data sets, in this case, twelve sets. The Weibull distribution may be applied to model a wide range of distributions, provided the shape and scale parameters for a given set of data are known. The challenge that can arise when dealing with failure data is that some components may have been taken out of service before the end of their lifespan, or the number of failures may be small. In that case, [35], [81]
recommend using the MLE for the parameter estimation. It is assumed that the transformers used in the present work had reached their end of lifespan because the DP was ≤ 200. Normally, a DP of ≤ 200 is an indication of complete loss of mechanical integrity of the insulation material [68], [96].
The model that has been advanced (proposed) successfully trended the failure risk of physical assets. It could be used as a planning tool and as a measure of improvements brought about by the application of strategies or technologies associated with the strategies. It is flexible because the shape and scale parameters that are estimated are unique to the type of failure data used. The parameter estimates can be used to show dominant failure modes during the component life cycle based on the plots of the PDF and hazard rates.
When using the model, analysts may choose any empirical failure function that best represents the failure data. For example, [15], [16] presented a transformer failure function based on Perks’
formula that may be used as follows:
btp p bt
e e t A
F
1
(3.28)where F(t) is the instantaneous failure rate, A represents the frequency of random events such as lightning and collisions, b, αp and μp are constants estimated using the Bayesian method in [15], and t is the time of operation.
In this chapter, it was possible to assign modelling equations to systems thinking approaches in management. Most of these approaches lack quantitative capabilities [2], [26], [28], [33], [34]. The MLE of Weibull parameters, Markov processes, and bathtub curve inferences are well known, but their integration in the systems thinking philosophy is remarkable, as it enhances the quantitative capability of the philosophy. The integration was facilitated by the appropriate application of the dynamic hypothesis. This underscores the importance of the dynamic hypothesis in the successful advancement of system education.
Models are supposed to break down complexity [58]. The proposed model did that by utilizing a small number of times to failure for modelling component reliability and risk. In the power utility AM this is significant, because data unavailability is one of the main barriers to successful statistical analysis as alluded to in [2]. Application of the MLE for parameter estimation guaranteed accuracy in dealing with the small sample sizes encountered in the study. The MLE is also a reliable method when handling data that is censored [35], [81].
It is worth mentioning that systems thinking helps to reveal the root causes of problems so that optimization models can be applied to deal with the specific problems that have been identified. In this chapter, the MLE was the model used to optimize the Weibull distribution parameter estimates.
The focus of the study (chapter) was on modelling of the risk profile changes (as opposed to maintenance optimization) with respect to the application of renewal strategies. This is very important in providing a strategic direction for a risk-based asset management planning process. It can be a useful tool for analyzing the risk level and to overcome some of the shortfalls of the risk matrix approaches which were examined in Section 2.5.2. When applying the model, asset managers may integrate models like SMDP or MDP, which were discussed in Section 2.6.3.2, in order to optimize other parameters like inspection rates and maintenance policies. These are beyond the scope of the current research, but will be the focus of future studies.