Modelling and Decentralized Model Predictive Control of Drinking Water Networks 09-04

Modelling and Decentralized Model Predictive Control of Drinking Water Networks 09-04
IRI-TR-09-04
Modelling and Decentralized
Model Predictive Control
of Drinking Water Networks
The Barcelona Case Study
Valentina Fambrini∗
Carlos Ocampo-Martinez
Abstract
In this report, MPC strategies have been designed and tested for the global centralized and
decentralized control of drinking water networks. Test have been performed in order to highlight
the advantages of having a partition of a complex network in several subsystems. Despite
the possible suboptimal solution of the optimization probles from the global point of view,
the clear gain related to the computation times and loads has been demostrated by means of
the simulations and test developed here. The high correlation between system elements, i.e.,
the strong coupling of the network, makes impossible to have independient subsystems to be
controlled by using a set of decoupled MPC controllers. Moreover, the necessity of a hierarchy
scheme is discussed and interesting results are obtained from the mixture of techniques giving
rise to a control law sharing decentralized and hierachical features.
∗
Valentina Fambrini is with the Department of Information Engineering, University of Siena,
Via Roma 56, 53100 Siena, Italy.
Institut de Robòtica i Informàtica Industrial (IRI)
Consejo Superior de Investigaciones Cientı́ficas (CSIC)
Universitat Politècnica de Catalunya (UPC)
Llorens i Artigas 4-6, 08028, Barcelona
Spain
Tel (fax): +34 93 401 5750 (5751)
http://www-iri.upc.es
Corresponding author:
C. Ocampo-Martinez
tel: +34 93 401 5786
[email protected]
http://www-iri.upc.es/people/cocampo
© Copyright IRI, 2009
CONTENTS
1
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Drinking Water Systems . . . . . . . . . . . . . . . . . . .
1.3 Model Predictive Control . . . . . . . . . . . . . . . . . .
1.4 WIDE Project and Distributed Model Predictive Control
1.5 Outline of the Report . . . . . . . . . . . . . . . . . . . .
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and Main Concepts
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26
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5 Concluding Remarks
5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
43
44
A Parameters of the ABN Case Study
46
B Aggregated Barcelona Drinking Water Network Model Equations
48
2 Background
2.1 Drinking Water Networks: Description
2.2 Model Predictive Control . . . . . . .
2.2.1 MPC Strategy . . . . . . . . .
2.2.2 Basic MPC Formulation . . . .
2.3 Decentralized MPC . . . . . . . . . . .
2.3.1 Preliminaries . . . . . . . . . .
2.3.2 General Formulation . . . . . .
3 Mathematical Modelling and MPC in Water
3.1 Modelling of Drinking Water Networks . . . .
3.2 MPC Problem Formulation . . . . . . . . . .
3.2.1 General Considerations . . . . . . . .
3.2.2 Problem Formulation . . . . . . . . .
3.2.3 Control Objectives . . . . . . . . . . .
3.2.4 Optimal Control Problem Formulation
3.3 Small Example Demonstration . . . . . . . .
3.3.1 Case Study Description . . . . . . . .
3.3.2 Control Objectives . . . . . . . . . . .
3.3.3 Scenarios and Simulations . . . . . . .
3.3.4 Results . . . . . . . . . . . . . . . . .
Networks
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4 Barcelona Drinking Water Network: the Case
4.1 Case Study Description . . . . . . . . . . . . .
4.1.1 The Barcelona Drinking Water Network
4.1.2 Aggregated Case Study . . . . . . . . .
4.2 Scenarios . . . . . . . . . . . . . . . . . . . . .
4.3 Centralized MPC . . . . . . . . . . . . . . . .
4.4 Decentralized MPC . . . . . . . . . . . . . . . .
4.5 Simulation of Scenarios . . . . . . . . . . . . .
4.5.1 Comparison with the CMPC . . . . . .
4.5.2 Main Results . . . . . . . . . . . . . . .
Study
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2
CONTENTS
Section 1
1
Introduction
3
Introduction
1.1
Motivation
Water, essential element for the survival of all known forms of life, is an increasingly important
global problem because it is becoming a scarce resource. Water demand already exceeds supply
in many parts of the world, and as world population continues to rise at an unprecedented rate,
many more areas are expected to experience this imbalance in the near future. As the economy
of a country becomes richer, a larger percentage of its people tend to have access to drinking
water, that is water that is of sufficiently high quality so that it can be consumed or utilized
without risk of immediate or long term harm. Such water is commonly called potable water.
Access to drinking water is measured by the number of people who have a reasonable means of
getting an adequate amount of water that is safe for drinking, washing, and essential household
activities. Like the overpopulation and scarcity of water resources, another contributing factor
of poor access to drinking water is the climate changes, that could have significant impacts on
water resources.
According to the problem presented above, limited water supplies, conservation and sustainability policies, as well as the infrastructure complexity for meeting consumer demands with appropriate flow pressure and quality levels make water management a challenging control problem.
Decision support systems provide useful guidance for operators in complex networks, where best
action resources management are not intuitive [6]. Optimization and optimal control techniques
provide an important contribution to strategy computing in drinking water management. Similarly, the problems related to modelling and control of water supply and distribution have been
the object of important research efforts in the last few years [3].
This report focuses on studying an approach and implementation of predictive control in drinking
water networks. Water network systems, that carry drinking water from sources to consumers,
consist of pipes, pumps, valves and storage tanks, and so, because of the interconnectivity of
these element, the complexity of the control increases in the sense of its management.
1.2
Drinking Water Systems
Water networks are generally composed of a large number of interconnected pipes, reservoirs,
pumps, valves and other hydraulic elements which carry water to demand nodes from the supply areas, with specific pressure levels to provide a proper service to consumers. Water supply,
treatment, transport and distribution are often operated separately, by different authorities.
Planning and management of these subsystems have different goals and time-scales. Additionally, the hydraulics involved differ considerably from one to another, in particular, between
large, spatially-distributed open channel areas and pressurized water sections with distribution
to consumers. In many water systems, network operation is carried out based on heuristic approaches, operator judgement, etc., which may be very complex in large-scale interconnected
systems. Decision support systems, which are based on mathematical network and operation
models may efficiently contribute to the optimal management of water networks by computing
control strategies ahead in time, which optimize management goals. According to the discussion
presented earlier, drinking water systems present some specific characteristics which make them
especially challenging from the point of view of analysis and control. Water systems complex
dynamics are usually comprised of:
• Compositional subsystems containing both continuous-variable elements.
• Storage and actuator elements with operational constraints, which are operated within a
specific physical range.
4
Modelling and Decentralized MPC of Drinking Water Networks
Operational
objective
determination
Set−points
determination
(MPC))
Control trajectories
realization
(PID controllers)
Management Level
Global Control Level
demand
Adaptation
Local Control Level
Application
water network
measurements
(flows, pressures,
volumes, levels,
etc.)
Information
exchange
Water Network
Figure 1: Hierarchical structure for RTC of drinking water system. Adapted from [18]
• Distributed, large-scale architecture, since water systems may have hundreds or even thousands of sensors, actuators and local controllers.
All the features mentioned before should be taken into account not only in the topology design
of a drinking network but also in the definition of an adequate control strategy in order to fulfil
a set of given control objectives.
1.3
Model Predictive Control
The control strategies in water networks deal with the problem of generating control strategies
ahead of time, to guarantee a good service in the network, while achieving certain performance
goals, which may include one or more of the following, according to the needs of a specific
utility: minimization of supply and pumping costs, maximization of water quality, pressure
regulation for leak prevention, etc. Such methodologies have to handle multivariable models
(since there are many sensors and actuators within a water network), and can consider physical
constraints and non-linear behaviour. Thus, within the field of water network control, there
exists a suitable strategy which fits the particular issues of such system. This strategy is known
as Model Predictive Control (MPC), which more than a control technique, is a set of control
methodologies that use a mathematical model of a considered system to obtain a control signal
which minimizes a cost function related to selected indexes related to the system performance.
MPC is very flexible regarding its implementation and can be used over almost all systems since
it is set according to the model of the plant [5]. Moreover, MPC has some features to deal
with complex systems as water networks: big delays compensation, use of physical constraints,
relatively simple for people without deep knowledge of control, multivariable systems handling.
So, according to [18], such controllers are very suitable to be used in the global control of
networks related to the urban water cycle within a hierarchical control structure. This global
control structure is depicted in Figure 1, where the MPc determines the references for the local
Section 1
Introduction
5
controllers located on different elements of the network. The Management level is used to provide
to MPC the operational objective, which is reflected in the controller design as the performance
indexes to be minimized.
1.4
WIDE Project and Distributed Model Predictive Control
In water distribution networks, the high costs of transportation, storage, treatment and distribution limit the accessibility of water for a large portion of the world. Innovation is needed to
enable efficient resource operation of such distribution infrastructures in order to keep the natural
environment from being further depleted . The emerging wireless sensor network (WSN) technologies enable now the deployment of a pervasive plant-wide information gathering system that
permits coordination and optimization across all hierarchical layers of plant-wide management.
The challenges of integrating WSNs in plant-wide management are at the core of the recently
funded European Project WIDE, with aims of developing a framework for advanced control and
real-time optimization of large-scale and spatially distributed processes. Using wireless sensor
networks to gather information and validing the approach in a real water distribution system
combined with the issue of size and uncertainty of measurements, makes the control problem
hard to solve with classical control techniques. One of the best solutions is the application of
MPC as it allows for the taking into account a lot of information about the plant to tune the control law. Since MPC is substantially an optimization problem, whose size is directly proportional
to the plant size, large-scale systems are a hard challenge to deal with MPC. The reason lies in
the fact that the optimization algorithms are hard to parallelize. Increasing computation power
is often not enough. A possible solution to adopt an MPC approach in control of large-scale
systems is the distribution of the model, that is the decomposition of a plant into independent
subsystems. Those subsystems will be the model that MPC controllers will use to compute the
control action. This implies a much smaller solution time for the MPC problems and also the
possibility to parallelize the computations. This approach is referred to as Distributed MPC
(DMPC) in the literature [1]. Since the growth of the complexity of the MPC computation
is usually exponential with the growth of the number of variables, the gain of implementing
DMPC increases with the size of the problem which presented a trade-off between performance
and complexity, since DMPC solutions provide a decrease in closed-loop performance due to the
sub-optimal approach.
1.5
Outline of the Report
This report is organized in the following way:
Section 2: Background This section proposes the main idea and concepts about the different
topics considered in this report. The section presents the theory and the definitions about
drinking water networks, MPC strategy and DMPC approach.
Section 3: Mathematical Modelling and MPC in Water Networks Once the structure
and operational mode of water networks are introduced, a modelling methodology for control
design and analysis is required. This section introduces the modelling principles for water
networks. Moreover, this section deals with the formalization of the control problem with
equality and inequality constraints, known disturbances and multi-objective cost function related
to the optimization problem. The section ends with the description of the DMPC strategy
followed in this report, formalizing the related control problem.
6
Modelling and Decentralized MPC of Drinking Water Networks
Section 4: Barcelona Drinking Water Network Case Study This section presents and
describes in detail the case study of this report on which the control techniques and methodologies will be applied. The case study corresponds to a representative version of the water network
of the city of Barcelona. The particular mathematical model is obtained and calibrated using
real data from forecasts demands of water occurred in Barcelona during the year 2007. After
having described the case study, the section proposes the description of the implementation of
both the techniques MPC and DMPC. Simulations and final results are provided abundantly
for the two strategies. Moreover, a comparison between the two implementations is shown in
order to highlight the advantages and disadvantages of each approach.
Section 5: Concluding Remarks This section summarizes the contributions made in this
report and discusses the ways for future research.
Appendices The appendices contain the parameter tables, the model equations and the description of the functions used in the Matlabr implementation of the case study.
2
Background
This section introduces briefly the main topics of this report, describes in a general way the
concepts and the definitions of drinking water networks, and introduces a standard formulation
of predictive control. For this purpose, a state space setting is used–it is the most convenient for
multivariable problem, and it includes all other approaches. Moreover, within the framework
of the predictive control, the decentralized strategy is introduced and a general mathematical
formalization of this control problem is outlined.
2.1
Drinking Water Networks: Description and Main Concepts
The potable or drinking water comes from untreated water (also called raw water that may come
from groundwater sources or surface waters such as lakes, reservoirs, and rivers) that is usually
transported to a water treatment plant, where it is processed to produce treated water (also
known as potable or finished water). The degree to which the raw water is processed to achieve
potability depends on the characteristics of the raw water, relevant drinking water standards,
treatment processes used, and the characteristics of the distribution system. Customers of a
water supply system are easily identified - they are the reason that the system exists in the
first place. Homeowners, factories, hospitals, restaurants, golf courses, and thousands of other
types of customers depend on water systems to provide everything from safe drinking water to
irrigation. Customers and the nature in which they use water are the driving mechanism behind
how a water distribution system behaves. Water use can vary over time both in the long-term
(seasonally) and the short-term (daily), and over space. Good knowledge of how water use is
distributed across the system is critical to accurate modelling. Moving water from the source to
the customer requires a network of pipes, pumps, valves, and other hydraulic elements. Storing
water to accommodate fluctuations in demand due to varying rates of usage or fire protection
needs requires storage facilities such as tanks and reservoirs. Piping, storage, and the supporting
infrastructure are together referred to as the drinking water network.
Water networks are generally composed of a large number of interconnected pipes, reservoirs,
pumps, valves and other hydraulic elements which carry water to demand nodes from the supply
areas, with specific pressure levels to provide a good service to consumers. The hydraulic
elements in a network may be classified into two categories: active and passive [7]. The active
elements are those which can be operated to control the flow and/or the pressure of water in
specific parts of the network, such as pumps and valves; passive elements are tanks and pipes.
Section 2
Background
7
A set of these typical elements are described below. The figures presented are taken from the
Barcelona drinking water network, which is described in Section 4 as the case study of this
report.
Pipes In a drinking water system, a pipe is the element used to convey fluids from one location
to another, thus it is considered only as the connection between network pieces. In Figure 2,
the pipes of the Barcelona drinking water network are shown.
Figure 2: Drinking Water Network Pipes. Taken from [11]
Tanks Water tanks are water storage containers which accumulate water for human consumption. In this case, a water tank provides for the storage capabilities of drinking water. A tank
has physical limits that are related to the maximum and minimum capacity of storage water.
Tanks, as pipes, may be classified as passive elements of the network, as they receive the effects
of the operation of the other active elements, in term of pressure and flow, but they can not be
directly acted upon [7]. Figure 3 shows a Barcelona water network tank.
Figure 3: Barcelona Drinking Water Network Storage Device. Taken from [11]
8
Modelling and Decentralized MPC of Drinking Water Networks
Nodes These elements correspond to points where water flows are either merged or propagated. Propagation means that the node has one inflow and multiples outflows. On the other
hand, merging means that more than one inflow merges or splits to one or more than one outflow respectively. The nodes represent network connections and mass balance relations between
inflows and outflows.
Pumping Stations Pumping stations are needed to take the water that can not flow by
gravity. The pumps, presented into the drinking water network, are of two types:
• those that draw from the underground sources;
• those that are used to carry the water where there is an elevation difference between two
different elements.
The Figure 4 shows one of the pumping stations which belong to the Barcelona water drinking
network.
Figure 4: Drinking Water Network Pumping Station. Taken from [11]
Valves A valve is a device that regulates the flow of the water by opening, closing, or partially
obstructing various passageways. Valves are technically pipe fittings, but are usually discussed
as a separate category. They may be operated manually, either by using a hand wheel, lever or
pedal. Valves may also be automatic, driven by changes in pressure, temperature or flow. More
complex control systems using valves requiring automatic control based on an external input
(i.e., regulating flow through a pipe to a changing set point) require an actuator. An actuator
will stroke the valve depending on its input and set-up, allowing the valve to be positioned
accurately, and allowing control over a variety of requirements. In the Barcelona case study,
these elements are active within the network but, unlike the pumps, they are not able to drive
the water from a hydraulic element to another with different elevation. The valves can only let
the water pass through or not, and establish the strength of the flow. It is important to notice
that there is always a valve coupled to an exterior source to control the bheviour of the sources.
2.2
Model Predictive Control
Model Predictive Control (MPC) is one of the most advanced control methodologies which has
made a significant impact on industrial control engineering. The reason for this success can be
attributed to the fact that MPC is, perhaps, the most general way of posing the process control
problem in the time domain. MPC does not designate a specific control strategy but a very
Section 2
Background
9
ample range of control methods which make an explicit use of a model of the process to obtain
the control signal by minimizing an objective function. The MPC can handle multivariable
control problems, it can take into account actuator limitations and allows the operation within
constraints. The main ideas appearing in the predictive control are basically [14] [5]:
• Explicit use of a model to predict the process output at future time instants (horizon).
• Calculation of a control sequence to minimize an objective function.
• Receding strategy, so that at each instant the horizon is deplaced toward the future, which
involves the application of the first control signal of the sequence calculated at each step.
MPC presents a series of advantages over other methods, amongst which stand out that:
• It is possible to staff with only a limited knowledge of control because the concepts are
very intuitive and at the same time the tuning is relatively easy;
• it can be used to control a great variety of processes, from those with relatively simple
dynamics to other more complex ones;
• the multivariable case can easily be dealt with;
• it introduces feed forward control in a natural way to compensate measurable disturbances;
• its extension to the treatment of constraints is conceptually simple and these can be
systematically included during the design process;
• it is very useful when future references are known;
This control strategy has also its drawbacks. One of these is that although the resulting control
law is easy to implement and requires no significant computation load, its derivation is more
complex and when the constraints are considered, the amount of computation required is even
higher. Although this, with the computing power available today,it is not an essential problem
but depends of the case. Even so, the greatest drawback is the need for an appropriate model
of the process to be available. The design algorithm is based on a prior knowledge of the model
and the performance is related to the quality of the plant representation.
2.2.1
MPC Strategy
The methodology of all the controllers belonging to the MPC family is characterized by a set of
common elements, that are the following:
(i) Prediction model, which should capture all process dynamics and allows to predict the
future behaviour of the system.
(ii) Objective function, which is, in general form, the mathematical expression that penalizes
deviations of the predicted controller outputs from a reference trajectory. The objective
function can take into account and penalize more than just one term and it allows multiobjective cases that represents performance indexes of the system studied.
(iii) Control signal computation.
(iv) Constraints, which can represent physical limit of the palnt as well as constraints on the
control signals, or manipulated variables, and on the outputs.
10
Modelling and Decentralized MPC of Drinking Water Networks
Reference
Trajectory
Past Inputs
and Outputs
Predicted
Outputs
Model
+
−
Future
Inputs
Optimizer
Cost
Function
Future Error
Constraints
Figure 5: Basic structure of MPC
The methodology of all the controllers belonging to the MPC family is based on iterative, finite
horizon optimization of a plant model. At the time k, the current plant state is sampled and
a cost minimizing control strategy is computed for a relatively short time in the future T time
steps. Specifically, an on-line calculation is used to explore state trajectories that emanate from
the current state and find a cost-minimizing control strategy until time k + T . Only the first
step of the control strategy is implemented, the plant state is sampled again and the calculation
is repeated starting from the now current state, yielding a new control and new predicted state
path. The prediction horizon keeps being shifted forward and for this reason MPC is also called
receding horizon control, and this approach is called receding horizon strategy [14]. In order to
implement this strategy, the basic structure shown in Figure 5 is used.
A model is used to predict the future plant outputs, based on past and current values and on the
proposed optimal future control actions. These actions are calculated by the optimizer taking
into account the cost function (where the future tracking error is considered) as well as the
constraints.
2.2.2
Basic MPC Formulation
In most of the cases shown in the research literature, the MPC formulation is expressed in state
space and presents a generic and simple representation of the strategy. The standard MPC
problem based on the linear discrete-time prediction model is considered [14]:
x(k + 1) = Ax(k) + Bu(k),
y(k) = Cx(k),
(1)
where x(k) ∈ Rn is the state vector and u(k) ∈ Rm is the vector of command variables at time
step k, and y(k) ∈ Rp is the vector of the measured output. Following the formalism of [14] for
the basic formulation of a predictive control, the cost function is assumed to be quadratic and
Section 2
Background
11
the constraints are in the form of linear inequality. Thus, the following optimization problem
has to be solved:
Hp −1
T
min V (k) = min x(k + Hp ) P x(k + Hp ) +
U
U
X
x(k + j)T Qx(k + j)+
(2a)
j=0
+ u(k + j)T Ru(k + j),
s.t
x(k + 1) = Ax(k) + Bu(k),
(2b)
x(0) = x(k),
(2c)
umin ≤ u(k) ≤ umax ,
k = 0, · · · , Hp − 1,
(2d)
where Hp is the prediction horizon, x(0) is the initial condition of the state vector, umin and
umax are known vectors defining the saturation constraints on inputs variables (operational
ranges), and “≤” denotes componentwise inequality. Problem (2) can be recast as a Quadratic
Programming (QP) problem, whose solution
U ∗ (k) , [u(k)∗T · · · u(k + Hp − 1)∗T ]T ∈ RHp m×1
(3)
is a sequence of optimal control inputs that generates an admissible state sequence. At each
sampling time k, Problem (2) is solved for the given measured (or estimated) current state x(k).
Only the first optimal move u(k)∗ of the optimal sequence U ∗ (k) is applied to the process,
uM P C (k) = u(k)∗ ,
(4)
the remaining optimal moves are discarded and the optimization is repeated at time k + 1.
2.3
2.3.1
Decentralized MPC
Preliminaries
Complex dynamical networks have attracted increasing attention from control scientists in recent
years [9]. Recently, much attention has been directed towards the study of control methodologies for large-scale systems that can be often characterized by a set of multiple interconnected
subsystems with constraints on information flows between them. The desired goal of structuring
a distributed information and decision framework for large-scale systems does not mesh with the
available centralized methodologies and procedures associated with classical and modern control
theory, thus providing impetus for a decentralized control scheme. The last two decades have
seen the widespread diffusion of MPC techniques, which are now recognized as a very useful approach to deal with control problems involving several inputs and outputs and under constraints
on such variables, as is typically the case in the process industry. However, centralized MPC
(CMPC) is largely viewed as impractical for control of large-scale systems due to the need of
converging all the measurements in one single location, where the optimization is solved, and
the computation time needed to solve the large optimization problem within a sampling step.
Decentralized MPC (DMPC) is a decomposition of a single CMPC problem into a set of M
subproblems, and each subproblem is assigned to a different MPC controller (a basic scheme of
this strategy is shown in Figure 6).
In decentralized control of multivariable systems, the achievement of a global control task is
obtained by the cooperation of many controllers, each one computing a subset of control commands individually under a possibly limited exchange of information with the other controllers
[1]. Compared to centralized schemes, decentralized control has the disadvantage of inevitably
leading to a loss of performance, however it has a twofold technological advantage:
12
Modelling and Decentralized MPC of Drinking Water Networks
Past inputs
and Outputs
Reference
Trajectory
Plant
+
Predicted
Outputs
−
Inputs
DMPC 1
DMPC 2
Error
DMPC M
Figure 6: Basic scheme of DMPC
(i ) no need for a high-performance central processing unit performing complex global control
algorithms that take into account the overall system dynamics, as it is replaced by several
simpler (and therefore cheaper) units;
(ii ) all process measurements (command variables) do not need to be conveyed to (issued from)
a single unit, therefore limiting the exchange of information between spatially distributed
components of the process.
More recently, the interest in decentralized control schemes was revived by the interest in increasingly complex distributed sensing and actuation system. The control of a multitude of
sensing and actuator devices is naturally tackled by decentralized control designs, conceived to
ensure both local self-properties and global coordination. From a theoretical control point of
view, a centralized approach with complete knowledge of the overall system has the potential
of providing significant properties like stability and optimality [16]; a similar advantage arises in
optimization while comparing a global solution to a collection of locally optimal solutions.
2.3.2
General Formulation
Several approaches to decentralized control design have been proposed in the literature. They
differ from each other in the assumption made on the kind of iteration between different components of the overall system, the model of information exchange between subsystems, and the
control design technique used for each subsystem. A very promising design approach to decentralized control was proposed in the context of MPC [12]. Motivated by describing problems of
networks of independently actuated systems, most contributions focused the attention on dynamically decoupled systems, i.e. dynamical systems decomposed into distinct subsystems that
can be independently actuated. For each subsystem, a distinct MPC controller computes the local control action based on the measurements (and predictions) of the states of its corresponding
subsystem and of its neighbours.
Section 2
Background
13
Decentralized Prediction Model Prediction is based on a linear-discrete time model of
each subsystem. Let the system to be controlled be described again by the process model
(1). Matrices A, B will have a certain number of zero components corresponding to partially
dynamically decoupled subsystems, or may even be block diagonal in case of total dynamical
decoupling (this is the case for instance of independent moving agents each one having its own
dynamics).
Let M be the number of decentralized control actions that we want to design, for example M = m
in the case that each individual actuator is governed by its own controller. For i = 1, · · · , M ,
x i ∈ Rni as the vector collecting a subset Ix i ⊆ 1 , · · · , n of the state components,

xi1


xi = WiT x =  ...  ∈ Rni ,
xini

(5)
where Wi ∈ Rn×ni collects the ni columns of the identity matrix of order n corresponding to
the indices in Ixi , and, similarly,

ui1


ui = ZiT u =  ...  ∈ Rmi ,
uimi

(6)
as the vector of input signals tackled by the i-th controller, where Zi ∈ Rm×mi collects mi
columns of identity matrix of order m corresponding to the set of indices Iui ⊆ 1, · · · , m. Note
that
WiT Wi = Ini , ZiT Zi = Imi , ∀i = 1, · · · , M.
(7)
By left-multiplying (1) by WiT , we obtain
xi (k + 1) = WiT x(k + 1) = WiT Ax(k) + WiT Bu(k).
(8)
An approximation of (1) is obtained by changing WiT A into WiT AWi WiT and WiT B into WiT BZi ZiT ,
therefore getting the new prediction model of reduced order
xi (k + 1) = Ai xi (k) + Bi ui (k),
(9)
where matrices Ai = WiT AWi ∈ Rni ×ni and Bi = WiT AZi ∈ Rmi ×mi are submatrices of the
original A and B matrices, respectively, describing in a possibly approximate way the evolution
of the states of i-th subsystem. Model (9) has a smaller size than the original process model (1).
The choice of the dimension ni , mi and of matrices Wi , Zi are a tuning knob of the proposed
decentralized procedure and should be inspired by the inspection of zero or negligible entries in
A, B (or in other words by physical insight on the process dynamics).
A controlled is designed for each set of moves ui according to the prediction model (9) and based
on feedback on xi , for all i = 1, · · · , M . Note that in general different states xi , xj and different
inputs ui , uj may share common components. In particular, to avoid ambiguities on the control
action to be provided to the process, is imposed that only a subset of input signals computed
by i-th controller is actually applied to the process, with the conditions that all actuators are
commanded and the condition that each actuator is commanded by only one controller. For the
sake of simplicity of notation, is assumed that M = m and that each controller i only controls
the i-th input signal.
14
Modelling and Decentralized MPC of Drinking Water Networks
Decentralized Optimal Control Problems
finite-time constrained optimal control problem
For all i = 1, · · · , M , consider the following
Hp −1
min Vi (k) = min
ui (0)
ui (0)
X
x(k + j)iT WiT QWi xi (k + j) + u(k + j)iT ZiT RZi u(k + j)i ,
s.t xi1 (k + 1) = Ai xi (k) + Bi ui (k),
i
(10a)
j=0
(10b)
i
(10c)
umin ≤ u(k) ≤ umax .
(10d)
x (0) =
WiT x(k)
i
= x (k),
At time k, each MPC controller MPC i measures (or estimates) the state xi (k) (usually corresponding to local and neighbouring states), solves Problem 10and obtains the optimal sequence
u(0)∗i = [u(0)∗i1 , · · · , u(0)∗ii , · · · , u(0)∗imi ]T ∈ Rmi . In the simplified case, M = m, only the
i-th sample of u(0)∗i
ui (k) = u∗ii
i
(11)
will determinate the i-th component ui (k) of the input vector actually implemented to the
process at time k. The inputs u(0)∗ij , j 6= i, j ∈ Iui to the neighbours are discarded, their only
rule is to provide a better prediction of the state trajectories x(k)i , and therefore a possibly
better performance of the overall system.
The collection of the optimal inputs of all the M MPC controllers,
uM P C (k) = [u(0)∗11 · · · u(0)∗ii · · · u(0)∗mm ]T ,
(12)
is the actual input of the model (1). The optimization problem (10) is repeated at time k + 1,
based on the new states xi (k + 1) = Wik x(k + 1), according to the usual receding horizon
control paradigm.
3
Mathematical Modelling and Model Predicitive Control in Water
Networks
Predictive control is model-based in the sense that it uses an explicit internal model to generate
predictions of future plant behaviour. Thus, in predictive control, and in general in the control
of dynamical system, great importance lies on the definition of the model for the considered
systems. This section proposes a mathematical description of a water system, the modelling
methodology and control strategy for MPC design. In the same way, the equations that describe
the dynamical behaviour are defined. The modelling methodology is inspired in previous works
reported in the literature where the study of complex networks related to the urban water cycle
were done [18], [17]. Moreover, this section proposes and describes an illustrative case study
based on the Barcelona drinking water system.
3.1
Modelling of Drinking Water Networks
Several modelling techniques have been presented in the literature that deal with drinking networks, see [3] [7] and [6]. The modelling approach used in this study is based on a flow-only
model, where only the control variables are required to compute the change in the state of the
networks produced by a control action and it follows the principles of the approach presented
in [2]. There, the model just considers the mass conservation law related to water flows, so the
equations that describe the system dynamics are linear. A further extension of the model would
Section 3
Mathematical Modelling and MPC in Water Networks
15
be, for instance, to include the non-linear relations between flow and pressure, for example.
However, this will lead to a non-linear model which is not considered in this report.
General Aspects In order to find the mathematical model that characterizes the network,
this section introduces the modelling principles for drinking water network used for the main
case study presented in this report.
• Tanks. A water tank provides the entire system with the storage capability of drinking
potable water. The state of water network is related to tank volumes so the maximum
and minimum capacity of the tanks are considered as model constraints. Moreover, tank
water volumes represent the output variable of the model.
• Actuators. There are two types of control actuators: pumps and valves. The manipulated
flows through the actuators represent the control input variables of the model. Both pumps
and valves have lower and upper bounds that are also model constraints. It is assumed that
there is a local controller that ensures that the flow required in the actuator is satisfied.
• Nodes. These elements correspond to the points where water flows are merged or divided within the network. Thus, the nodes represent mass balance relations and they are
modelled as equality constraints related to inflows and outflows of the nodes.
• Sectors of Consume. A sector of consume represents the water demand made by the
network users. It is considered as a known disturbance in the model since there exists a set
of profiles which forecast the behaviour of these demands characterized using a statistical
approach, see [8] and [13].
Model Equations A convenient description of the model of a water network could be obtained
by considering the set of flows through the actuator elements as the vector of control variables,
the set of reservoir volumes as a vector of observable state variables and, since the model is
used for predictive control, a set of flows of demands as a vector of demand forecasts, obtained
through appropriate prediction models (see Section 3.1).
• xi : volume associated to the i-th tank;
• ui : flow through the i-th actuator;
• di : water demand associated to the i-th sector of consume;
• ∆t: sampling time.
Using this description, the following elementary models are introduced:
Tank. The mass balance expression relating the stored volume, the manipulated inflows and
outflows and the demands can be written as the difference equation:


X
X
(13)
qout,j (k) ,
qin,i (k) −
xi (k + 1) = xi (k) + ∆t 
i
j
where qin,i (k) and qout,j (k) correspond to the i-th inflow and the j-th outflow, respectively
(Figure 7), given in m3 /s. The physical constraint related to the range of tank volume
capacities is expressed as:
xmin
≤ xi (k) ≤ xmax
,
(14)
i
i
where xmin
and xmax
denote the minimum and the maximum volume capacity, respectively,
i
i
3
given in m . As this constraint is physical, it is impossible to send more water to a tank
than it can store.
16
Modelling and Decentralized MPC of Drinking Water Networks
Figure 7: Scheme of a network storage deposit
Node. The expression of this element can be treated as a mass balance relation. Its equation is
simply an equality of the inflows (from other tanks through valves or pumps) and outflows,
represented not only by manipulated flows but also by demand flows. The static equation
that expresses the mass conservation, can be written as:
X
X
qin,i (k) =
qout,j (k)
(15)
i
j
where qin,i (k) and qout,j (k) are the the i-th inflow and the j-th outflow respectively, given
in m3 /s , (see Figure 8).
Figure 8: Scheme of a network node
3.2
3.2.1
MPC Problem Formulation
General Considerations
One of the most effective and accepted control strategies for complex system control problems
is MPC [14]. The objective to use the optimal/predictive control in the controller design of a
drinking water network is to calculate in a predictive way the best control inputs in order to
achieve the optimal performance of the network according to a given set of control objectives
and predefined performance indexes.
The predictive controller in drinking water networks is usually thought to occupy the middle
level of a hierarchical control structure where on the top, the states are estimates and the water
demand is predicted over the control horizon. This information is the input into the MPC problem while the outputs are the reference values for local controllers that implement the calculated
set-points. See [18] for reference where this hierarchical structure is followed.
Associated with the drinking water network control problem, there are more than one control
objectives, thus the optimization problem of the MPC controller has multiple objectives as
well. Like it is well outlined in [18], the most common approach to solve multi-objective control
problems is to form a scalar cost function, composed of a linearly weighted sum of expressions
associated with each objective. The priority and the importance that the objectives have is
reflected on the choice of the weights. Generally, the selection of weights is done by heuristic
Section 3
Mathematical Modelling and MPC in Water Networks
17
trial and error procedures involving a lot of numerical simulation. For this report the choice of
the weights for the multi-objective cost function has been based on [4] and on heuristic valuation.
3.2.2
Problem Formulation
The MPC model used in this study, given the element-based modelling discussed beforehand, is
a constrained one (mass balance relations) with measured disturbances (known demands).
MPC model and constraints The model used for this study has the following discrete time
representation in state space:
x(k + 1) = Ax(k) + Bu(k) + Bp d(k),
y(k) = Cx(k),
(16a)
(16b)
where x(k) ∈ Rn is the state vector corresponding to the water volumes of the tanks, u(k) ∈ Rm
represents the vector of manipulated flows through the actuators, d(k) ∈ Rp corresponds to the
vector of the known disturbances which, in this case, are the water demands, and y(k) ∈ Rn is
the vector of outputs and these measured output variables are the same of state variables. A,
B, Bp and C are the system matrices of suitable dimensions and k is the discrete time instant.
Since the demands are supposed to be known, d(k) is a known vector containing the system
measured disturbances. Therefore, (16a) can be rewritten as
x(k + 1) = Ax(k) + B̃ ũ(k),
(17)
where B̃ = [B Bp ] and ũ(k) = [u(k)T d(k)T ]T .
Following the considerations made in Section 3.1, system constraints are related to:
• Mass balance relations at the network nodes (relations between manipulated inputs and,
in some cases, measured disturbances). These are equality relations of the form
E1 ũ(k) = E2 .
(18)
• Bounds in system states and measured inputs expressed by the inequalities
umin 6 u(k) 6 umax ,
x
min
6 x(k) 6 x
max
,
(19a)
(19b)
where umin and umax are vectors containing the lower and upper limits of the actuator,
respectively.
Hence, using (16b), (17), (18) and (19), the constrained model of the system for MPC design
purposes is expressed as

x(k + 1) = Ax(k) + B̃ ũ(k),




 y(k) = Cx(k),
(20)
E1 ũ(k) = E2 ,

min 6 u(k) 6 umax ,


u

 min
x
6 x(k) 6 xmax .
18
Modelling and Decentralized MPC of Drinking Water Networks
Multi-objective Optimization Like it has been discussed beforehand, the control problem
associated to the MPC controller has more than one objective. A survey of multi-objective
optimization can be found in [15]. In general, such a problem can be formulated as:
min [f1 (z), f2 (z), · · · , fr (z)],
z∈Z
(21)
where z ∈ Z is a vector containing the optimization variables, Z ⊆ Rm is the admissible set of
optimization variables, fi are scalar-valued function of z, and r is the number of the objectives.
A common approach to solving multi-objective optimization problems is by scalarization, see [15]
and conversion of the problem into single-objective optimization problem with a scalar-valued
objective function. A common way to obtain a scalar objective function is to form a linearly
weighted sum of functions fi ,
r
X
Wi f i ,
(22)
i=1
where the priority of the objective is reflected by the weight Wi
3.2.3
Control Objectives
Optimal control in water network deals with the problem of generating flow control strategy
from the sources to the consumer areas in order to satisfy the demand of water, while optimizing
performance goals such as network safety volumes and flow control stability. Thus, the following
operational objectives should be satisfied by the MPC controller by order of priority:
(i) Water production and transport cost. The economical costs are associated to water production (chemical treatments and legal canons) and the electrical costs associated to the
water transport using pumps. Hence, each pump and water source has a given economical
factor depending on the water flow, energy consumption (electricity) and water quality
and availability. For this study, this control objective can be described by the expression
f1 = ||α1 u(k)||2Wα + ||α2 (k)u(k)||2Wα
1
2
= α1 u(k)T Wα1 u(k) + (α2 (k)u(k))T Wα2 (α2 (k)u(k)),
(23)
where α1 corresponds to a known vector related to the economical costs of the water
according to the selected source (treatment plant, dwell, etc.) and α2 (k) is a vector of
suitable dimensions associated to the economical cost of the flow through certain actuators
(pumps only) and their control effort (pumping). Notice the k-dependence of α2 since the
pumping effort has different values according to the moment of the day (electricity costs).
Weight matrices Wα1 and Wα2 penalize the control objective related to the economical
costs in the optimization process, and can have the same or different value according to
the case.
(ii) Security term. The satisfaction of water demands should be satisfied at any time instant.
This is guaranteed through the equality constraints of the water mass balances at demand
sectors. However, some infeasibility avoidance mechanisms should be introduced in the
management of the tank volumes such that this volume does not fall below a security
amount resulting in demands which can not be satisfied. This leads to the management
of the tank volumes above a certain security volume, which ensures that the demand flows
can always be supplied by the network. Preliminary approaches to this control objective
within a quadratic framework suggest an expression of the form
f2 = ||x(k) − β xmax ||2Wx
= (x(k) − β xmax )T Wx (x(k) − β xmax ),
(24)
Section 3
Mathematical Modelling and MPC in Water Networks
19
where β is a factor which determines the security volume to be considered for the control
law computation. Notice that this term induces to a regulation around this critical volume,
which is not completely realistic. Some approaches are currently being analyzed in order
to improve this aspect.
(iii) Stability of control actions. Pumps and valves should operate smoothly in order to avoid
big transients in the pressurized pipes that can lead to their damage. To obtain such
smoothing effect, the MPC controller includes in the objective function a third term that
penalizes control signal variation △u(k),
△u(k) = u(k) − u(k − 1).
(25)
This term additionally infers stability in the level/volume of the header deposits and to
the entire network. It can be expressed as
f3 = ||∆u||2Wu
= ∆u(k)T Wu ∆u(k),
(26)
where △u(k) are the changes of the inputs vector and Wu corresponds to the the weight
matrix of suitable dimensions.
Therefore, the cost function V (k) considering the aforementioned control objectives will be of
the form
V (k) =
HX
u −1
δu(k + i|k)T Wu △u(k + i|k)
i=0
Hp
+
X
(x(k + i|k) − βxmax )T Wx (x(k + i|k) − βxmax )
i=1
HX
u −1
+ α1
+
u(k + i|k)T Wα1 u(k + i|k)
i=0
Hu−1
X
(α2 (k + i|k)u(k + i|k))T Wα2 (α2 (k + i|k)u(k + i|k)),
(27)
i=0
where Hp and Hu correspond to the prediction and control horizons, respectively. Note that for
this study, the control horizon Hu and the prediction horizon Hp will have the same value.
3.2.4
Optimal Control Problem Formulation
In order to solve the predictive control problem, now it is described the approach followed to
compute the optimal value of the controller variables [14]. Considering the system model (16a),
the prediction model can be written in the following form:
X = σ + Λ U + Υ D,
with
(28)
20
Modelling and Decentralized MPC of Drinking Water Networks

x(k + 1|k)
x(k + 2|k)
..
.




X = 
 x(k + Hu |k)


..

.
x(k + Hp |k)

0
B
..
.
B
AB
..
.
0
0
..
.




Λ = 
 AHu −1 B AHu −2 B · · ·


..
..

.
.
···
H
H
p −1
p −2
A
B A
B ···

Bp
ABp
..
.
···
···





,









σ = 
 AHu

 ..
 .
AHp

0
0
..
.
···
···
B
..
.
···
PHp −Hu i
···
AB
i=0
0
Bp
..
.
0
0
..
.
···
···
A
A2
..
.
0
0
..
.




···
Υ = 
 AHu −1 Bp
·
·
·
B
0
p ···


..
..
.
.

.
.
· · · . . ..
AHp −1 Bp AHp −2 Bp · · · · · · Bp





 x(k),









,






U = 







,




u(k|k)
u(k + 1|k)
..
.
u(k + Hu − 1|k)
d(k|k)
d(k + 1|k)
..
.




D = 
 d(k + Hu − 1|k)


..

.
d(k + Hp − 1|k)



,






,




where X ∈ RnHp is the state vector over the prediction horizon, σ ∈ RnHp ×n is the matrix with
the past part of the system model, Λ ∈ RnHp ×mHu is the matrix associated to the control signal
and part of the future of the model, U ∈ RmHu is the manipulated inputs vector over the control
horizon, Υ ∈ RnHp ×pHp is the matrix associated to the disturbances and part of the future of
the model and D ∈ RpHp is the known vectors of disturbances (demands) over the prediction
horizon. Now considering the expressions (18) and (19), the constraints over the prediction
horizon can be formulated in the following way:
∆U (k)
U
X
E
= 0, F
≤ 0, G
≤ 0,
(29)
1
1
1
where ∆U is defined analogously to U and E, F , G are matrices of suitable dimensions to
represent inequality and equality constraints. The entire optimization problem is formulated
considering as controlled variable the slow rate of the manipulated inputs ∆U . So the expressions
(29) are rewritten in terms of ∆U . Let
E = [E1 , E2 ] ,
F = [F1 , f ] ,
G = [G1 , p] ,
(30)
where E1 , F1 , G1 , are matrices of suitable dimensions and E2 , f , p are the last vectors of the
respective matrices. According to (25):
U = Θ∆U + Πu(k − 1),
(31)
where Θ and Π are matrices of suitable dimensions to calculate the expression (25) over the
prediction horizon. Thus, the expression (28) in terms of ∆U is
X = ρ + (ΛΘ)∆U + ΥD,
(32)
Section 3
Mathematical Modelling and MPC in Water Networks
21
where
ρ = σ + ΛΠu(k − 1).
(33)
Considering (31) and (32), (29) finally has the following form:
F1 Θ
G1 ΛΘ
E1 Θ
∆U ≤
∆U =
−F1 Πu(k − 1) − f
−G1 (ρ + ΥD) − p
−E2 − E1 Πu(k − 1)
,
(34)
.
The same procedure described for the constraints is done for the cost function (27). All the
control objectives expressions are changed in terms of ∆U , and the final expression has the
following form:
Hp −1
X
V (k) =
∆U (k + i|k)T Φ∆U (k + i|k) + φT ∆U (k + i|k).
(35)
i=0
where Φ is the Hessian of the objective function and φ is a vector of suitable dimensions for the
linear term.
Considering (34) and (35), the problem to be solved will be a standard optimization problem
known as QP (Quadratic Programming) and hence will be convex [14]. Finally, the optimization
problem has the following form:
min V (k)
(36a)
△u
s.t
H △U = h
Ω △U ≤ ω
(36b)
where
H = [E1
Ω =
F1 Θ
G1 ΛΘ
Θ],
,
h = [−E2 − E1 Φu(k − 1)],
ω =
F1 Φu(k − 1) − f
G1 (ρ + ΥD) − p
.
(37)
Hence, the optimal set of future optimization variables1 is:
△U ∗ = [∆u∗1 (k|k) · · · ∆u∗1 (k + Hu − 1|k) · · · ∆u∗m (k|k) · · · ∆u∗m (k + Hu − 1)]T ,
being m the number of control inputs of the system.
3.3
Small Example Demonstration
In this section is presented a motivational example useful to understand the approach proposed
and to show its effectiveness. The particular drinking water system used as case study is a
small but significant example containing the representative elements of a water network system
(water sources, tanks, actuators —pumps and valves—, nodes and demands, see Figure 9) and
it is extracted from the Barcelona drinking water network.
1
Remember that only the part of this solution corresponding to the first step is used, in accordance with the
receding horizon strategy.
22
Modelling and Decentralized MPC of Drinking Water Networks
Figure 9: Small portion of the Barcelona drinking water network
3.3.1
Case Study Description
The network, as depicted in Figure 9, is composed by three tanks, six actuators (three pumps
and three valves), four sectors of consume and two nodes and it is described by the discrete-time
equation (16a), where




1 0 0
0 0 0 1 1 0
A =  0 1 0  , B = ∆t  0 0 0 0 0 1  ,
0 0 1
0 0 1 0 0 0


−1 0 0
0
Bp = ∆t  0 0 −1 0  .
0 0 0 −1
The system constraints according to Section 3.2.2 are:
(i) Physical range related to physical limits of the system variables, such as:
xmin
6 xi 6 xmax
, i = 1, 2, 3
i
i
0 6 uj 6 umax
, j = 1, · · · , 6
j
(ii) Mass Balance constraints in network nodes:
u1 (k) = u2 (k) + u3 (k) + u6 (k),
u2 (k) = u5 (k) + d2 (k).
Section 3
Mathematical Modelling and MPC in Water Networks
23
Tables 1, 2 and 3 describe the values of the parameter for the different elements considered in
this small case study.
Table 1: Tanks and their lower and upper bounds
Name
Model Variable
Min Volume [m3 ]
Max Volume [m3 ]
d125PAL
d110PAP
d54REL
x1
x2
x3
150
375
800
470
960
3100
Table 2: Valves and their lower and upper bounds
Name
Model Variable
Min flow [m3 /s]
Max flow [m3 /s]
VALMA
VALMA45
VALMA47
u1
u2
u3
0
0
0
1.297
0.05
0.12
Table 3: Pumps and their lower and upper bounds
3.3.2
Name
Model Variable
Min flow [m3 /s]
Max flow [m3 /s]
bMS
CPIV
CPII
u4
u5
u6
0
0
0
0.015
0.0317
0.022
Control Objectives
For simplicity, in the example presented in this section only the stability term is included since
this example has been considered in order to illustrate the modelling principles and the general
methodology proposed for in MPC control of water distribution networks. Thus, the cost function used in the optimization control problem has only the objective term (26) and its expression
is:
Hp−1
X
V (k) =
∆u(k + i|k)T Wu ∆u(k + i|k)
i=0
3.3.3
Scenarios and Simulations
The implementation of this example has been made using Matlabr and for the simulation of this
case study the quadprog2 solver has been used that attempts to solve the QP-problem associated
to the MPC controller (see the Matlabr help for further details on the solver quadprog).
2
quadprog
24
Modelling and Decentralized MPC of Drinking Water Networks
Profile sector of consume d1
0.014
0.013
3
demand forecasted [m /s]
0.012
0.011
0.01
0.009
0.008
0.007
0.006
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 10: Periodical profile of sector of consume d1
The scenario used has the prediction horizon Hp set as 24, which is equivalent to 24 hours (with
the sampling time ∆t = 3600s), to take into account the current operational need as well as
the transport delays between the supplies and the consumer sites. The scenario has a number
of iterations that emulate the system behaviour of two days (48 hours). The weights for the
optimization control problem have all the same value, set at one. The initial condition of the
tanks is chosen as the 80% of their maximum range (see Table 4), while the initial position of
the actuators is their lower bound (see Table 2 and 3).
Table 4: Initial condition of the tanks
Name
Model Variable
Initial Condition[m3 /h]
d125PAL
d110PAP
d54REL
x1
x2
x3
376
768
2480
The demands episode used in the example scenario is provided by AGBAR3 , the company that
controls the water distribution in Barcelona and refers to the year 2007. The demands have
a periodical profile like depicted in Figure 10, and the variables that represent the sectors of
consume are listed in the Table 5.
Table 5: Sectors of consume
3
Name
Model Variable
c125PAL
c70PAL
c110PAP
c10COR
d1
d2
d3
d4
AGBAR: SOCIEDAD GENERAL DE AGUAS DE BARCELONA, S.A.
Section 3
Mathematical Modelling and MPC in Water Networks
25
Trajectory tank x
1
500
450
MATLAB
min=150
max=470
volume level [m3/s]
400
350
300
250
200
150
100
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 11: Tank x1 trajectory
3.3.4
Results
Due to the demonstrative aim of this example, it has been used only the outlined scenario
to running the simulation. The optimal solutions are computed in 10.7080 seconds and the
computational time refer to Matlabr implementation running on a Intelr Pentiumr Core
Duo 2.4 Ghz machine.
The optimization problem solver finds the optimal solution, for all the iteration, that minimizes
the particular cost function, satisfying the mass balance equation at the network nodes, the
physical constraints and the volume balance. Let kf in be the total number of iterations made in
the simulation, hence the volume balance is defined as follows:
kf in
VT (0) +
X
kf in
Vin (i) = VT (kf in ) +
X
Vout (i),
(38)
i=1
i=1
where Vin is the volume of water taken from the sources during the simulation, VT (0) is the initial
volume of the tanks , Vout is the volume of water required by the sectors of consume during all
the simulation and VT (kf in ) is the final volume of the tanks. The results of the predictive control
application on the simplified network for the test scenario described beforehand are shown in
the following figures. Figure 11 shows a deposit storage device behaviour that evolves regularly,
as suit the operational needs with respect to the physical constraints. Figure 12 shows the
behaviour of two inputs, and they evolve without abrupt changes like it is requested by the
optimization problem. As the tanks, also the controlled actuators work within their range of
operability, satisfying the physical usage limits.
Trajectory actuator u
Trajectory actuator u6
1
0.025
0.025
0.02
0.02
MATLAB
min=0
max=1.297
0.01
0.005
0.01
0.005
0
0
−0.005
−0.005
−0.01
1
5
9
13
17
21
25
29
33
37
41
MATLAB
min=0
max=0.022
0.015
flow value [m3/s]
flow value [m3/s]
0.015
45
48
−0.01
1
5
9
13
time (samples)
17
21
25
29
33
37
41
45
48
time (samples)
(a) Flow behaviour of actuator u1
(b) Flow behaviour of actuator u6
Figure 12: Actuator flows trajectories
To validate some way the correct working of Matlabr simulation, the results are compared with
26
Modelling and Decentralized MPC of Drinking Water Networks
results provided form another well tested informatic tool, called Plio4 , which minimizes a similar
cost function and it is based on predictive control [10]. The results obtained from Plio simulation
are made with the same scenario, parameters and set-up of initialization as Matlabr . Figure 13
shows the behaviour of the volume of deposit x2 resulting from the manipulated flows provided
from the controlled actuators and the flow through the actuator u6 . The results are very similar
to the Plio ones.
PLIO vs MATLAB: trajectory tank x
PLIO vs MATLAB: trajectory actuator u
2
1000
0.02
900
0.015
800
volume level [m3]
flow value [m3/s]
6
0.025
MATLAB
PLIO
min=0
max=0.022
0.01
0.005
700
600
500
0
400
−0.005
−0.01
MATLAB
PLIO
min=375
max=960
1
5
9
13
17
21
25
29
33
37
41
45
300
48
1
5
9
13
17
21
time (samples)
25
29
33
37
41
45
48
time (samples)
(a) Flow behaviour of actuator u6
(b) Volume behaviour of tank x2
Figure 13: Comparison within the tool Plio (red) and Matlabr (blue)-based approach
The final interesting thing to observe is the behaviour of the cost function of both Plio and
Matlabr simulation. Figure 14 shows how the controllers work almost always in the same way.
PLIO vs MATLAB: cost function evaluation
−5
9
x 10
8
7
MATLAB
PLIO
min V(k)
6
5
4
3
2
1
0
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 14: Cost function behaviour of tool Plio (red) and Matlabr (blue)
4
Barcelona Drinking Water Network: the Case Study
This section proposes and describes a case study based on a real drinking water network derived
from the Barcelona water distribution system. Using such a case study, the MPC and DMPC
techniques, outlined and discussed in Section 2, are applied to the system and their associated
advantages and problems are discussed through the analysis of different situations related to the
system and the controller design.
4
Plio uses a GAM S r solver
Section 4
Barcelona Drinking Water Network: the Case Study
27
Figure 15: AGBAR Control Centre
4.1
4.1.1
Case Study Description
The Barcelona Drinking Water Network
The city of Barcelona has a drinking water network that covers a territorial extension of 425
km2 , with a total length of 4470 km. It supplies 237,7 hm3 of drinkable water to a population
of more than 2.8 million of inhabitants and it supplies water not only to Barcelona city but
also to the metropolitan area. The network is managed by the company Aguas de Barcelona
(AGBAR). Since 1976, the network has a centralized tele-control system, organized in a twolevel architecture. At the upper level, a supervisory control system installed in the control
centre of AGBAR (see Figure 15) is in charge to optimally control the whole network by taking
into account operational restrictions and consumer demands. This upper level provides the
set-points for the lower-level control system. On the other hand, the lower level optimizes
the pressure profile to minimize losses by leakage and provide sufficient pressure, e.g., for high
rise buildings. The whole control system responds to changes in network topology (ruptures),
typical daily/weekly profiles, as well as major changes in demand. The Barcelona water network
is comprised of 200 sectors with approximately 400 control points. At present, the Barcelona
information system receives, in real time, data from 200 control points, mainly through flow
meters and a few pressure sensors. Sensor measurements are sent to the operational database of
the telecontrol information system via telephone XTC network or GSM radio using the ModBus
communication protocol [11].
The Sources Currently, Barcelona drinking water supply comes from three main sources:
the water stored in the Sau and Susqueda reservoirs form the river Ter, that is carried to the
purification plant in Cardedeu and later supplied to Barcelona through the Trinitat water tanks;
the Llobregat River, whose waters are harnessed in the area close to Sant Joan Despı̀ to be later
distributed through the urban piping network; the subsurface flow of the Llobregat River, a
strategically important water reserve that can now be used (although in limited quantities in
order to prevent overexploitation) [11].
Balance between supply and demand indicates that security margins are already below the
desirable levels in Barcelona, like in all the Catalonia region, and is prone to a water deficit that
is likely to become significant. The main problems affecting water supply within the area of
28
Modelling and Decentralized MPC of Drinking Water Networks
Barcelona are related to the periodic droughts that deplete the water reserves from rivers and
ironically, despite all these concerns surrounding droughts, the flooding, that has also become
an increasing occurrence. Floods and droughts are a common characteristic of Mediterranean
water cycles in the Barcelona region. Due to the climatological and hydrological peculiarities
of Mediterranean ephemeral streams, the rivers suffer from extended periods of very low or
non-existent flow occasionally altered by much shorter and violent flooding episodes.
4.1.2
Aggregated Case Study
From the whole drinking water network of Barcelona, which has described beforehand, this
report considers an aggregated version of this model that is a completely representative version
of the entire network. Aggregated means that some sectors of the network are collected in a
unique part, hence some tanks are collected in a single representative tank and the respective
actuators in a single representative pump or valve. This operation has been made to simplify
the complexity of the model in order to have a more manageable but at the same time significant
system in which the control strategy of this study was apply. Hence, this representative network
is selected to be the case study of this report, where a calibrated and validated model of the
system, following the methodology explained in Section 3, is available as well as the demands
data. According to this approach, the equivalent system obtained is presented in Figure 16.
The Aggregated Barcelona drinking water network (ABN) is comprised of 17 tanks, 61 actuators
divided in 26 pumps and 35 valves. Among of the pumps, five (bMS, bCast, bPousCAST,
bPousE, bPousB) draw water from the underground sources and the rest are used to carry the
water where an elevation difference between two elements of the network exists. Moreover, the
network has 25 sectors of consume and 11 nodes. Both the demands episode and the calibration
set-up of the network are provided by AGBAR. The demands episode used for the simulation
of the ABN and the design of the strategy is based on forecasts demands data. All the physical
parameters and ranges of tanks and actuators used for the calibration of this case study are
reported in Appendix A.
4.2
Scenarios
Currently, there is only one demands episode available from AGBAR, and it is comprised of
the forecasts demands provided referred to the water consume of the year 2007. Using it, some
scenarios are considered by modifying the simulation and the controller parameters setup. These
parameters are the following:
(i) the security volume level, denoted with β, in the objective term (24) of the cost function.
The security levels considered are the following:
– the 80% of the maximum capacity of the tanks xmax , that is denoted as µ. This level
is purely illustrative to show the effectiveness of the MPC controller;
– the minimal tank volumes requested to satisfy the demands, except for the tanks
dPLANTA, d54REL and d10COR that are considered as sources because of their
strategical location in the network and so the volumes have to be kept as big as
possible. Let us call this set of values η. This second security level is more realistic
and actual, because it is required to keep the volumes of the tanks as low as possible,
satisfying at each step the demands. In this way the controller tries to satisfy the
consumer demands using the minimal quantity of water. Less water means less costs
and less usage of the actuators, that lead to less ruptures as well as less need of
electricity. Both of these aspects have relevant weight in an economical consideration.
All the specific security level values used in each scenario are listed in Table 6.
Section 4
Barcelona Drinking Water Network: the Case Study
29
Figure 16: Barcelona Drinking Water Network: Aggregated case
(ii) the weight matrices for the optimization control problem. Let (ω∆u , ωx ) be the couple of
weights associated to the weight matrices used for the term (26) and for the term (24),
respectively. The weight matrices are diagonal with the i-th diagonal element representing
the specific weigh of the associated variables. As the weights referred to the state variables
30
Modelling and Decentralized MPC of Drinking Water Networks
Table 6: Security volume levels used in the scenarios
Name
Model Variable
β = µ [m3 ]
β = η [m3 ]
d125PAL
d110PAP
d115CAST
d80GAVi80CAS85
dPLANTA
d54REL
d100FCE
d10COR
d200BLL
d130BAR
d176BARsud
d70BBEsud
d200ALT
d100BLLnord
d200BARnord
d101MIR
d120POM
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
356
768
3096
2600
11560
2480
52160
9396
5840
12800
828
78433
3392
30160
5840
3930
1428
172.40
399.83
256.01
693.10
14450
3100
18917
11745
1005.3
5449.3
333.266
26787
755.82
8113.5
1404.7
2070.5
160.54
or inputs are the same for each type of variables, instead of a matricial expression this
couple of weights matrices are denoted with a couple of numbers that represent the weights
used for the term (26) and the (24) respectively. According to this, in this case study are
used two couple of weights that are (1, 1) and (0.1, 1) . These particular values of the
weights are carefully selected, according to a previous study, based on trial and error
tuning procedure [4].
According to this, the case study has the following scenarios:
Scenario 1 : the security level is β = µ, the weights are (ω∆u , ωx ) = (1, 1);
Scenario 2 : the security level is β = µ, the weights are (ω∆u , ωx ) = (0.1, 1);
Scenario 3 : the security level is β = η, the weights are (ω∆u , ωx ) = (1, 1);
Scenario 4 : the security level is β = η, the weights are (ω∆u , ωx ) = (0.1, 1).
The duration of the simulations is selected as 48 samples or 48 hours (since ∆t=1h) as the demands have profiles that are periodic, so it is not necessary to have a large horizon of simulation.
The initial conditions of the state variables are chosen to be the 50% of their maximum range,
as reported in Table 7.
All the used scenarios have the control and prediction horizons selected as 24 samples, what
is equivalent to 24 hours, with a sampling time of 3600 seconds. This selection has been done
according to the heuristic knowledge of the AGBAR control center and field tests made in their
water network.
4.3
Centralized MPC
The related system model of ABN has 17 state variables corresponding to the volumes in the
17 tanks, 61 control inputs corresponding to the manipulated actuator flows and 25 measured
Section 4
Barcelona Drinking Water Network: the Case Study
31
Table 7: Initial condition of the ABN tanks
Name
Model Variable
X (′) [m3 ]
d125PAL
d110PAP
d115CAST
d80GAVi80CAS85
dPLANTA
d54REL
d100FCE
d10COR
d200BLL
d130BAR
d176BARsud
d70BBEsud
d200ALT
d100BLLnord
d200BARnord
d101MIR
d120POM
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
222
480
1935
1625
7225
1550
32600
5873
3650
8000
517
49021
2120
18850
3650
2456
892
disturbances corresponding to the measurements of the consumers demands. Moreover, the
model has 11 mass-balance constraints corresponding to the nodes and inequality constraints
due to the ranges of both inputs and state variables. According to the modelling methodology
proposed in Section 3, the tables in Appendix A summarize the description of the case study
variables as well as their ranges while, in the Appendix B, are reported the model and the
nodes equations. The implementation of CMPC controller on the ABN follows the modelling
formalization made in Section 3. For this case study, the control objective relative to the
economical cost is omitted, thus the cost function only take into account the terms (24) and
(26).
Simulations and Results Discussion
The implementation of the ABN model has been made using Matlabr . Because of the complexity
of the model, the simulation of this case of study took an elevated computation time, which
make impossible to simulate the optimization control problem with the solver quadprog, as it
has been done in the motivational example of Section 3. Hence, to solve the problem considering
the computational time issue, the simulations have required a more sophisticated solver, and in
this case study it is been used the Cplex5 solver under the Tomlabr6 platform. All the data
that will be shown in the next sections are referred to this Matlabr implementation running on
a Intelr CoreT M Duo 2.26 Ghz machine. For all the simulation the optimization problem
solver finds the optimal solution, satisfying at each step the requests of water, the mass balance
equations at the network nodes, the physical constraints and the volume balance (38).
5
Cplex is an optimization software package.
Tomlabr Optimization Environment is a powerful optimization platform for solving applied optimization
problems in Matlab. Tomlab/Cplex efficiently integrates the solver package Cplex with Matlab and Tomlab.
6
32
Modelling and Decentralized MPC of Drinking Water Networks
Input Trajectory u21
Input Trajectory u22
0.8
2.5
2
1.5
1
u21
0.5
u21max
0
u min
45 4821
1
5
9
13
17
21
25
29
33
37
41
flow value [m3/s]
3
flow value [m /s]
3
0.6
0.4
u22
0.2
u22max
0
1
5
9
13
time (samples)
Input Trajectory u23
25
29
33
37
41
u min
45 4822
Input Trajectory u24
4
3
2.5
flow value [m /s]
3
flow value [m /s]
21
time (samples)
3
2
1.5
1
u23
0.5
0
17
3
2
u24
1
u23max
1
5
9
13
17
21
25
29
33
time (samples)
37
41
u min
45 4823
u24max
0
1
5
9
13
17
21
25
29
33
37
41
u min
45 4824
time (samples)
Figure 17: Flow behaviour for actuators considering Scenario 1.
Results Scenario 1 and Scenario 2 The simulations of these scenarios are very useful to
better understand the strategy of the MPC controller and to validate the strategy on a more
complex network model. The duration of the simulations scenario has a CPU time on average
of 207 seconds. Figure 17 shows the optimal control strategy for four actuators with weights
setted to (1,1). As it can be seen, the actuators respect the range and the controller tries to
keep them as constant as possible.
The same thing happens even if the controller strategy has a lower weighted cost function for the
term (26). Figure 18 shows this behaviour for four actuators, that have a smooth flow evolution
flow except for the u48 that has a peak in the first iteration. The control strategy, moreover,
follows with the expected behaviour the volume references of the 80% of the maximum states
capacity in both the scenarios ( Figure 19 show this behaviour for the Scenario 2).
Hence, an improvement performance on the inputs has be observed weighting more the objective
of ∆U , according to the fact that the control strategy penalizes more the variation on the inputs,
as depicted in Figure 20, but it is not significant.
No important performance improvement with regard to the state variables is observed between
the two scenarios (see Figure 21).
According to this, there is no very significant improvement choosing one of the two weights on
the ∆U term.
Comparing Scenario 2 and Scenario 4 The duration of the simulations scenario has a
CPU time on average of 211 seconds. Figure 22 shows the evolution of the reservoir x17 in
Scenario 4 and can be observed that the evolution is what is expected from the controller
action, since no extreme condition are reached while satisfying all the demands. The simulations
of this Scenario 2 and Scenario 3 have a more relevant importance, due to its more realistic
objectives. Like the Scenario 1 and Scenario 2, no significant performance improvement
exists between the two different weights on the objective (26), but significant differences can be
seen comparing the scenarios with a different security level. According to this, without lost of
generality, from now on, the weights combination used will be (0.1, 1), making the comparison
between different security level.
Barcelona Drinking Water Network: the Case Study
Input Trajectory u45
−3
flow value [m3/s]
5
x 10
Input Trajectory u46
4
3
2
u45
1
0
u45max
1
33
1.5
5
9
13
17
21
25
29
33
37
41
flow value [m3/s]
Section 4
u min
45 4845
1
u46
0.5
u46max
0
1
5
9
13
time (samples)
Input Trajectory u47
25
29
33
37
41
u min
45 4846
Input Trajectory u48
0.025
0.6
0.4
u
47
0.2
u47max
1
5
9
13
17
21
25
29
33
37
41
flow value [m3/s]
3
flow value [m /s]
21
time (samples)
0.8
0
17
u47min
45 48
0.02
0.015
0.01
u
48
0.005
0
u48max
1
5
9
13
time (samples)
17
21
25
29
33
37
41
u48min
45 48
time (samples)
Figure 18: Flow behaviour for actuators considering Scenario 2.
Normalized Tanks Trajectories
1
Normalized Volum Level
0.9
0.8
0.7
0.6
W∆u = 0.1
0.5
Wx = 1
0.4
0.3
0.2
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 19: Normalized Tank Trajectories Scenario 2.
The comparison shows, according to the choice of the minimal level of security, the gain in
terms of quantity of water used to satisfy the demands in the Scenario 4 with respect to the
Scenario 2. In fact, keeping the volume capacities on the minimum level η allows a lower
request of water from the sources consequently a lower usage of the actuators than the security
level of Scenario 2 requires. Figures 23 and 24 show this gain on four actuators, while the
respective level of security is followed, as expected, for both the scenarios.
4.4
Decentralized MPC
The ABN case study of this report is a very complex network and, as it is a large-scale dynamically interconnected system with constraints, the complexity of on-line optimal control problem
34
Modelling and Decentralized MPC of Drinking Water Networks
Normalized Input Trajectory: u46
0.6
0.5
0.5
Normalized Flow
Normalized Flow
Normalized Input Trajectory: u45
0.51
0.49
0.48
0.47
0.4
0.3
0.2
W =1
W =1
∆u
0.46
1
5
9
13
17
21
25
29
33
37
∆u
W =0.1
∆u
45 48
41
0.1
1
5
9
13
17
time (samples)
25
29
33
37
41
W =0.1
∆u
45 48
time (samples)
Normalized Input Trajectory: u47
Normalized Input Trajectory: u48
1
1
0.9
0.8
Normalized Flow
Normalized Flow
21
0.8
0.7
0.6
0.6
0.4
0.2
W =1
W =1
∆u
∆u
0.5
1
5
9
13
17
21
25
29
33
37
W =0.1
∆u
45 48
41
0
1
5
9
13
time (samples)
17
21
25
29
33
37
41
W =0.1
∆u
45 48
time (samples)
Figure 20: Flow behaviour actuators: Scenario 1 (blue) and Scenario 2 (red).
Normalized Tank Trajectory x1
Normalized Tank Trajectory x2
1
Noramlized Volume
Noramlized Volume
1
0.9
0.8
W∆u=1
0.7
W∆u=0.1
0.6
0.5
1
5
9
13
17
21
25
29
33
37
41
0.9
0.8
W∆u=1
0.7
0.6
0.5
45 48
W∆u=0.1
1
5
9
13
time (samples)
Normalized Tank Trajectory x3
25
29
33
37
41
45 48
Normalized Tank Trajectory x4
1
0.85
0.8
0.75
W∆u=1
0.7
W∆u=0.1
0.65
0.6
0.55
1
5
9
13
17
21
25
29
33
time (samples)
37
41
45 48
Noramlized Volume
Noramlized Volume
21
time (samples)
0.9
0.5
17
0.9
0.8
W∆u=1
0.7
W∆u=0.1
0.6
0.5
1
5
9
13
17
21
25
29
33
37
41
45 48
time (samples)
Figure 21: Volume level evolution: Scenario 1 (blue) and Scenario 2(red).
Section 4
Barcelona Drinking Water Network: the Case Study
35
Tank Trajectory x
17
1800
1600
x17
1200
3
Volume level [m ]
1400
x max
17
x min
1000
17
ηe
800
600
400
200
0
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 22: Volume level evolution tank x17 in Scenario 4.
Normalized Input Trajectory u57
Normalized Input Trajectory u58
0.2
0.3
0.2
0.1
0
ηe
1
5
9
13
17
21
25
29
33
37
41
Flow value [m2/s]
Flow value [m3/s]
0.4
0.15
0.1
0.05
ηe
0
80%
45 48
1
5
9
13
time (samples)
Normalized Input Trajectory u59
25
29
33
37
41
80%
45 48
Normalized Input Trajectory u60
1
Flow value [m3/s]
Flow value [m2/s]
21
time (samples)
1
0.8
0.6
0.4
0.2
0
17
ηe
1
5
9
13
17
21
25
29
time (samples)
33
37
41
80%
45 48
0.8
0.6
0.4
0.2
ηe
0
1
5
9
13
17
21
25
29
33
37
41
80%
45 48
time (samples)
Figure 23: Flow behaviour actuators: comparison between Scenario 2 (red) and Scenario 1
(blue).
could causes practical difficulties, i.e., it requires a significant computation load, like it has been
outlined beforehand. Therefore, ordinary MPC technique based on centralized optimization
problem may involve limits on its applicability on complex high dimensional systems. Thus, it
becomes very important to apply on this large-scale system computationally efficient algorithms
with the attractive less computational load. Like it has been discussed in Section 2, one way to
cope with computational issues is to use decentralized control schemes. The main idea of the
DMPC algorithm is that the on-line optimization of MPC for large-scale system can be converted into a small-scale optimization of more that one MPC controller. Hence, according to the
distributed MPC approach reported in [1], for this case study a DMPC controller is implemented
in order to make a decentralized control that takes into account the overall system dynamics,
replaced by several simpler units, and on the other hand takes care about computational prob-
36
Modelling and Decentralized MPC of Drinking Water Networks
Normalized Tank Trajectory x
17
0.9
0.8
Normalized Volume
0.7
x17, ηe
0.6
ηe
x17, 80%
0.5
80%
0.4
0.3
0.2
0.1
0
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 24: Volume level evolution tank x17 in Scenario 2 (red) and in Scenario 4 (blue).
lems. Even if this distributed approach described has been followed, due to the complexity of
this case study and its interconnected elements, it has been not possible to partition the system
in independent subsystems, but they exchange information through some actuators. Hence, it
is been necessary to introduce an approach of hierarchical control.
System Partitioning: Algorithm and Heuristics
The decomposition of the ABN is carried out in two steps: first partitioning algorithm sensitivitybased [2] on the aggregated system is applied and then, to improve the partition obtained, some
heuristic methods are used. The algorithm works with the following parameters:
• the topology of the network, that are the matrices A, B and E of the model expression
(20). Using this algorithm, it is necessary to provide these matrices in the form:
B
A 0
, Balgorithm =
Aalgorithm =
0 0
E
where E is the matrix of equality constraints of the expression (29) considering Hp = 1.
• the usage of each actuators. This parameter is an optional parameter but it is very useful
because it can provide a more accurate partition. Unfortunately, despite its utility, this
parameter has a drawback which is the necessity of a previous simulation and computation
of inputs behaviour. In this case study, to calculate the usage of the actuators is used a
previous centralized optimal simulation and the relative value of the inputs that allows,
making the average of the normalized value of the optimal inputs, to obtain the normalized total amount of water flowed through each actuator. These parameter offers to the
algorithm a criteria to evaluate how important is a single actuator.
• the threshold of the actuators magnitude. This parameter, together with the usage, is
used by the algorithm in order to neglect same actuators that have less importance in the
entire system behaviour.
Once all the input parameters are provided to the algorithm, a trial and error procedure is made,
changing to threshold value of the magnitude, in order to find an acceptable partition of the
ABN case study. Obtaining a reasonable partition of the system, the second step is to improve
the partition with heuristics. Due to the possibility of the algorithm to neglect actuators, it is
necessary a heuristic procedure that makes this partition feasible with the approach followed in
this case study, where any of the actuators can be considered as neglected. Finally a partition
in three subsystems is found, as depicted in Figure 25. The subsystems are:
Section 4
Barcelona Drinking Water Network: the Case Study
37
Table 8: Dimension comparison between the subsystems and the entire system
Elements
Subsystem 1
Subsystem 2
Subsystem 3
Entire System
Tanks
Actuators
Demands
Nodes
2
5
4
2
5
22
9
3
10
34
22
6
17
61
25
11
1. Subsystem 1: composed by tanks xi , i ∈ {1, 2}, inputs uj , j ∈ {1, 2, 3, 4, 5}, demands dl ,
l ∈ {1, 2, 3}, and nodes nq , q ∈ {1, 2}. It is represented in Figure 6 with red color.
2. Subsystem 2: composed by tanks xi , i ∈ {3, 4, 5, 12, 17}, inputs uj ,
j ∈ {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 25, 26, 32, 34, 40, 41, 47, 48, 56, 60},
demands dl , l ∈ {4, 5, 6, 7, 15, 18, 22}, and nodes nq , q ∈ {3, 4, 7} with blue color.
3. Subsystem 3: composed by tanks xi , i ∈ {6, 7, 8, 9, 10, 11, 13, 14, 15, 16}, inputs uj , j ∈
{6, 17, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 42, 43,
44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61}, demands dl , l ∈ {8, 9, 10, 11, 12,
13, 14, 16, 17, 19, 20, 21, 23, 24, 25}, and nodes nq , q ∈ {5, 6, 8, 9, 10, 11}. It is represented in
Figure 25 with green color.
Thus, taking into account all the actuators, the subsystems have to exchange informations from
each others (the shared elements are represent with the colour violet in Figure 25 and in Figure
26). Hence, also in this situation, a heuristic procedure has been followed in order to establish
a criteria for the control of these actuators. The shared elements (see Figure 26) are:
Subsystem 2–Subsystem 3 : u18 , u20 , u21 , u32 , u34 , u40 , u47 , u56 , u60 ;
Subsystem 1–Subsystem 3 : u6 .
These actuators can have a entrance direction or an exiting direction according the considered
subsystem. The input shared actuators in the i-th subsystem are considered inputs and they
are controlled by the i-th MPC. Otherwise, the shared actuators that have an exiting direction
from the i-th subsystem are considered as sectors of consume and, at each step, the value of
the known demand is the optimal input calculated previously by another controller in which
this element is a manipulated variable. According to these considerations, the total number
of variables for each subsystem increases. Table 8 summarizes the amount of variables of each
subsystem as well as of the entire model.
This heuristic procedure makes necessary to consider the distributed DMPC with a hierarchical
issue, as the control of the subsystems has to be executed with a preestablished order. The order
of the DMPC execution is in the following way:
1. MPC Subsystem 3. It needs, for the first step, the initial values for the shared elements
u18 , u32 , u34 , u40 , u47 , u56 , u60 , that are considered as demands, values that in the next iterations will be provided by the optimal inputs calculated by the MPC of Subsystem 2.
These values are given by the equivalent simulation of CMPC. The MPC of this subsystem at each step generates the optimal inputs that will represent the value of known
disturbances u20 , u21 , u6 for the Subsystem 2 and Subsystem 1.
38
Modelling and Decentralized MPC of Drinking Water Networks
Figure 25: Partition of ABN
2. MPC Subsystem 2. It considers as demands the elements u20 , u21 . At each step, the
value of these elements are provided by the previous execution of the MPC of the Subsystem 3. Moreover, this MPC provides for the next step the values for the actuators
u18 , u32 , u34 , u40 , u47 , u56 , u60 , that are considered as demands in the Subsystem 3.
Section 4
Barcelona Drinking Water Network: the Case Study
39
Subsystem 1
u6
Subsystem 3
u 20
u 21
Subsystem 2
u 18
u 32
u 34
u 40
u 47
u 56
u 60
Figure 26: Shared actuators between Subsystem 1, Subsystem 2 and Subsystem 3
3. MPC Subsystem 1. It considers the element u6 like demand. At each step, the value of
this element is provided by the previous execution of the MPC of Subsystem 3.
4.5
Simulation of Scenarios
The implementation of the DMPC controller has been made using Matlabr . In order to compare
the computational load with the CMPC, it is been used the Cplex solver under the Tomlabr
platform. All the data that will be shown from now on are referred to this Matlabr implementation running on a Intelr CoreT M Duo 2.26 Ghz machine. For all the simulations the
optimization problem solver finds the optimal solution that satisfy the request of water, the
mass balance equations at the network nodes, the physical constraints and the volume balance
(38).
4.5.1
Comparison with the CMPC
To evaluate the strategy of the DMPC controller against the CMPC, the behaviour of the
manipulated inputs, the state variables, as well as the cost function evolution are compared for
each scenario. In Figures 27 and 28 are reported all the tanks trajectories obtained with the
Scenario 2 (in blue the Centralized behaviour and in red the Decentralized one).
As depicted in these figures, even if there is an inevitable loss of performances in the DMPC
framework. The gap with the optimal solution of the CMPC is not significant for the states
(in some reservoirs the evolution is similar), while for the manipulated inputs, this loss of
performance is relevant (see Figure 30).
4.5.2
Main Results
To compare the strategy of the DMPC controller against the CMPC over the simulation scenarios, values related to the control objectives were calculated for each scenario, as well as the
computational times. Moreover, in order to have a more exhaustive comparison, an economical
evaluation has been made even if both the controllers do not minimize the term referred to
40
Modelling and Decentralized MPC of Drinking Water Networks
Tank Trajectory x1
Tank Trajectory x2
1000
Volume level [m3]
3
Volume level [m ]
500
400
300
Centralized MPC
x max
1
200
x min
800
600
Centralized MPC
x max
2
400
x min
2
1
100
1
5
9
13
17
21
25
29
33
time (samples)
80%
37 Decentralized
41
45 48
MPC
200
1
5
9
13
21
25
29
33
time (samples)
Tank Trajectory x3
Tank Trajectory x4
4000
3
3
Volume level [m ]
4000
Volume level [m ]
17
80%
MPC
37 Decentralized
41
45 48
3000
2000
Centralized MPC
x3 max
1000
0
x3 min
1
5
9
13
17
21
25
29
33
time samples
3000
2000
0
37 80%
41
45 48
Decentralized MPC
Centralized MPC
x4max
1000
x4min
1
5
9
13
17
21
25
29
33
time (samples)
37 80%
41
45 48
Decentralized MPC
(a)
Tank Trajectory x5
Tank Trajectory x6
3
Volume level [m ]
3500
3
Volume level [m ]
15000
10000
Centralized MPC
x5max
5000
x6min
1
5
9
13
17
21
25
29
33
time (samples)
Centralized MPC
x6max
1500
1000
x6min
1
5
9
13
17
21
25
29
33
time (samples)
37 80%
41
45 48
Decentralized MPC
Tank Trajectory x8
12000
3
3
Volume level [m ]
x 10
5
4
Centralized MPC
x7max
3
2
x7min
1
2000
500
80%
41
45 48
Decentralized MPC
6
1
2500
Tank Trajectory x7
4
7
37
Volume level [m ]
0
3000
5
9
13
17
21
25
29
time (samples)
33
37 80%
41
45 48
Decentralized MPC
10000
8000
6000
Centralized MPC
x8max
4000
2000
0
x8min
1
5
9
13
17
21
25
29
time (samples)
33
37 80%
41
45 48
Decentralized MPC
(b)
Figure 27: Tanks Trajectories Scenario 2 form x1 to x8 : DMPC Vs CMPC
Section 4
Barcelona Drinking Water Network: the Case Study
41
Tank Trajectory x10
16000
7000
14000
Volume level [m ]
6000
3
3
Volume level [m ]
Tank Trajectory x9
8000
5000
4000
3000
Centralized MPC
x max
2000
9
1000
0
12000
10000
8000
6000
Centralized MPC
x max
10
4000
x min
x min
9
1
5
9
13
17
21
25
29
33
time (samples)
10
80%
37 Decentralized
41
45 48
MPC
2000
1
5
9
13
17
21
25
29
33
37
time (samples)
Tank Trajectory x11
Tank Trajectory x12
4
1200
10
80%
41
45 48
Decentralized MPC
x 10
3
Volume level [m ]
3
Volume level [m ]
9
1000
800
600
Centralized MPC
x11max
400
1
5
9
13
17
21
25
29
33
time (samples)
7
6
5
Centralized MPC
x12max
4
3
x11min
200
8
80%
37 Decentralized
41
45 48
MPC
2
x12min
1
5
9
13
17
21
25
29
33
time (samples)
37
80%
41
45 48
Decentralized MPC
(a)
Tank Trajectory x13
x 10
3
Volume level [m ]
3
Volume level [m ]
4
4000
3000
2000
Centralized MPC
x13max
1000
0
Tank Trajectory x14
4
5000
x13min
1
5
9
13
17
21
25
29
33
time (samples)
3
2
0
37 80%
41
45 48
Decentralized MPC
Centralized MPC
x14max
1
x14min
0
10
4080%
50
Decentralized MPC
30
time (samples)
Tank Trajectory x15
Tank Trajectory x16
5000
3
3
Volume level [m ]
8000
Volume level [m ]
20
6000
4000
Centralized MPC
x15max
2000
0
x15min
1
5
9
13
17
21
25
29
time (samples)
33
37 80%
41
45 48
Decentralized MPC
4000
3000
Centralized MPC
x16max
2000
1000
x16min
1
5
9
13
17
21
25
29
time (samples)
33
37 80%
41
45 48
Decentralized MPC
(b)
Figure 28: Tanks Trajectories Scenario 2 form x9 to x16 : DMPC Vs CMPC
42
Modelling and Decentralized MPC of Drinking Water Networks
Tank Trajectory x17
1800
1600
1400
3
Volume level [m ]
1200
Centralized MPC
x max
1000
17
x min
17
800
80%
Decentralized MPC
600
400
200
0
1
5
9
13
17
21
25
29
33
37
41
45
48
time (samples)
Figure 29: Tanks Trajectories Scenario 2 x16 : DMPC Vs CMPC.
Input Trajectory u21
Input Trajectory u22
0.8
3
2.5
Flow value [m /s]
3
Flow value [m /s]
3
2
1.5
1
Centralized MPC
u21max
0.5
0.6
0.4
u21min
0
5
10
15
20
25
30
35
Centralized MPC
u22max
0.2
0
40Decentralized
45
MPC
u22min
1
5
9
time (samples)
21
25
29
33
37 Decentralized
41
45 48
MPC
Input Trajectory u24
4
3
3
2.5
Flow value [m /s]
3
Flow value [m /s]
17
time (samples)
Input Trajectory u23
2
1.5
1
Centralized MPC
u23max
0.5
0
13
u23min
1
5
9
13
17
21
25
29
time (samples)
33
MPC
37 Decentralized
41
45 48
3
2
Centralized MPC
u24max
1
0
u24min
1
5
9
13
17
21
25
29
33
MPC
37 Decentralized
41
45 48
time (samples)
Figure 30: Actuators Behaviour Scenario 2: DMPC Vs CMPC
the economical cost in optimization problem. This economical evaluation has been made using
a water network simulation environment, developed in Simulinkr, that allows to evaluate the
performances of water network controllers as well as the economical cost of the strategy used
[4]. The main obtained results, derived from the numerical results summarized in Table 10 and
9, are now discussed.
In the first column the scenarios used for each simulation are reported. The more important
result, according to the reason of design an DMPC controller for the complex system, is the
computational time, as almost always the DMPC controller execute the control problem in
about half time than the CMPC controller with a suboptimality that is about 2%. Another
parameter investigated is the maximum time that the solver has spent to make a single iteration.
Also in this case, as the computational time, the DMPC controller executes, in the worst case,
a single iteration in a time that is almost always a half lower then the CMPC one. About
in the economical cost, as the suboptimality is about the 2% for all the scenarios, the gap
between CMPC and DMPC is not relevant. Thus, according to the DMPC approach, despite
Section 5
Concluding Remarks
43
Table 9: Time Results obtained for the considered control objective and scenarios
Scenario
Scenario
Scenario
Scenario
Scenario
CENTRALIZED MPC
Total time Max time
[s]
[s]
1
2
3
4
207.12
206.27
210.57
211.18
6.0866
7.0348
4.9057
5.5524
DECENTRALIZED MPC
Total time
Max time
[s]
[s]
128.2828
130.888
125.5362
126.0275
3.2086
3.3209
4.7260
2.8945
Table 10: Cost Results obtained for the considered control objective and scenarios
Scenario
Scenario
Scenario
Scenario
Scenario
1
2
3
4
CENTRALIZED
MPC
P
fi
Cost
[Euro]
DECENTRALIZED
MPC
P
fi
Cost
[Euro]
58.0787
57.5404
74.0044
74.3957
59.9548
59.1040
76.1662
78.4981
220080
219730
197850
199120
223490
223900
220210
200430
the decentralized controller has the disadvantage of inevitably leading to a loss of performance,
in this case study this disadvantage in terms of cost function and economical costs has not a
relevant impact compared to the advantages that the DMPC strategy have in terms of time and
computational load.
5
Concluding Remarks
This section summarizes the main contributions done and proposes future ways to continue the
line of the report.
5.1
Contributions
In this work, the basic idea was the design of MPC strategies for water systems able to calculate
in a predictive way the ammissible manipulated inputs, provided from the actuators, in order to
achive the best admisible solution for the flow management of a network. The main contributions
of the report are:
(i) A modelling methodology was developed for modelling of water networks. The proposed
methodology allows to represent each constitutive element of the network in the MPC
framework.
(ii) Once the modelling formalization was provided, a predictive control of water network has
been proposed. This fact allows to compute the global optimal solution of the associated
optimization problem when a QP-problem is stated. Before starting with the complex
process of modelling directly over the case study, a demonstrative example, comprised
of each constitutive elements of water networks, was implemented in order to show the
effectiveness of the approach used.
(iii) A suboptimal DMPC strategy was derived for reducing the computation time taken by
the solution of the MPC associated to the optimization problem. The strategy follows a
distributed approach and considers the addition of heuristcs procedures.
44
Modelling and Decentralized MPC of Drinking Water Networks
(iv) A comparative study between the MPC and DMPC control techniques and methodology
mentioned beforehand was developed on a particular water network, and this system, used
as case study, was an aggregated version af the entire Barcelona water nework. This study
has shown the advatages and disadvatages of both the strategies on control of complex
water networks.
5.2
Future Work
To continue the research proposed in this report, some ideas are outlined below.
• The implementation of control strategy do not consider the term referred to the economical
cost in the cost function of the associated optimal control problem. Thus, a further development would be to add economical penalization in the multi-objective control problem,
such this is a relevant topic in water networks management.
• Another important situation that should be investigated is related with considering the
tanks volume level references as dynamically varied values. The volume of the tanks should
follow a forecasts profile. The study and the generation of these variable references, that
is basically a prediction of which should be the tank capacity at each time in order to
completely fulfilled the request of water, is currently under development.
• The implementation of a suitable partitioning algorithm, inspired on the modelling formalization developed in this report for water network, should be done. It should take into
account automatically the heuristic procedures considered in this report, as well as the
hierarchical issue in the DMPC approach. Moreover, it could be allowed the possibility of
subsystems overlapping.
• The modelling formalization of water networks made in this report should be extended
considering non-linar dynamics and the network pressure issues. The optimization model
should contain a hydraulic model of the network, which would make possible to test the
effect produced by control action (flow through the actuators) on the network in terms of
pressure.
Finally, a further extension of the aggregated model is at the moment under developing, considering the entire drinking water network of Barcelona. Figure 31 shows the whole Barcelona drinking water network. Considering this new system topology, an improvement of the methodologies
and techniques proposed on this report could provide better results than the implementation
with the older aggregated version.
Section 5
Concluding Remarks
Figure 31: Barcelona Drinking Water Network
.
45
46
Modelling and Decentralized MPC of Drinking Water Networks
A
Parameters of the ABN Case Study
Table 11: Sectors of consume
Name
c125PAL
c110PAP
c100LLO
c70LLO
c140LLO
c176BARsud
c100FCE
c70FLL
c100BLLcente
c200ALT
c100BLLnord
c200BARnord
c135SCG
Model Variable
d1
d3
d5
d7
d9
d11
d13
d15
d17
d19
d21
d23
d25
Name
c70PAL
c115CAST
dc80GAVI80CAS
c200BLL
c10COR
c130BAR
c100BLLsud
c200BARsc
c70BBEsud
c176BARcentre
c120POM
c101MIR
Model Variable
d2
d4
d6
d8
d10
d12
d14
d16
d18
d20
d22
d24
Table 12: Tanks of the network and their lower and upper bounds
1
Name
Model Variable
Min Volume [m3 ]
Max Volume [m3 ]
d125PAL
d110PAP
d115CAST
d80GAVi80CAS85
dPLANTA
d54REL
d100FCE
d10COR
d200BLL
d130BAR
d176BARsud
d70BBEsud
d200ALT
d100BLLnord
d200BARnord
d101MIR
d120POM
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
150
375
198
480
0
800
16500
0
700
3840
200
22450
500
6000
700
1403
150
445
960
3870
3250
14450
3100
65200
11745
7300
16000
1035
98041
4240
37700
7300
4912
1785
unlimited valve; the limit for each valve is the maximum quantity of water the associated source can provide.
unlimited valves, so they can have a huge pyhsical flow.
3
This values has not been provided by AGBAR.
2
Section A
Parameters of the ABN Case Study
47
Table 13: Values and their lower and upper bounds
Name
Model
Variable
Max
[m3 /s]
Flow
VALVA
VALVA47
VALVA308
VCA
VSJD
VALVA50
VE
VZF
VCO
VT
VP
VCOA
VMC
VALVA56
VBNC
VALVA54
VALVA55
VALVA312
u1
u6
u12
u14
u18
u26
u30
u32
u37
u40
u43
u46
u49
u51
u54
u57
u59
u61
1,2971
1,2
5,341
0,065
0,75
0,1594
0,45
0,35
0,5249
1,3
0,15
1,35
0,24
1,7
0,392
1,7361
0,1852
6,276841
Name
Model
Variable
Max
[m3 /s]
VALVA45
VCR
VALVA48
VALVA309
VALVA64
VF
VRM
VB
VS
VCT
VBSLL
VPSJ
VALVA60
VALVA57
VALVA53
VALVA61
VCON
u2
u8
u13
u16
u25
u28
u31
u35
u39
u41
u44
u47
u50
u52
u56
u58
u60
0,05
0,03
0,22
2.51
15 2
0,29
3,5
0,15
1,2
1,2
0,15
0,55
152
0,4051
1,5001
152
0,035
Flow
Table 14: Pumps and their lower and upper bounds
.
Name
Model
Variable
Max
[m3 /s]
CPIV
CPII
bPousCAST
CB
bPousE
CPLANTA50
CE
CC100
CF200
CC70
CCO
CPR
CRO
u3
u5
u9
u11
u17
u20
u22
u24
u29
u34
u38
u45
u53
0,0317
0,0220
0,0056
0,0500
0,2300
1,8000
0,6200
3,1000
0,2600
0,4000
0,8500
0,0053
0,1342
Flow
Name
Model
Variable
Max
[m3 /s]
Flow
bMS
bCast
CCA
CPLANTA70
CGIV
PLANTA10
CRE
CC50
CC130
CF176
CA
CMO
bPousB
u4
u7
u10
u15
u19
u21
u23
u27
u33
u36
u42
u48
u55
0,0150
0,000013
0,1200
0,2900
0,0108
2,9000
3,0000
0,6000
0,09
0,1563
0,425
0,025
0,38
48
Modelling and Decentralized MPC of Drinking Water Networks
B
Aggregated Barcelona Drinking Water Network Model Equations
TANK 1
x1 (k + 1) = x1 (k) + ∆t(u3 (k) + u4 (k) − d1 (k))
(39)
x2 (k + 1) = x2 (k) + ∆t(u5 (k) − d3 (k))
(40)
x3 (k + 1) = x3 (k) + ∆t(u7 (k) + u10 (k) + u11 (k) − u8 (k) − d4 (k))
(41)
TANK 2
TANK 3
TANK 4
x4 (k + 1) =x4 (k) + ∆t(u8 (k) + u9 (k) + u13 (k) + u19 (k) − u10 (k)
− u11 (k) − u14 (k) − d6 (k))
(42)
TANK 5
x5 (k + 1) = x5 (k) + ∆t(u12 (k) + u16 (k) − u15 (k) − u20 (k) − u21 (k))
(43)
x6 (k + 1) = x6 (k) + ∆t(u6 (k) + u20 (k) + u27 (k) − u23 (k))
(44)
x7 (k + 1) = x7 (k) + ∆t(u17 (k) + u23 (k) + u24 (k) + u37 (k) − u18 (k))
(45)
x8 (k + 1) = x8 (k) + ∆t(u21 (k) − u24 (k) − u27 (k) − u33 (k) − u34 (k) − d10 (k))
(46)
x9 (k + 1) = x9 (k) + ∆t(u29 (k) + u33 (k) − u28 (k) − d8 (k))
(47)
TANK 6
TANK 7
TANK 8
TANK 9
TANK 10
x10 (k + 1) =x10 (k) + ∆t(u30 (k) + u38 (k) + u45 (k) + u51 (k) + u52 (k)
− u29 (k) − u36 (k) − u37 (k) − u42 − d12 (k))
(48)
x11 (k + 1) = x11 (k) + ∆t(u35 (k) + u36 (k) − d11 (k))
(49)
x12 (k + 1) = x12 (k) + ∆t(u41 (k) + u47 (k) + u56 (k) − u48 (k) − d18 (k))
(50)
x13 (k + 1) = x13 (k) + ∆t(u42 (k) − u44 (k) − d19 (k))
(51)
TANK 11
TANK 12
TANK 13
Section B
Aggregated Barcelona Drinking Water Network Model Equations
49
TANK 14
x14 (k + 1) =x14 (k) + ∆t(u55 (k) + u57 (k) + u58 (k) − u46 (k) − u53 (k)
− u54 (k) − d21 (k))
(52)
TANK 15
x15 (k + 1) = x15 (k) + ∆t(u49 (k) + u50 (k) + u53 (k) − d23 (k))
(53)
x16 (k + 1) = x16 (k) + ∆t(u54 (k) + u59 (k) − d24 (k))
(54)
x17 (k + 1) = x17 (k) + ∆t(u48 (k) + u60 (k) − d22 (k))
(55)
u1 (k) = u2 (k) + u5 (k) + u6 (k)
(56)
u2 (k) = u3 (k) + d2 (k)
(57)
u18 (k) = u13 (k) + d5 (k)
(58)
u14 (k) + u15 (k) + u26 (k) = u19 (k) + u25 (k) + d7 k
(59)
u22 (k) = u30 (k) + d9 (k)
(60)
u31 (k) = u39 (k) + u40 (k) + d14 (k)
(61)
u25 (k) + u32 (k) + u34 (k) + u40 (k) = u26 (k) + u41 (k) + d15 (k)
(62)
u39 (k) + u46 (k) = u45 (k) + u47 (k) + d17 k
(63)
u28 (k) + u49 (k) = u35 (k) + u43 (k) + u43 (k) + d16 (k)
(64)
TANK 16
TANK 17
NODE 1
NODE 2
NODE 3
NODE 4
NODE 5
NODE 6
NODE 7
NODE 8
NODE 9
50
REFERENCES
NODE 10
u43 (k) + u44 (k) = d20 (k)
(65)
NODE 11
u61 (k) =u50 (k) + u51 (k) + u52 (k) + u56 (k) + u57 (k) + u58 (k)
+ u59 (k) + u60 (k) + d25 k
(66)
References
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