Average worst 5 year engineering and environmental performance

(or the worst planning period) engineering and environmental performance, i.e., supply resilience and eco-deficit, within a single scenario whilst keeping the assessment across scenarios average to avoid planning for the worst case scenario only. By doing so the resilience and eco-deficit performance metrics reflect the worst planning period (i.e., 5 year) performance encountered during the planning time horizon (i.e., 50 years). The Equations 5-7 and 5-8 were changed as shown by Equations 5-13 and 5-14, respectively:

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝐴𝑣𝑒𝑟𝑎𝑔𝑒[𝑓𝑆𝑢𝑝𝑅𝑒𝑠 = max_𝑇 𝐷𝑗]

5-13

where j represents a five year period within the planning time horizon T, and Dj is the

failure duration in weeks in period j. The supply resilience metric minimizes the average maximum duration of a failure across all periods within the planning time horizon across all scenarios in the ensemble.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒_{𝐴𝑣𝑒𝑟𝑎𝑔𝑒}[𝑓_𝐸𝐶𝑂= max

𝑇 {(|𝐴𝑁𝑄70− 𝐴𝑆𝑄70| 𝐴𝑁⁄ 𝑄70)𝑗∗ 100%}]

5-14

where ANQ70 is the area under the naturalized flow duration curve (FDC) and ASQ70 is

the area under the simulated FDC. The final eco-deficit metric minimizes the average worst deficit across all periods that occurred in the planning time horizon across all scenarios in the ensemble.

133 Using this formulation results in most flexible Pareto optimal plans in all four sets implementing Meters in the first 10 years of the planning time horizon. However, using different bootstrapped scenario ensembles within the optimization creates different schedules present in the Pareto optimal sets, i.e. the recommendations are affected by the random resampling of scenarios. In practice that would mean that if decision makers used a single ensemble of resampled scenarios, they would obtain different solutions than if they used a slightly different bootstrapped ensemble generated in the same way. Figure 5-6 shows the four Pareto optimal schedules obtained using the worst five year performance assessment of the resilience and eco-deficit objectives. The difference between the sets is more significant that in Figure 5-5. In particular, the ESD implementation (yellow bars) spikes in 2040 for the ensemble 1 but gets distributed between 2040 and 2055 for the other three ensembles. The LRD implementation (light green bars) spikes in 2040 for ensembles 2 and 4, but is more evenly distributed for ensembles 1 and 3. The OCT and SLARS implementation (greenish-blue bars and light blue bars, respectively) spikes in different periods for each ensemble. The remaining intervention schedules also show some degree of difference between ensembles. This suggests that the worst 5 year performance occurs in different planning periods between the ensembles and assessing this performance as it stands is not suitable for

optimization across randomly resampled scenarios.

Figure 5-6. Schedules of interventions within the Pareto optimal plans obtained using the

bootstrapped scenario ensembles and average of the worst 5 year engineering and environmental performance. The bars and axes represent the same dimensions as shown in Figure 5-3 and Figure 5-5.

134

5.2.4.3. Combination of discounting and worst 5 year performance

Discounting only financial performance in time with the engineering and environmental performance temporally undistinguished delays investments that may prove effective in short term. This results in unequal consideration of the multiple criteria over time resulting in potentially biased solutions as demonstrated above. The financial performance is discounted here as water companies are required to discount the financial aspects of their strategies when planning for the next 25-30 years

(Environment Agency, 2012). Therefore, to reduce the temporal inequalities between objectives this study proposes to discount the engineering and environmental

performance. Pearce et al. argue that if people’s preferences count and if people perceive future risks to be regarded as of lower consequence than current risks, those preferences should be incorporated in the policy making.

The resilience and eco-deficit objective is discounted here for each five year period of the planning time horizon using the same discount rate of 4.5% as applied to the financial objectives. The resilience and eco-deficit objective value in a particular scenario then refers to the worst discounted five year performance value that occurs within the planning horizon. The final value across scenario ensemble is taken as the average value of the above. This allows for considering adverse events such as system failures that occur within near future to be of greater impact than events occurring in distant future as new technology or information may become available to prepare for adverse events more effectively.

Equations 5-13 and 5-14 were updated as follows: 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒_{𝐴𝑣𝑒𝑟𝑎𝑔𝑒}[𝑓_{𝑆𝑢𝑝𝑅𝑒𝑠} = max

𝑇 {𝐷𝑗 ∗ (1 + 𝑖𝐶) −𝑗_}]

5-15

where j represents a five year period within the planning time horizon T, Dj is the failure

duration in weeks in period j, and ic is the discount rate. The supply resilience metric

minimizes the average discounted maximum duration of a failure across all periods within the planning time horizon across all scenarios in the ensemble.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒_{𝐴𝑣𝑒𝑟𝑎𝑔𝑒}[𝑓_𝐸𝐶𝑂= max

𝑇 {[(|𝐴𝑁𝑄70− 𝐴𝑆𝑄70| 𝐴𝑁⁄ 𝑄70)𝑗∗ 100%] ∗ (1 + 𝑖𝐶) −𝑗_}]

5-16

where ANQ70 is the area under the naturalized flow duration curve (FDC), ASQ70 is the

135 minimizes the average discounted worst deficit across all periods that occurred in the planning time horizon across all scenarios in the ensemble.

This updated problem formulation ensures that the most flexible plans within the four Pareto optimal sets implement the ALC and Meters within the first decade of the

planning time horizon. The optimal schedules from the four optimizations using the four bootstrapped scenario ensembles shown in Figure 5-7 are more similar than when undiscounted worst 5 year engineering and environmental performance was considered (Figure 5-6). In particular, ALC (orange bars), Mains (light red bars), Meters (brown bars), and OCT (light green bars) are mostly implemented in 2020, ESD (yellow bars) in 2040, and LRD (green bars) in 2045 with small degree of variation between the rest of the considered interventions.

Figure 5-7. Schedules of interventions within the Pareto optimal plans obtained using the bootstrapped scenario ensembles and average of the discounted worst 5 year engineering and environmental performance. The bars and axes represent the same dimensions as shown in Figure 5-3, Figure 5-5, and Figure 5-6.