Objectives and constraints

The capital cost objective (fCapCost), as in Matrosov et al (2015), is the annualized capital

cost of implementing new supply and demand interventions in a portfolio normalized to each intervention’s expected design life. This is to provide equal comparison between interventions that have unequal design lives. For instance, it may be more practical to implement more expensive reservoir that remains functional for 80 years than less expensive desalination plant that would however need to be rebuilt after 25 years. The capital cost of each implemented intervention is therefore divided by its design life in years to assess how much it would cost per year if we assume its total capital cost requirement would be spread over the intervention’s life. The total annualized capital cost of a portfolio is minimized:

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓𝐶𝑎𝑝𝐶𝑜𝑠𝑡 = ∑[(𝐶𝑎𝑝𝐶𝑜𝑠𝑡𝑖⁄𝐷𝐿𝑖) ∗ 𝑌𝑖] 𝑖

3-2

where CapCosti is the capital cost of implementing intervention i and the DLi is the

design life of intervention i.

The supply deficit (fSupDef) objective represents the maximum annual deficit [%]

experienced by the London demand and is minimized: 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓𝑆𝑢𝑝𝐷𝑒𝑓 = max_𝑡 [((𝐷𝑇𝑡− 𝑀𝑡) 𝐷𝑇⁄ 𝑡) ∗ 100%]

75 where DTt is the London’s demand target for year t and Mt is the demand met during

year t.

Resilience is defined by how quickly the system recovers from a failure (Moy et al., 1986). The supply resilience objective (fSupRes) is assessed on the LAS node and the

failure occurs when the LAS storage level drops below the LTCD Demand level 3 threshold and the non-essential use ban is brought into effect (Figure 1-12). The objective aims to minimize the maximum time period over the whole time horizon required to recover from the failure, which refers to a period during which the Demand level 3 restrictions are in place:

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓_{𝑆𝑢𝑝𝑅𝑒𝑠} = 𝑚𝑎𝑥𝐷

3-4

where D is the failure duration in weeks.

The supply reliability objective (fSupRel) is also assessed on the LAS node and aims to

minimize the frequency of failures [%] (Hashimoto et al., 1982). This is similar to temporal reliability (Kiritskiy and Menkel, 1952), The reliability therefore maximizes the proportion of time over the whole time horizon when the LAS level is above the LTCD Demand Level 3 threshold and the Demand level 3 restrictions are not brought into effect:

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑓𝑆𝑢𝑝𝑅𝑒𝑙 = (1 − (𝐹𝑆⁄ )) ∗ 100% 𝑆

3-5

where Fs is the number of time-steps (weeks) during which the system was in failure,

and S is the total number of time-steps within the modeling time horizon.

The eco-deficit objective (fECO) (Vogel et al., 2007) represents the difference between

the naturalized low flows and simulated low flows [%] (low flows here denote the flows under Q70, i.e., flows that are not exceeded 70% of the record time) at the Teddington Weir on the River Thames. The naturalized flows here refer to the river flow where there are no TWUL’s abstractions; the objective therefore assesses direct impact of TWUL’s abstractions and return flows on the river itself. The higher the difference (i.e., deficit), the more the environmental conditions of the river deteriorate due to lower water levels than the natural state. Eco-deficit of 0% implies no deficit while 100% eco- deficit is the largest possible deficit:

76

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓_𝐸𝑐𝑜 = (|𝐴𝑁_𝑄70− 𝐴𝑆_𝑄70| 𝐴𝑁⁄ _𝑄70) ∗ 100%

3-6

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

the area under the simulated FDC. A flow duration curve (FDC) is a graphical

representation of the overall variation of a streamflow, usually showing the probability of exceedance on the horizontal axis and the magnitude of flow on the vertical axis. FDCs provide an estimate of the percentage of time of the considered record during which the flow exceeds a particular magnitude.

The energy objective (fEnergy) quantifies the cost of the average annual energy use of the

whole supply system including the existing and implemented possible supply interventions: 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓_{𝐸𝑛𝑒𝑟𝑔𝑦}= ((1 𝑇⁄ ) ∗ ∑ ∑ 𝐸𝑖,𝑡 𝑖 𝑇 𝑡=1 ) ∗ 𝑈𝑃 3-7

where Ei,t is the energy requirement to operate the supply intervention i over each year t,

T is the total number of years and UP represents the unit price of 1 kWh. The Ei,t is

based on the release of the particular supply intervention during year t: 𝐸𝑖,𝑡 = 𝑅𝑖,𝑡∗ 𝐸𝑅𝑖

3-8

where Ri,t is the release of the supply intervention i during year t (ML) and ERi is the

energy requirement for a mega liter of intervention’s i release (kWh/ML).

The constraints ensure satisfactory reliability of the aggregate surface storage (assessed on LAS) that complies with the TWUL specified LoS (Figure 1-12) and are based on the occurrence reliability definition (Kiritskiy and Menkel, 1952):

𝑐𝑘 = [1 − (𝐹𝑘⁄ )] ∗ 100% 𝑇

3-9

where k denotes a particular LoS level, Fk is the number of years during which the

storage volume dropped below the LoS level k. The constraints limit how often storage volumes drop below the LTCD demand levels to the maximum frequency of occurrence specified by TWUL’s Levels of Service (Figure 1-12).

77 The algorithm implements a constraint based tournament operator where feasible

solutions are always preferred to infeasible solutions. In general, simulations that do not meet these constraints are considered infeasible and are not passed into the archive of the MOEA. However, if all solutions are infeasible, the constrained tournament selection promotes solutions with the smallest aggregate constraint violations (Deb et al., 2002).