DECLARATION 2- PUBLICATIONS
4.7 Results and discussion
4.7.2 Probability plots and sensitivity analysis
This section presents results of probability plots and sensitivity analysis, that is, sensitivity of the risk factor with respect to operations and maintenance (O & M) costs. The MLE parameters from Table 4-3 were applied to compute and plot the PDF, CDF, hazard rate and risk-cost trends.
The computed parameters were fitted in the term Tin equation (4.24) in order to plot the cost and risk trends.
Plots of the PDF and CDF are presented in Figure 4-5, whereas the hazard rates (risks of failure) are shown in Figure 4-6. Figure 4-5 shows that the PDF and CDF are slightly skewed to the right with the point of inflection for the CDF occurring at 33.7 years of age. The hazard rate corresponding to that age (point of inflexion) is 0.07. The PDF is zero when the CDF is equal to unity; signifying the end of technical life (lifespan), with probability = 1 at 67 years of age. Thus by determining the CDF and PDF from parameter estimates, analysts can predict the life span of assets such as transformers.
Figure: 4-5. Plots of (a) PDF and (b) CDF
Figure: 4-6. Hazard rate characteristics
0 10 20 30 40 50 60 70
0 0.01 0.02 0.03
0.04 (a) PDF for =3.43
Probability density
Technical life, t [yr.]
0 10 20 30 40 50 60 70
0 0.5
1 (b) CDF for =3.43
Cumulative density
Technical life, t [yr.]
0 10 20 30 40 50 60 70
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.45 Hazard rate for =3.43
Technical life, t [yr.]
Magnitude
In another study on the degradation of the degree of polymerization (DP) of transmission transformer insulation paper, a life span of 70-80 years was projected [72]. These transformers were definitely under different operating conditions to the ones being studied in the current research, but their results portray the average life expectancy of the transformers in most electric grids.
Trends of planned and unplanned maintenance costs are displayed in Figure 4-7, showing the intersection of the two curves at point A. The data that was used for the computation of the costs came from Table 4-2 (Appendix B, Table B3 is the source of the data).
Figure 4-7: Maintenance cost profiles
Figure 4-7 demonstrates how maintenance costs vary with time. The planned costs are equal to unplanned costs at 29.9 years of age; shown as point A. Point B on the total cost curve that is overlaid on the figure is the total cost for point A. After point A, unplanned costs exceed the planned costs because the number failing due to the aging of items is higher than that of surviving items.
The point A in Figure 4-7 is very significant in the timing of refurbishment and major maintenance strategies. It points to the fact that the refurbishment strategies should be implemented just before point A, that is, before the breakdown maintenance costs exceed the planned maintenance costs. The timing represents 44.6% of the lifespan or just before mid-life-span. This is
0 10 20 30 40 50 60 70
0 0.5 1 1.5 2 2.5
3x 104 Maintenance costs
technical life, t [yr.]
Cost, US$
Planned preventive costs Unplanned costs Total cost
A C B
a simplified and innovative way of forecasting asset replacement and refurbishment timing and could be a valuable tool for power utility AM planning. The actual refurbishment could be planned to commence earlier, say at 40% of the lifespan.
Cumulative maintenance costs are presented in Figure 4-8. The two cost-curves intersect at 52.4 years of age (point B). At that time, decisions regarding end-of-life strategies like refurbishment and disposal should have been made so that breakdown costs are reduced (i.e., before they exceed the costs of preventive maintenance).
Figure 4-8: Cumulative maintenance costs over the lifetime
The rest of this section shows how the model, that is, equation (4.24), was applied in trending the risk and costs for the following three scenarios: risk with business as usual (unmitigated risk), risk with mid-life refurbishment/renewal, and risk with end-life refurbishment. The values used in the analysis were as follows: ϕ = 3, γ = 20, σ = 6, λ= λr = 0.02, and μ = 0.03. Then, the sensitivity of the risk factor to changes in ρ, τ and ω were analyzed.
Risk trends for the three scenarios are presented in Figures 4-9 and 4-10, with plots of risk reductions due to the application of the strategies provided beneath them. It is worth noting that the
0 10 20 30 40 50 60 70
0 2 4 6 8 10 12 14 16
18x 105 Cumulative maintenance costs (transmission), =3.43
Technical life, t [yr.]
Cost, US$
Cumulative costs
Costs with surviving components Costs with failed components
A B
abscissa of the plot of risk trends is given in age group ζ as explained in Chapters three and four [97]. The reason for this is that the key lifecycle decisions and actions in AM are not done yearly, but, on average, in five-year strategic time frames [94], [97].
Figure 4-9 shows the plots of risk trends (on top) and magnitudes of risk reductions (beneath) for a case where major end-life renewal efforts are implemented.
Figure 4-9: Risk profiles and reductions (for end-life strategies)
0 2 4 6 8 10 12 14
-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
Figure 4-10: Risk profiles and reductions (for mid-life strategies)
Figure 4-10 indicates plots similar to those in Figure 4-9, but for a scenario where major mid- life renewal efforts are applied. The risk reductions are of prime importance as they measure the magnitudes of risk mitigation required by asset managers in order to invoke AM decisions.
Figure 4-11 shows the risk reductions that were presented beneath Figures 4-9 and 4-10 as well as their cumulative values. Later on (in Figures 4-12 and 4-13), these risk reductions are expressed in terms of cost benefits of risk mitigation (attenuations). This is achieved by superimposing the cost model, that is, equation (4.23) onto the risk trending model, that is, equation (4.24) as demonstrated in Figures 4-12 and 4-13. This is of great significance to the asset manager as it quantifies the impact of the risk-profiling (trending) in monetary terms, which is more meaningful than using the risk factors only.
0 2 4 6 8 10 12 14
-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
Figure 4-11: Risk reduction and cumulative values (a) end-life (b) mid-life
Figures 4-12 and 4-13 outline trends of risk as well as the cost benefits associated with the various AM strategies. Figure 4-12 shows the cost benefits accrued when the major end-life renewal strategy is carried out, whereas Figure 4-13 indicates the cost benefits associated with the major mid-life renewal.
0 2 4 6 8 10 12 14
-0.5 0 0.5 1 1.5 2 2.5
3 (a) Cumulative reduction in risk factor(major end-life renewal)
Age group,
Risk factor
Mid-life cumulative Endlife cumulative End-life reduction Mid-life reduction
0 2 4 6 8 10 12 14
-0.5 0 0.5 1 1.5 2
2.5 (b) Cumulative reduction in risk factor(Major mid-life renewal)
Age group,
Risk factor
Mid-life cumulative Endlife cumulative End-life reduction Mid-life reduction
Figure 4-12: Risk-cost trends (major end-life renewal strategies)
Figure 4-12 shows that end-life renewal strategies yield substantial cost benefits (savings), but the benefits are accrued late in the lifecycle. In contrast, Figure 4-13 portrays that mid-life renewal strategies accrue cost benefits early enough to be re-invested in the business. From an AM point of view, the mid-life renewal case is the best option. The model is vital for a risk-based power distribution AM as it expresses the risk profile in terms of savings in O & M costs. Since the financial benefits of refurbishment are hard to show, the findings from this study can be used to emphasize the importance of refurbishment strategies. The study shows that although refurbishment does not necessarily add capacity that is required for generating more revenue, it is a means of achieving tangible financial benefits in the form of O & M savings.
0 2 4 6 8 10 12 14
-1 0 1 2 3 4 5 6 7
8x 104 Risk-cost superimposition (major end-life renewal)
Age group,
Cost, US$
Business as usual With mid-life renewal With end-life renewal Mid-life cost savings End-life cost savings Cumulative mid-life savings Cumulative end-life savings
Figure 4-13: Risk-cost trends (major mid-life renewal strategies)
In this chapter, systems thinking (theory) (extended from Chapter three) was used in determining cause and effect relationships in a complex power utility AM system. Causal links that are difficult to detect by statistical or analytical techniques were determined by systems thinking.
That augmented the capability of statistical inferences. It validated the notion that statistics may show correlations, but not necessarily causality [35]. Furthermore, equipment failure data was applied to compute the Weibull parameters using the MLE and MOM. Then, the parameters were employed in reliability analysis and in risk-cost trending. That was done in order to evaluate the cost benefits of component risk trending.
Therefore, the risk trending model was successfully applied to evaluate the cost benefits of the risk trending process. This is of great significance as the quantitative capability of the model can help physical asset managers in exploring, treating, monitoring and reviewing risks associated with their physical assets over the entire lifespan. The ability to show the quantitative benefits of refurbishment (renewal) can change the managers’ perception of refurbishment projects, so that the projects are seen as adding economic value in terms of the O & M cost-savings. In order to
0 2 4 6 8 10 12 14
-1 0 1 2 3 4 5 6
7x 104 Risk-cost superimposition (major mid-life renewal)
Age group,
Cost, US$
Business as usual With mid-life renewal With end-life renewal Mid-life cost savings End-life cost savings Cumulative mid-life savings Cumulative end-life savings
augment the model, asset managers may conduct a comparative NPV analysis for the three strategic options, namely: business as usual, mid-life renewal and end-life renewal as advanced by [64].
The multi-criteria risk evaluation approach presented in this chapter has only considered preventive and breakdown maintenance costs in the cost benefit analysis. Future research should explore ways of adopting other types of costs. The model developed could assist asset managers in the assessment of their physical assets during the lifecycle. Section 4.8 provides a summary of the overall model formulation process.