Thames Water abstracts water from the lower reaches of the River Thames for the purpose of public water supply via a number of large reservoirs to the west of London. Transfers are also made to reservoirs in the Lea Valley found to the northeast of London. Left unconstrained, these abstractions could have a deleterious effect on the downstream environment. Accordingly, these are undertaken in agreement with the English Environment Agency (EA) environmental regulator, under Section 20 of the Water Resources Act 1991 [11
]. This agreement describes the Lower Thames Control Diagram (LTCD), which is used to control the level of abstraction permitted as a function of current reservoir storage. Thames Water seeks to optimise the LTCD, with a view to maximising the deployable output of the system as a whole. Deployable output is considered to be the maximum output capacity (i.e., demand that could be supplied) of one or more commissioned water sources that can achieve a prescribed level of service as constrained by factors such as, inter alia, hydrological yield, licence constraints and treatment and transport and pumping capacity.
In addition, a second optimisation scenario was evisaged in which aggregate could be extracted from an existing reservoir to facilitate additional storage capacity for the system. This was to be run as a separate analysis to determine what impact such a change would have on the deployable output of the system as a whole.
2.1. Lower Thames Control Diagram
The LTCD controls abstraction principally by defining a target environmental and navigational flow that must reach the tidal reaches of the Thames at Teddington Lock: the Teddington target flow (TTF). The TTF matrix is illustrated in Figure 1
where each month/operating band has a minimum flow target.
As can be seen, when reservoir storage is full, Thames Water are obligated to ensure that a minimum of 800 Ml/day is discharged into the tidal reach. This figure diminishes as reservoir storage becomes lower, with the constraints becoming more relaxed in the late spring and early summer months.
The solid lines on the LTCD represent the points at which the various demand-saving measures, agreed with the environmental (EA) and economic (Ofwat) regulators and outlined in the appropriate act and statutory instruments [11
], are implemented:
Level 1: intensive media campaign.
Level 2: sprinkler/unattended hosepipe ban and enhanced media campaign.
Level 3: temporary use ban, ordinary drought order (non-essential use ban).
Level 4: emergency drought order (e.g., standpipes and rota cuts).
In addition, the “crossing” of these demand-saving lines also triggers the implementation of further schemes, such as transfers of water from neighbouring water resource zones and the use of the Thames Gateway desalination plant at Beckton [13
For the purposes of this analysis, the existing LTCD [14
], which dates back to 1980 and was last updated in 1997, is considered to give a deployable output of 2285 Ml/day. The shape of the curves was derived by iteratively applying a water resource model over the historical draw-down record and adjusting the profiles to account for violations of the level of service constraints.
The deployable output (DO) is defined as being the maximum demand that the system can supply whilst meeting the terms of the level of service. The level of service criteria, measured over a time horizon of 100 years, agreed with the regulator for the system are:
Level 1 events should occur at a frequency of no more than 1 in 5 years.
Level 2 events should occur at a frequency of no more than 1 in 10 years.
Level 3 events should occur at a frequency of no more than 1 in 20 years.
Level 4 events are considered unacceptable and thus any solution must not allow such an event.
The permitted occurrence of Level 2 and Level 3 events is complicated by the impact of the Flood and Water Management Act 2010 [15
], which stipulates that there should be periods of 14 and 56 days of public consultation, respectively, in advance of these measures being implemented. Accordingly, it is required that 14 days elapse between a Level 1 and a Level 2 event starting, and 56 days between the start of Level 2 and Level 3 events. The existing LTCD DO of 2285 Ml/day did not consider these additional constraints. As at present, the lines defining the implementation of the demand-saving levels were to be considered coincident with the boundaries between the respective TTF bands.
Further constraints were agreed with the environmental regulator for the production of the new LTCD, which included ensuring that the boundary between the TTF800 and TTF600-700 bands (coloured blue and green, respectively, in Figure 1
) should be no higher than its current implementation. In addition, the definition of the Level 4 curve is changed to represent 30 days of storage at the prevailing DO and thus this line will represent a greater storage capacity for higher demand scenarios; the revised form of the Level 4 curve for the baseline scenario is shown as the horizontal line in Figure 2
The addition of each of these constraints increases the complexity of the problem to such an extent that it is difficult to imagine an efficient mechanism for deriving a workable solution, let alone a good one, without the use of optimisation tools.
2.4. Genetic Algorithm Optimisation
Genetic algorithms (GAs) are a powerful optimisation technique which can be applied to a wide variety of problems without any prerequisite knowledge of the problem domain. They perform a directed search of the decision space, which also contains a stochastic component, based on the “survival of the fittest” principle. The methodology takes advantage of the simulation model, i.e., Aquator in this case, ensuring that each potential solution is tested using a realistic representation of the water resource system being analysed. The most important advantage of GA over any other optimisation techniques is its flexibility in simulating different decision variables, objectives and constraints, due to the fact that any potential solution can be assessed directly in the model without the need for the derivation of specific mathematical properties (e.g., linearity) or expressions (e.g., derivatives), which present the main drawbacks to classic optimisation methods. A multiobjective GA [18
] that can easily handle multiple constraints was used as part of the AquatorGA software.
Two objectives were specified for the production of the new LTCD: to maximise the deployable output of the system and to minimise the complexity of the produced curves in order to make them acceptable to practitioners by reducing their jaggedness. Although this latter objective is, strictly speaking, not a genuine operational requirement, this objective was included as a result of the discussions with the client and consideration of the practicability of the implementation of the solution. Past applications of the AquatorGA software demonstrated that practitioners find it easier to relate to and explain control rules and curves when they are presented as smooth curves rather than more jagged ones, even though these may be perfectly valid solutions and represent mathematically “better” results.
The shape of the LTCD is represented by 48 decision variables representing the monthly values for each of the four profile curves, as shown in Figure 2
. Each variable was defined with a nominal precision of 1 decimal place and was permitted to vary between the level of the Level 4 line and the current boundary between the TTF800 and TTF600-700 bands. In order to accommodate the curve complexity objective, each of the 48 curve shape decision variables was coupled with a Boolean decision variable, which determined whether the point was considered as part of the curve or not. In this way, by “switching off” the curve points, the optimisation can easily simplify the shape of the curves.
One further decision variable was used to define the requested DO for the solution, hence the unusual situation where the DO was both an objective a decision variable. This approach was adopted because the total demand on the system was, along with the curve profile shapes, an input value submitted to the Aquator simulation model. The long run-times of the Aquator model meant that it was important that the number of infeasible solutions evaluated was minimised. To this end, once a feasible set of profiles had been identified, the DO decision variable was gradually increased in subsequent generations in order to determine the maximum valid DO for the combination of curve profiles specified. This was achieved by dynamically constraining the allowed range of the DO decision variable. If a feasible solution was subsequently found to have a valid higher DO, then the minimum value of the DO decision variable for this solution would be set to the new high DO. Similarly, if the evaluation of a higher DO proved not to be feasible, then the maximum value of the DO would be pegged to that higher value, so that higher values could no longer considered for that solution. Over time, the population of solutions gradually migrated to their true DO values. “Immature” solutions whose maximum DO has yet to be determined were protected from being removed from the population.
The combination of decision variables and constraints gives rise to a solution space consisting of 48 curve shape decisions, each of which can take on 1000 different values (0–100% at 1 decimal place = 100048 options) plus 48 boolean decisions (248 options) plus a single integral decision variable representing DO which is allowed to vary between 1800 and 2350 Ml/day (550 options), which gives 100048 × 248 × 550 = 1.5 × 10161 possible solutions to the problem. The use of a genetic algorithm allowed this huge space to be efficiently sampled and evaluated, using of the order of 120,000 solutions. Nevertheless, with each solution taking around 1 h to simulate on a high-specification PC (2015), it was necessary to employ some form of parallelisation in order to reduce the optimisation run-times to a manageable length. To this end, the AquatorGA software used in this optimisation included a distributed-processing system in order to militate against the extended run-times that are a common issue when optimising evolution algorithms applied to hydroinformatics problems. The software employs the industry standard message passing interface (MPI) protocol to execute many Aquator simulation models in parallel. This system permits the concurrent evaluation of a large number of potential solutions either on local processors or to other computers on a local area network.
For the purposes of this optimisation, the software was deployed across a cluster of five workstations, each equipped with two Intel Xeon E5645 CPU packages, which comprise six cores running at 2.4 GHz for a total of 60 processor cores. In addition, this hardware architecture can take advantage of hyper-threading technology, which improves the performance of identical threads running on multiple cores by around 10–20%. Accordingly, the run-time of the optimisation model was reduced, in total, from around 13 years to around 3 weeks when deployed to 120 virtual processor cores.