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Mathematical optimisation of strategies for the
realisation of sustainable urban water
management
Optimisation math?matique des strat?gies pour la r?alisation de la
gestion durable des eaux urbaines
Kaufmann I.*; Schmitt T.G.*, Meyer T.**, Kalsch M.**, Hamacher
H.W.**
* Institute of Urban Water Management, University of Kaiserslautern
Paul-Ehrlich-Stra?e 14, 67663 Kaiserslautern, Germany
ikaufman@rhrk.uni-kl.de
** Department of Mathematics, University of Kaiserslautern
Paul-Ehrlich-Stra?e 14, 67663 Kaiserslautern, Germany
kalsch@mathematik.uni-kl.de
RESUME
Actuellement, le drainage et l?approvisionnement en eau urbains dans les pays d?ve-
lopp?s sont domin?s par des syst?mes centralis?s qui ne sont incontestablement pas
conformes aux conditions durables. Si les technologies pour une gestion des eaux de
pluies naturelle ou un assainissement d?centralis? remplacent au moins partiellement
les syst?mes existants les travaux de reconstruction intensifs seront essentiels.
L?expos? pr?sente le d?veloppement et l?impl?mentation pratique d?un outil math?ma-
tique afin d??laborer une strat?gie optimis?e pour la r?alisation de concepts de drai-
nage et d?assainissement alternatifs et plus d?centralis?s dans les zones urbaines
existantes. La succession des mesures de construction pour la totalit? de la p?riode
consid?r?e (environ 50 ? 100 ans) a ?t? d?termin?e sur base d?un mod?le d?opti-
misation math?matique ? condition que le futur ?tat favoris? soit connu. Le mod?le
d?crit les interd?pendances complexes du cycle urbain de l?eau et permet la minimi-
sation des efforts financiers et des impacts ?cologiques sur le chemin vers l??tat futur.
ABSTRACT
Urban drainage and water supply in developed countries are at present dominated by
centralised systems, which unquestionably do not comply with sustainable require-
ments. If technologies for a natural stormwater management or decentralised sanita-
tion should at least partially replace existing systems, intensive reconstruction work
becomes essential. This paper presents the development and practical implementa-
tion of a mathematical tool to find an optimised strategy for the realisation of alterna-
tive and more decentralised drainage and sanitation concepts in existing urban areas.
The succession of construction measures for the whole period of consideration (about
50 to 100 years) is determined based upon a mathematical optimisation model on
condition that the favoured future state is known. The model describes the complex
interdependencies of the urban water cycle and enables the minimisation of both fi-
nancial efforts and ecological impacts on the way towards the future state.
KEYWORDS
Change in urban drainage, cost-benefit consideration, mathematical modelling,
optimisation of strategies, sustainable urban water management.
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1 INTRODUCTION
1.1 Background
Urban drainage and water supply in industrialised countries are at present dominated
by centralised systems and structures. As these concepts do unquestionably not
comply with sustainable requirements, alternative concepts of sustainable drainage
and sanitation have become more and more significant in recent years. To revise
relying on conventional piped drainage systems in stormwater management has been
a recent trend. Developed areas are drained in a more natural way, using the
infiltration and storage capacities of semi-natural devices such as infiltration trenches,
swales and ponds (Butler and Davies, 2004) ? Sustainable Urban Drainage Systems
(SUDS). In Germany, in most federal states an obligation to implement SUDS exits
for development areas. In existing areas the realisation of SUDS devices will be more
difficult. A disconnection of 10 to 15 % of paved areas from the piped systems is
regarded as realistic.
In recent years alternatives for the disposal of foul water are also discussed (e.g. Hi-
essl, 2005; Starkl et al., 2004). Concepts of decentralised sanitation and reuse (DE-
SAR) should close urban water and nutrient cycles and conserve water resources
(Lens et al., 2001). Devices for water reuse and decentralised treatment of sanitary
wastewater are investigated in numerous field studies. The realisation of such
concepts in existing areas would cause extensive financial and constructional efforts
and would be more difficult ? particularly because of residents? acceptance ? than the
implementation of sustainable stormwater management.
1.2 Open questions and aim of the study
Decision support approaches to the selection of sustainable drainage systems or
DESAR are investigated in some studies (e.g. Ellis et al., 2006, Huang et al., 2004).
But to find strategies how the future state can be reached in an optimal way is not
investigated so far. If devices of SUDS and DESAR should at least partially replace
existing centralised systems, intensive reconstruction work becomes essential. The
conceptual change is superposed by a high demand for rehabilitation in water supply
and sewer networks. An open question is how these two requests can be reconciled
under economical and ecological aspects. At all stages of a transition a reliable water
supply and disposal of wastewater have to be guaranteed. A conversion can only be
realised successively over a long period due to high constructional and financial
expenses and requires new strategies for ?hot plug-in?. ?Manually? the optimal strategy
to attain the favoured future state is difficult to develop. Therefore, an urgent need for
research exits for a strategy for the implementation of decentralised and sustainable
drainage und sanitation devices in existing urban areas on condition that the favoured
future state is known. The paper will present the development and implementation of
a mathematical optimisation tool to determine the succession of the construction of
devices under minimal ecological impacts and economical efforts.
2 MATERIAL AND METHODS
2.1 Mathematical Approach
2.1.1 Optimisation problem
As financial efforts (economical costs) as well as ecological impacts (ecological costs)
should be minimised on the way to more sustainable systems, the mentioned problem
belongs to the field of multi-criteria optimisation. Essential constraints of the
optimisation problem are to ensure the proper functioning of waste water disposal and
to meet standards in regulations at any time. Therefore such feasible strategies of
SESSION 8.1
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reconstructing measures should be found, which could not be enhanced in both
criteria (economical end ecological costs). Generally, not only one solution of the op-
timisation problem exits but numerous reasonable Pareto-optimal solutions (see e.g.
Ehrgott, 2005). Only the subjective weighting of the different criterions or the discus-
sion of local deciders can lead to the definite choice of solution, namely the strategy
of conversion that should be applied.
2.1.2 Design of the model
In a first step the functional progression within the optimising procedure was defined
and is shown in Figure 1. The mathematical modelling is based upon the scale of
subcatchments and simplified networks of drainage elements (functioning network).
The subcatchments are connected due to flow directions and all interrelationships of
the main elements are represented. This allows besides the temporal succession of
appropriate measures a spatial consideration.
succession
balances
feasibility
economical costs
present state
measures
impact
evaluation
analysis
installation
period
functioning
standards in
regulations
environment
ecological costs
useful life -span
future state
requirements
constraints
choice of
measures
minimi-
sation
optimised strategy for realisation
Figure 1. Scheme of mathematical optimisation model.
Based upon the boundary conditions of the present state and the favoured future
state potentially realisable measures are provided for each subcatchment by using an
own decision support tool (Schildw?chter, 2006). At this, devices for SUDS, DESAR
as well as drainage elements (sewers and surficial drains) are suggested depending
on numerous parameters, e.g. topography, subsurface conditions, land use, space
requirements or population density. For all measures investment costs and operating
costs as well as installation periods were calculated in all subcatchments. Further-
more information about impacts on ?flows?, discharge and pollution are linked to each
measure allowing a simplified balancing of volumes and loads in different flow types
and discharge paths (e.g. waterbodies, WWTP effluent, soil). The environmental im-
pact is estimated with ecological costs expressing negative as well as positive eco-
logical impacts. Within the optimising process the feasibility of the systems is verified
in each time step, which decisively affects the succession of conversation measures.
Due to the requirements of the favoured future state now expedient measures are in
such a way chosen, that on the one hand the hydraulic and legally allowed function-
ing of the systems is ensured at any time. On the other hand the succession of
measures should cause the minimal economical and ecological costs.
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2.2 Mathematical Modelling
The mathematical model itself is formulated as a bi-criteria mixed-integer program
(MIP). The structure was build as a complex network of ?nodes? and ?arcs?. All transfer
points of drainage systems (manholes, overflow structures, outfalls) as well as sur-
face types (area of roofs, streets, yards, unpaved area) and foul water components
were represented as nodes. Further nodes are the measures and devices and bal-
ancing nodes such as infiltration or evaporation. Arcs represent all possible connec-
tions between nodes and are characterised by capacities, costs and installation peri-
ods. All possible flow paths of dry and wet weather flow and their pollution are de-
scribed by that way. That means e.g. that the surface type roof is linked to possible
nodes as green roof, rainwater utilisation, infiltration, open drainage or stormwater
sewer. But starting from one surface type many more arcs are necessary to describe
the complex interdependencies (see Figure 2). The nodes-arc-network can easily
reach dimensions of about 100 nodes and 300 arcs for each subcatchment.
rainw.
utilisation
infiltration
swale
flat roof
comb.
sewer
stormw.
sewer
surface
drain
infiltration evaporation
use of
rainwater
balancing nodes
surface types
measures and devices
(selection)
drainage (transport )
nodes
comb.
sewer
stormw.
sewer
surface
drain
Construction of or existing
transport elements
balancing arcs
construction of devices
(arcs with costs, installation
periods , etc.)
connection arcs with
different capacities for
flows and pollution
drainage (transport )
nodes
green
roof
Figure 2: Model structure of nodes and arcs starting from one surface type.
Based on this structure a simultaneous project scheduling and network flow problem
is defined. The challenge and specific is on the one hand, that not all specified expe-
dient measures have to been chosen but just those measures should been selected,
which lead to a Pareto-optimal strategy. On the other hand the network for the sched-
uling and flow problem is time dependent, as with the construction of different devices
arcs are opened due to installation periods and closed when elements are replaced
by new devices. By the implementation of different variables and adequate con-
straints within the mathematical modelling procedure all paths of the network are
scant in order to find a feasible optimal solution under the consideration of economi-
cal and ecological costs, the objective functions of the model.
The economical costs at every time step are the result of investment costs (?) of de-
vices with beginning of construction in the regarded time step and operating costs
(?/year) of all installed measures. Furthermore rehabilitation and reinvestment costs
respectively (?) as well as costs for repairing damage caused by flooding (?/year) are
SESSION 8.1
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considered. Also a WWTP effluent charge (?/year) is balanced in a simplified way.
The economical costs for the whole period under consideration are calculated as total
project costs. The total project costs are the sum of money which is necessary for fi-
nancing all measures based on today?s cost level with a real interest rate of 3 percent.
The ecological costs are not accounted monetarily but by a number resulting from a
point system. Positive costs represent an environmental ?damage? whereas negative
numbers express a benefit. The costs are calculated ?on-line? by simplified methods.
The different criterions can be weighted individually. Table 1 lists the implemented
ecological parameters, but as a matter of course others are possible as well. A more
detailed description of the mathematical model can be seen in Kaufman et al. (2006).
impacts on criterions calculation of points per year
water cycle
distance from natural water cycle for
infiltration rate, evaporation rate, use of
rainwater
obliged value (%) - present value (%)
resources
distance from favoured resources pro-
tection for rate of reducing use of pota-
ble water, rate of water reuse, rate of
fertiliser production
obliged value (%) - present value (%)
emitted pollution loads in water bodies pollution load (kg/year) / 100
discharge rate for overflows (ATV, 1992) present rate - admissible rate emissions
dilution rate in overflow discharge (required rate - present rate) ? 2
Table 1: Evaluation of ecological costs.
3 IMPLEMENTATION OF THE MODEL
3.1 Catchment and boundary conditions
The model has been implemented for a suburb of Kaiserslautern in Germany, a rural
catchment of about 3,000 inhabitants. A section of the locality is shown in Figure 3.
I
II
III
VI
IV
V
CSO 2
CSO 1
sewers represented
in functioning network
Figure 3: Catchment for implementation of the model (section).
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In a detailed preprocessing 32 subcatchments were determined. For a first model im-
plementation they were merged to 6 subcatchments. The entire catchment has a
drainage area of about 90 ha and implies 35 ha of paved area. About 30 % are
drained by separate systems whereas the rest consists of combined sewer systems.
Two combined sewer overflow devises (CSOs, see Figure 3) and one final sewer
overflow tank (not included in section of Figure 3) are installed in the sewer system. A
business park in the south of the suburb has an area of about 20 ha and its effluent
shows the characteristics of domestic waste water. Dry weather flow amounts to
11.5 L/s and consist of 6.0 L/s foul sewage and 5.5 L/s infiltration water. The pollution
of dry weather flow is 560 mg COD/L. Within the optimisation model (so far) only the
parameter COD is implemented.
In this paper as an example of numerous potential future states it is determined that
stormwater runoff and wastewater should not be mixed any more and a natural
stormwater management should be achieved. The implementation of DESAR tech-
niques should lead to a decentralised treatment of black-water (faeces and urine)
whereas grey-water (all the wastewater produced in a house except by the WC)
should be treated centrally in the WWTP. Additionally obliged values for water bal-
ance and resources protection for the future state are chosen as follows (Table 2).
impacts on criterions obliged value in future state
water cycle
infiltration rate
evaporation rate
use of rainwater
30 %
55 %
10 %
resources
rate of reducing water consumption
rate of grey-water reuse
rate of fertiliser production
45 %
0 % (should be treated at WWTP)
25 %
Table 2: Obliged values for future state.
For this example of implementation two different specifications are made for the opti-
misation process. In a first specification (S 1) both the ecological and the economical
costs are equally weighted. In a second specification (S 2) the ecological costs are
secondary weighted (half of S 1) and the future state should primarily be reached with
minimal financial efforts.
3.2 Results and discussion
Figure 4 illustrates the succession of measures for the optimal strategy of transition
for a subcatchment. The beginning of construction as well as installation periods (grey
bars) and phases of rehabilitation (white bars) are shown for the two specifications.
S 1
S 2
S 1
S 2
S 1
S 2
S 1
S 2
S 1
S 2
S 1
S 2
S 1
S 2
S 1
S 2
0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55
period under consideration [years]
reduction of water consumpt.
combined sewer (used for
stormwater)
decentr. black-water treatment
foul water sewer
green roof
infiltration swale
rainwater utilisation
unpaving
Figure 4: succession of measures for a subcatchment (No. I) as result of mathematical modelling
SESSION 8.1
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At S 1 some devices are built earlier since negative ecological impacts are reduced
more efficiently right from the start of the consideration period. In this subcatchment in
both specifications the same devices are chosen to be implemented by the optimisa-
tion model, but the measure ?unpaving? is realised for a larger paved area in S 1.
The economical costs are demonstrated as annually costs (columns) and as summa-
tion curve for the period of consideration (lines) in Figure 5. Due to the real interest
rate in S 2 many devices are implemented rather late being ?cheaper?. As ecological
costs here are weighted lower there is no reason to build them earlier. All in all, in S 1
32.5 million ? and in S 2 25.0 million ? result as total costs. The higher weight on eco-
logical costs causes 23 % more economical costs in S 1 than in S 2.
0
1
2
3
4
5
6
7
0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55
period under consideration [years]
i
n
ve
st
m
e
n
t
a
n
d
o
p
e
r
a
t
i
o
n
a
l
co
st
s
[
m
ill
io
n
?
/
5
y
e
a
r
s
]
0
5
10
15
20
25
30
35
su
m
ec
o
n
o
m
i
ca
l
co
st
s
[
m
il
lio
n
?
]
econom. costs S 1
econom. costs S 2
sum S 1
sum S 2
Figure 5: Economical costs for S 1 and S 2.
Annually ecological costs are reduced more rapidly in S 1 and accumulate to 2,720
?points? whereas in S 2 3,390 ?points? result for the whole time. That means that the
strategy in S 2 causes nearly 25 % more negative impacts. Taking e.g. the emitted
COD loads at the stormwater tank into consideration in S 1 242 tons are emitted in
the 55 years under consideration and in S 2 286 tons. Similarly the earlier adjustment
to obliged values for water balance and resources protection is more distinctive.
As mentioned in chapter 2.1.1 you can see that more than one optimal solution exists.
The results of both scenarios are each an optimal strategy for the implementation of
more decentralised devices in regard to the favoured future state. The weighting of
both kinds of costs (economical and ecological) as well as the weight of the ecological
criterions among each other and of deficits influences the optimal solution. Only local
deciders can make the final decision for a strategy.
4 CONCLUSION AND OUTLOOK
Numerous alternatives for stormwater drainage are established in recent years
whereas the use of components of domestic wastewater as resources is under con-
sideration. This present change in exposure to wastewater causes intensive recon-
struction work for existing centralised drainage systems. To ensure that every step of
reconstruction ecologically and economically benefits the future an optimised strategy
for the transition of systems should be investigated. A first tool to find such strategies
was developed as a bi-criteria optimisation model and implemented for a rural area in
Germany. The mathematical approach necessitates many simplifications due to the
high complexity of interdependencies in the urban water and nutrient cycle. Neverthe-
less, the results are plausible and optimal strategies for the sequence of measures to
more sustainable systems under ecological and economical aspects can be found. At
the moment the enhancement of the solving process is in progress to achieve a more
detailed consideration.
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More reliable strategies could be developed if many more constraints, for instance the
detailed consideration of wastewater treatment processes and receiving water or
population development, are taken into account. The hitherto investigations have also
shown, that current guidelines and regulations could increase the price of or even in-
hibit favoured restructuring of drainage systems. In the meantime there will be states
where not all regulations could be fulfilled. Therefore standards should be adapted to
changing systems. Furthermore it is essential to define the requirements and condi-
tions of favoured future states, such as an obliged water balance or admissible emis-
sions. They also have an important influence on costs and impacts of reconstruction
measures.
The mathematical optimisation has been turned out to be an adequate instrument to
find strategies for the realisation of sustainable urban water management. The devel-
oped tool possibly will be a support for decision-making processes. The potential of
the approach will rise with the complexity of the specific application. For complex sys-
tems an optimal solution for transition to a favoured future state cannot be found
manually.
ACKNOWLEDGEMENT
The authors thank the ?Stiftung Rheinland-Pfalz f?r Innovation? (Foundation for Inno-
vation in Rhineland-Palatinate) for funding the project OptionS and the municipal
wastewater enterprise of the City of Kaiserslautern for providing data for the imple-
mentation case study.
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