1
Application of Predictive Control Strategies to the
Management of Complex Networks in the Urban
Water Cycle
Carlos Ocampo-Martinez
∗
, Vicenc¸ Puig, Gabriela Cembrano, and Joseba Quevedo
The management of the urban water cycle (UWC) is a subject of increasing interest because of its
social, economic and environmental impact. The most important issues include the sustainable use of
limited resources and the reliability of service to consumers with adequate quality and pressure levels, as
well as the urban drainage management to prevent flooding and polluting discharges to the environment.
Climate change is expected to produce regional changes in water availability in the 21st century.
For example, Northern and Southern Europe are expected to experience, respectively, an increase and
a decrease in mean precipitation, as well as an increase in the magnitude and frequency of extreme events
[1]. These changes will have direct consequences through impacts on the availability and quality of water
in the water cycle. Optimal management strategies for the systems in the water cycle can contribute to
reduce the vulnerability of urban water systems (UWS) to climatic variability and change.
An UWC is mainly comprised of the following systems:
(i) Supply/production: water supply from superficial or underground sources and treatment to achieve
necessary quality levels,
(ii) transport networks, which use natural or artificial open-flow channels and/or pressurized conduits to
deliver water from the treatment plants to the consumer areas,
(iii) water distribution to consumers, involving pressurized pipeline networks, storage tanks, booster
pumps and pressure/flow control valves,
(iv) urban drainage and sewer systems carrying waste- and rain water together to wastewater treatment
plants (WWTP), before returning it to the receiving environment.
In urban environments, drinking water is provided by means of a drinking water network (DWN) to
consumers and industry, and sanitation/urban drainage is achieved through a sewer network (SN). In a
large number of cities, DWNs are managed using telemetry and telecontrol systems which provide, in real
time, pressure, flow, quality and other measurements at several key locations within the network. Flow,
pressure and storage control elements are operated from a central dispatch in a centralized or decentralized
scheme.
In some cases, advanced urban drainage systems also include sewage control infrastructure, such as
detention tanks, pumps, gates and weirs. All these elements are monitored and controlled by using
telemetry/telecontrol systems, which involve rain-gauge networks, wastewater level and/or flow meters
in the sewers and actuators at the valves, pumps and weirs, a communication network, and monitory
and control software. The control system manages the flows and the storage in the network in order to
minimize the risk of untreated water overflows to the streets or to the receiving environment.
The use of optimal control for managing water systems to achieve energy efficiency, cost minimization
and environmental protection is summarized in this article. Applying optimal control concepts to water
systems requires the development of control-oriented dynamic models to represent open-channel systems
(such as rivers, canals, aqueducts or SNs), pressurized pipes or combinations of both, which have nonlinear
responses to control actions, such as changing modes at different operating points. Those systems also
contain storage and control elements, such as tanks and valves, with a pre-determined operational range,
∗
Corresponding Author: Technical University of Catalonia (UPC), Institut de Rob`otica i Inform`atica Industrial (CSIC-UPC), Llorens
i Artigas, 4-6, 08028 Barcelona (Spain), cocampo@iri.upc.edu
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which leads to the inclusion of physical constraints in the model. Additionally, some on-off elements such
as pumps or valves may exist.
The management of UWS must be carried out predictively. Control actions must be computed ahead
in time, with an appropriate time horizon, based on real measurements and on state estimation, as well
as predictions of the stochastic variables involved in the models such as consumer demands in drinking
water systems or rain intensities in urban drainage systems. For water distribution networks, the prediction
horizon is usually of 24 hours. Longer horizons are chosen for water supply and treatment management.
For real-time control of urban drainage systems, the horizons depend on the average sewage transport
time between the discharge points and the final collection/treatment/discharge points. Thus, the length of
those horizons results to be particular for each case-study application, mainly depending on topographic
and physical characteristics of the terrain and the sewers. Predictive and optimal control techniques are a
smart option to compute control strategies for these complex dynamic systems. In order to achieve certain
control goals, one or more optimization problems are posed using a cost function to represent control
goals and a set of constraints to take into account the system dynamics and physical and operating limits.
Predictive and optimal control techniques also allow the user to establish priorities among the different
control objectives, whenever these cannot be achieved simultaneously [2].
Over the past few years, Model Predictive Control (MPC) has proven to be one of the most effective
and accepted control strategies for large-scale complex systems [3], [4]. The objective of using this
technique for controlling UWS is to compute, in a predictive way, the manipulated inputs in order
to achieve the optimal performance of the network according to a given set of control objectives and
predefined performance indices. As shown in [5], [6], [7], [8], such controllers are suitable to be used in
the global/supervisory control of networks related to the urban water cycle. Figure 1 shows a conceptual
scheme for a hierarchical structure considered on the control of networks related to the UWC.
This article summarizes the real-time global optimal management of two systems of the UWC, both
of them located in Barcelona, Spain: its DWN – specifically the transport network – and a representative
portion of its SN. Real-time control (RTC) of both types of UWC systems has received special attention
during the last few years, due to the increasing demand for improved system performance to meet consumer
and regulatory needs, often at reduced cost [5], [11]. The main goal to be achieved in DWNs is to reduce
pumping costs – for instance, by filling tanks in low tariff periods – while maintaining adequate system
pressure to meet fluctuating consumer demands [12]. Similarly, in urban drainage management, the goals
are to minimize flooding and combined sewer overflow to the receiving environment (CSO) by controlling
flow within the wastewater system, through for example, inline storage [13] or using underground detention
tanks, gates and pumps [6], [14].
I. CONTROL-ORIENTED MODELING PRINCIPLES
Complex nonlinear models are very useful for off-line operations (for instance, calibration and simula-
tion). Detailed mathematical representations such as the Saint-Venant equations for describing the open-
flow behavior in SNs [15] or pressure-flow models for DWNs allow the simulation of those systems with
enough accuracy to observe specific phenomena, useful for design and investment planning. However, for
on-line computation purposes such as those related to global management, a simpler and control-oriented
model structure must be conveniently selected. This simplified model includes the following features:
(i) Representativeness of the main network dynamics: It must provide an evaluation of the main
representative hydrological/hydraulic variables of the network and their response to control actions
at the actuators.
(ii) Simplicity, expandability, flexibility and speed: It must use the simplest approach capable of achieving
the given purposes, allowing very easily to expand and/or modify the modeled portion of the network.
(iii) Amenability to on-line calibration and optimization: this modeling approach must be easily calibrated
on-line using data from the telemetry system and embedded in an optimization problem to achieve
the network management objectives.
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Fig. 1. Hierarchical structure for RTC system. Adapted from [9] and [7]. Here, the MPC, as the global control law, determines the references
(set points) for the local controllers placed at different elements of the networked system. These references are computed according to
measurements taken from sensors distributed around the network. The management level provides the MPC with its operational objectives,
which are reflected in the controller design as the performance indices to be enhanced, which can be either minimized or maximized,
depending on the case. Finally, water systems control requires the use of a supervisory system to monitor the performance of the different
control elements in the networks (sensors and actuators) and to take appropriate correcting actions in the case where a malfunction is detected,
to achieve a proper fault-tolerant control [10].
This section deals with the control-oriented model principles for the DWN and SN systems described in
this article. The distinction is done at the stage of component description while the structure of the model
from the merging of elements is described and discussed in a unified way for both systems, determining
then the correspondence of their variables with the common variables established in the control theory.
A. Drinking Water Networks (DWN)
Several modeling techniques dealing with DWNs have been presented in the literature; see, e.g., [16],
[15]. Here, a control-oriented modeling approach that considers a flow-model is outlined, which follows
the principles presented by the authors in [5], [6] and [17]. The extension to include the pressure-model
can be found in [16], [15], [18]. A DWN generally contains a set of pressurized pipes, water tanks at
different elevation, and a number of pumping stations and valves to manage water flows, pressure and
elevation in order to supply water to consumers.
The DWN model can be considered as composed of a set of constitutive elements, which are presented
and discussed below. Figure 2 shows, in a small example, the interconnection of typical constitutive
elements.
1) Tanks: Water tanks provide the entire DWN with the storage capacity of drinking water at appropriate
elevation levels to ensure adequate water pressure service to consumers. The mass balance expression
relating the stored volume v in the n-th tank can be written as the discrete-time difference equation
v
n
(k + 1) = v
n
(k) + ∆t
X
j
q
jn
in
(k) −
X
h
q
nh
out
(k)
!
, (1)
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Fig. 2. Example of a basic topology of a generic drinking water transport network. Notice the interaction of the main constitutive elements
shown here: sources supply water to the system by means of pumps or valves, depending of the nature of the particular source (superficial
or underground). Water is moved by using manipulated actuators in order to fill detention tanks and/or supply water to demands sectors.
where q
jn
in
(k) denotes the manipulated inflows from the j-th element to the n-th tank, and q
nh
out
(k) denotes
the manipulated outflows from the n-th tank to the h-th element (which includes the demand flows as
outflows). Moreover, ∆t corresponds with the sampling time and k the discrete-time instant. The physical
constraint related to the range of admissible storage volume in the n-th tank is expressed as
v
n
≤ v
n
(k) ≤
v
n
, for all k, (2)
where v
n
and
v
n
denote the minimum and the maximum admissible storage capacity, respectively. Notice
that v
n
might correspond with an empty tank; in practice this value can be set as nonzero in order to
maintain an emergency stored volume.
For simplicity, the dynamic behavior of these elements is described as a function of volume. However,
in most cases the measured variable is the tank water level (by using level sensors), which implies the
computation of volume taking into account the tank geometry.
2) Actuators: Two types of control actuators are considered: valves and pumps, or more precisely,
complex pumping stations. A pumping station generally contains a number of individual pumps with fixed
of variable speed. In practice, it is assumed that the flow through a pumping station is a continuous variable
in a range of feasible values. The manipulated flows through the actuators represent the manipulated
variables, denoted as q
u
. Both pumping stations and valves have lower and upper physical limits, which
are taken into account as system constraints. As in (2), they are expressed as
q
u
m
≤ q
u
m
(k) ≤
q
u
m
, for all k, (3)
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where q
u
m
and
q
u
m
denote the minimum and the maximum flow capacity of the m-th actuator, respectively.
Since this modeling is stated within a supervisory control framework, it is assumed that a local controller
is available, which ensures that the required flow through the actuator is obtained.
3) Nodes: These elements correspond to the network points where water flows are merged or split.
Thus, nodes represent mass balance relations, modeled as equality constraints related to inflows – from
other tanks through valves or pumps – and outflows, the latter being not only manipulated flows but also
demand flows. The expression of the mass balance in these elements can be written as
X
j
q
jr
in
(k) =
X
h
q
rh
out
(k), (4)
where q
jr
in
(k) denotes inflows from the j-th element to the r-th node, and q
rh
out
(k) denotes outflows from
the r-th node to the h-th element. From now on, node inflows and outflows will be denoted by q
in
and
q
out
, even if they are manipulated variables (denoted by q
u
).
4) Demand Sectors: A demand sector represents the water demand of the network users of a certain
physical area. It is considered as a measured disturbance of the system at a given time instant. The
demand can be anticipated by forecasting algorithms, which are integrated within the MPC closed-loop
architecture. For the cases of study in this paper, the algorithm proposed in [19] is considered. This
algorithm typically uses a two-level scheme composed of
(i) a time-series model to represent the daily aggregate flow values, and
(ii) a set of different daily flow demand patterns according to the day type to cater for different
consumption during the weekends and holidays periods. Every pattern consists of 24-hourly values
for each daily pattern.
The algorithm runs in parallel with the MPC algorithm. The daily series of hourly-flow predictions
are computed as a product of the daily aggregate flow value and the appropriate hourly demand pattern.
Regarding the daily demand forecast, its corresponding flow model is built on the basis of an ARIMA
time-series modeling approach described elsewhere [20]. Then, the structure of the daily flow model for
each demand sensor may be written as
y
p
(k) = −b
1
y(k − 1) − b
2
y(k − 2) − b
3
y(k − 3) − b
4
y(k − 4) − b
5
y(k − 5) − b
6
y(k − 6) − b
7
y(k − 7). (5)
where the parameters b
1
, . . . , b
7
are estimated based on historical data. The 1-hour flow model is based
on distributing the daily flow prediction provided by the time-series model in (5) using an hourly-flow
pattern that takes into account the daily/monthly variation as follows:
y
ph
(k + i) =
y
pat
(k, i)
24
X
j=1
y
pat
(k, j)
y
p
(k), i = 1, . . . , 24, (6)
where y
p
(k) is the predicted flow for the current day k using (5) and y
pat
(k) is the prediction provided by
the flow pattern with the flow pattern class day/month of the current day. Demand patterns are obtained
from statistical analysis.
B. Sewer Networks (SN)
Sewer are open canals. The Saint-Venant equations, based on physical principles of mass conservation
and energy, allow the accurate description of the open-canal flow in sewer pipes [15] and therefore also
allow to have a detailed nonlinear description of the system behavior. These partial-differential equations
constitutes a nonlinear system, which is in general solved by iterative numerical procedures. For an
arbitrary geometry of the sewer pipe, these equations may not have an analytic solution. Notice that
these equations describe the system behavior in high detail. However, this level of detail is not useful for
