Learning-Based Tuning of Supervisory Model Predictive Control for Drinking Water Networks

Learning-Based Tuning of Supervisory Model Predictive Control for Drinking Water Networks
Learning-Based Tuning of Supervisory Model Predictive Control for Drinking
Water Networks
J.M. Grosso∗ , C. Ocampo-Martínez, V. Puig
Institut de Robòtica i Informàtica Industrial (CSIC-UPC). Llorens i Artigas 4-6, 08028. Barcelona, Spain.
Abstract
This paper presents a constrained Model Predictive Control (MPC) strategy enriched with soft-control techniques as
neural networks and fuzzy logic, to incorporate self-tuning capabilities and reliability aspects for the management
of drinking water networks (DWNs). The control system architecture consists in a multilayer controller with three
hierarchical layers: learning and planning layer, supervision and adaptation layer, and feedback control layer. Results
of applying the proposed approach to the Barcelona DWN show that the quasi-explicit nature of the proposed adaptive
predictive controller leads to improve the computational time, especially when the complexity of the problem structure
can vary while tuning the receding horizons.
Keywords: model predictive control, neural networks, fuzzy logic, self-tuning, reliability, drinking water networks,
multilayer controller.
1. Introduction
Drinking Water Networks (DWNs) are large-scale multi-source/multi-node flow systems which must be reliable
and resilient while being subject to constraints and to continuously varying conditions with both deterministic and
probabilistic nature. Optimal management of these systems is a complex task and has become an increasingly environmental and socio-economic research subject worldwide (Biscos et al., 2003; Cembrano et al., 2011; El Mouatasim
et al., 2012; Pascual et al., 2011; Vieira et al., 2011), with special attention to efficient handling of energetic and
natural resources in dense urban areas.
Model Predictive Control (MPC) is becoming a strong approach to deal with these challenging multi-criteria problems of supply-chain dynamic systems, see, e.g., Aggelogiannaki et al. (2008); Alessandri et al. (2011); Perea-Lopez
et al. (2003). The MPC framework is usually preferred among advanced control techniques because it incorporates
performance indices and it is able to cope with constraints and modelling errors in an explicit way (Qin and Badgwell,
2003; Tatjewski, 2007).
Nevertheless, the ever growing complexity of networked systems (i.e., dimensionality, information structure constraints, non-linearities, uncertainty) and the higher performance requirements turn these kind of problems costly to
solve for real-time control applications as in transport water networks, where current MPC algorithms are not prepared
enough to face the important computational burden when design parameters (i.e., set-points, bounds, prediction and
control horizons, tuning weights, and system topology size) have to be continually redefined.
The tuning task of MPC controllers has been widely investigated and general guidelines are available in the literature (Garriga and Soroush, 2010; Schwartz and Rivera, 2006; Shah and Engell, 2011; Toro et al., 2011; Wojsznis
et al., 2003). Some methods propose heuristics while others are based on stability criteria, closed-loop frequencydomain analysis, optimisation-based algorithms, genetic programming, on-line process identification, among others.
The general approach in tuning procedures is to fix MPC parameters off-line for all the system operation but this fact
could lead to decrease the system performance due to manoeuvrability reduction. Other methods generate the complete Pareto frontier and select the best solution according to an extra criterion, but this approach is computationally
Email address: [email protected] (J.M. Grosso∗ )
Preprint submitted to Engineering Applications of Artificial Intelligence
March 7, 2013
prohibitive in fast dynamic or large-scale systems. Therefore, implementation of adaptive structures and tractable
on-line tuning procedures are necessary to assure economic efficiency and safety of complex multi-variable systems
as DWNs.
In order to face the aforementioned design issues, MPC algorithms have been extended or replaced with softcomputing techniques in different control architectures (Tatjewski and Ławryńczuk, 2006). Most of these approaches
are intended to improve performance by using expert-guidance or iterated experiments in order to simplify models of
non-linear systems or to approximate and generalise by learning-based techniques the solution of optimal controllers
(Åkesson and Toivonen, 2006; Ali, 2003; Aswani et al., 2012; Parisini and Zoppoli, 1995; Valera García et al., 2012).
Intelligent control systems are able to replicate aggressive manoeuvres while performing adaptation, function approximation, knowledge modelling and massive parallel processing. Nevertheless, the main drawback of replacing MPC
controllers with predictive soft-controllers is that they may not guarantee safety, stability or robustness due the lack
of feedback correction mechanisms to face disturbances so the performance is subject to the limited scenarios used in
the learning process.
The main contribution of this paper is the synergy of artificial intelligence, supply chain theory and automatic
control to devise an adaptive and robust MPC controller for the management of DWNs. Here, a Learning-based
Supervisory MPC (LB-MPC) controller with a hierarchical multilayer structure is proposed, where soft-computing
techniques (i.e., neural networks and fuzzy logic) are used not to approximate an MPC controller but to tune its design
parameters on-line by learning expectation of performance. In contrast to general tuning approaches, the LB-MPC
presented in this paper not only adapts the weights of the multi-objective optimisation problem that takes place within
the MPC but also the prediction horizon according to the current situation of the plant. Furthermore, operational
output constraints are dynamic and governed according to the non-stationary uncertainty of the DWN disturbances
to assure service reliability while optimising economic resources. The proposed control scheme is a quasi-explicit
MPC with less on-line computational burden, because most of the heavy computations are converted into non-linear
explicit modules using neural networks. The benefits, i.e., flexibility and reliability, of this LB-MPC controller as a
decision-support tool for the management of DWNs, are shown in this paper through a real case study: the Barcelona
DWN.
The paper is organised as follows. Section 2 describes a control-oriented flow-based model of DWNs. Section
3 concerns to the enhanced MPC strategy addressed in this paper. Section 4 describes the case study where the
effectiveness of the proposed approach is analysed via simulations. Section 5 highlights the concluding remarks that
can be drawn from the results presented in this paper as well as some future research directions.
2. Problem Statement
2.1. DWN Control Problem
Let N = {1, 2, ...} be the set of natural numbers and N0 , N ∪ {0}. Consider a DWN being represented as the
interconnection of n tanks, m actuators, p demands and q intersection nodes (Ocampo-Martinez et al., 2009, 2010).
This system may be generally described by the following control-oriented discrete-time dynamic model for k ∈ N0 :
S (x, u, d) =
(
x(k + 1) = Ax(k) + Bu(k) + B p d(k),
0 = E1 u(k) + E2 d(k),
(1a)
(1b)
where x ∈ Rn is the state vector of water stock volumes in m3 ; u ∈ Rm is the vector of manipulated flows in m3 /s ;
d ∈ R p corresponds to the vector of disturbances (water demands) in m3 /s; index k represents the current time instant;
A ∈ Rn×n , B ∈ Rn×m and B p ∈ Rn×p are state-space system matrices for mass balances in tanks (1a); and E1 ∈ Rq×m
and E2 ∈ Rq×p are matrices for the mass balance in nodes (1b). All vectors and matrices are dictated by the network
topology.
Consider also that in DWNs control objectives are evaluated by the following performance indices:
JE (k) , k(α1 + α2 (k))T u(k)k1 ,
(2a)
JS (k) ,
kε(k)k22 ,
(2b)
JU (k) ,
k∆u(k)k22 ,
(2c)
2
where JE represents the economic cost of network operation taking into account water production cost (α1 ) and water
pumping cost (α2 ) which change every time instant according to the variable electric tariff; JS is a performance
index which penalises the amount of volume ε that goes down from a safety volume value x s ; and JU represents the
penalisation of control signal variations ∆u(k) , u(k) − u(k − 1) to extend actuators life and assure a smooth operation.
The performance indices considered in this paper may vary or generalised, with the corresponding manipulation, to
include other control objectives.
Assumption 1. The states and control inputs are subject to hard constraints due physical limits in tanks and operational limits in actuators, i.e.,
x(k) ∈ X , {x ∈ Rn : xmin ≤ x(k) ≤ xmax },
u(k) ∈ U , {u ∈ R : umin ≤ u(k) ≤ umax },
m
∀k,
∀k.
(3a)
(3b)
Assumption 2. For safety and service reliability, states are subject to management soft constraints
x(k) ≥ x s (k) − ε(k) ≥ 0,
∀k,
(4)
where x s ∈ Rn is a vector of base stocks in m3 (minimal volume in each tank to avoid stock-outs) and ε , max {0, xs − x},
ε ∈ Rn , is a slack variable which represents the amount of volume that goes down from the desired base stocks.
Assumption 3. Disturbances are non-stationary but follow a periodic behaviour and can be forecasted H p times
ahead. The prediction error ed is known to lie in a heteroscedastic distribution D, as follows:
ed (k) = [d(k) − d̃(k)] ∼ D(µ(k), σ(k)),
(5)
where d̃ is the vector of disturbances prediction for the time instant k, and µ and σ are vectors of the means and
standard deviations of the errors, respectively.
Assumption 4. The states x and disturbances d of the plant are exactly measured at each time instant k.
Assumption 5. Every source can supply its underlying demand.
Assumption 6. All elements operate with a common review period and storage tanks are subject to the same replenishment policy.
The main goal of the operational control of water transport networks is to satisfy the demands at consumer sectors,
but optimising, at the same time, management policies expressed as a multi-objective control problem. Hence, MPC is
a suitable technique to control a DWN because its capability to deal efficiently with multivariable dynamic constrained
systems and predict the proper actions to achieve the optimal performance according to a user-defined cost function.
Moreover, the MPC design follows a systematic procedure (Maciejowski, 2002), which generates the control input
signals to the plant by combining a prediction model and a Receding-Horizon (RH) strategy.
Problem 1 (MPC for DWNs). Consider the system (1) at a measured condition. Given a prediction horizon H p ∈ N,
and the control objectives in (2) aggregated in a performance index J : RnH p ×m(H p −1) → R, the MPC problem for
DWNs consists in solving a finite horizon optimal control problem (FHOCP) given by
J ∗ , min
∗ ∗
u ,ε
k+H
p −1
X
i=k
i
h
k(α1 + α2 (i|k))T u(i|k)k1,We + kε(i + 1|k)k22,Wx + k∆u(i|k)k22,Wu ,
(6)
subject to:
x(i + 1|k) = Ax(i|k) + Bu(i|k) + B p d(i|k),
(7)
E1 u(i|k) + E2 d(i|k) = 0,
xmin ≤ x(i + 1|k) ≤ xmax ,
(8)
(9)
umin ≤ u(i|k) ≤ umax ,
x(i + 1|k) ≥ x s (k) − ε(i + 1|k) ≥ 0,
(10)
(11)
(x(k|k), d(k|k)) = (x(k), d(k)).
(12)
3
Then, according to the RH strategy, apply only the first column vector u∗ (k|k) of the optimal sequence u ∗k (xk ) ,
→
−
h
i
u∗ (k|k), . . . , u∗ (k + H p − 1|k) . At the next time instant, the prediction horizon is shifted one time instant and the
optimisation is restarted with new feedback measurements to compensate unmeasured disturbances and model inaccuracies. This scheme is repeated at each future time instant.
Remark 1. Despite the intuitive formulation of the RH strategy, on-line tuning of an MPC controller is not trivial or
systematic. The MPC tuning parameters for the given cost function usually are prediction horizon H p and weighting
matrices We , W x , Wu . This paper also addresses the adaptation of the relaxed boundary x s according to time-variant
disturbances.
♦
2.2. Safety Volumes and Service Reliability Problem
There is the need to guarantee a safety water stock in each tank of the network in order to decrease the probability
of shortages (when a tank or a node has not sufficient water to satisfy external demands or the transfer request coming
from other tank/node) due to uncertain events. To determine the amount of safety water stocks, an inventory planning
strategy is addressed here to enrich previous control approaches (Pascual et al., 2011; Toro et al., 2011) with replenishment policies. The goal is to dynamically allocate the minimal volume in each storage unit to avoid stock-outs for
a given period of time.
Problem 2 (Safety risk and base-stocks). Consider the system (1) with initial condition x(k) = x0 and take into
account stochastic disturbances d̃ with a given forecast error probabilistic distribution N(µ(k), σ2 (k)) and a leadtime vector τ(k) ∈ Rn (the time from the moment a supply requirement is placed to the moment it is received). The
safety risk problem consists in finding the minimal base-stock vector x s (k) that assures a given customer service level,
named γ, for any predicted demand in a short horizon of N time instants ahead.
Inventory management for supply chains literature, even in multi-stage multi-echelon schemes, suppose a hierarchical and descendant flow of products, in a way that predicted safety stock changes are easily communicated
backwards in order for supporting availability of quantities when they are needed (Kanet et al., 2010), but this behaviour is not true in real large-scale supply networks (i.e., the Barcelona DWN), because a meshed topology with
multi-directional flows between tanks and nodes prevails instead of spread tree configurations.
To circumvent the aforementioned difficulty, an optimal inventory planning is developed here to dynamically
allocate proper amounts of water in the tanks and avoid stock-outs. The planning is based on an economic linear
programming (LP) problem that estimates, in each time instant k, future flows û s (k) , [û s (k|k), . . . , û s (k + N − 1|k)]
→
−
over a short-term prediction of demand variance, as follows:
û (k) = arg min
u
→
−s
k+N−1
X
JE (i|k),
(13)
i=k
subject to (7), (8), (9), (10) and (12).
The resultant sequence of estimated flows û s (k), allows to virtually decouple the interconnections between tanks
→
−
and estimate their net demand d̂net , and mean forecast demand d̂avg , for a short horizon N ≤ H p , N ∈ N, as follows:
d̂net (i|k) = |Bout û s (i|k) + B p d̂(i|k)| ∀i ∈ [k : k + N − 1] ,
Pk+N−1
d̂net (i|k)
,
d̂avg (k) = i=k
N−1
(14)
(15)
where Bout û s (i|k) represents the estimated endogenous demands, i.e., the outflow of the tanks caused by water requirements from neighbouring tanks or nodes, and B p d̂(i|k) the estimated exogenous demands for a given time instant.
Even if the net demand is predicted with a strong confidence level in a short horizon, it is required to store safety
stocks s(k) ∈ Rn in m3 to face uncertainties. The amount of safety volumes is related with the stochastic nature of
demands and lead-times, and also with the goodness of the forecasting method. Hence, s(k) is given by
s(k) = Φ−1 (γ)σ(k),
4
(16)
where Φ−1 (·) is the inverse cumulative normal distribution, γ ∈ (0, 100]% is the desired customer service level (percentage of customers that do not experience a stock-out) and σ(k) , [σ1 (k), . . . , q
σn (k)]T ∈ Rn is the vector of
total forecast deviations, where each tank forecast deviation is given by σ j (k) , σ2d, j (k)τ j (k) + σ2τ, j (k)d̂avg, j (k),
∀ j ∈ [1, . . . , n], taking into account the sample standard deviation σd, j of the net demands, and the sample standard
deviation στ, j of the lead-times error.
Finally, the vector x s (k) ∈ Rn of water base-stocks in m3 is computed as follows to solve Problem 2:
x s, j (k) = τ j (k)d̂avg, j (k) + s j (k), ∀ j ∈ [1, . . . , n] ,
x s (k) , x s,1 (k), . . . , x s,n (k) T .
(17)
(18)
The value of x s (k) is re-computed at each time instant before the MPC algorithm is executed and, as stated in Assumption 2, it is introduced in (11) to lead the tank states to be greater than such base stocks (when possible) and let
the system employ safety volumes s(k) to face uncertainties (when needed) but penalising the used amount of safety.
Remark 2. This strategy deals specifically with tank reliability (assuming its faulty behaviour as the inability to
satisfy its own demands), which is affected by both the capacity and reliability of the system supplying it. If the
supply capacity is less than the average demand, no tank will be large enough to provide a sustained service.
♦
3. Learning-based Tuning and Supervisory MPC for DWN
In order to achieve a flexible and reliable controller as a decision-support tool for the management of DWNs, a
learning-based tuning approach and a supervisory MPC (LB-MPC) with adaptive capabilities and less on-line computational burden is proposed in this section. LB-MPC is a quasi-explicit controller that combines the advantages of
conventional MPC with those of knowledge-based soft-control.
3.1. Control System Structure
The control strategy addressed in this paper is based on a multilayer (hierarchical) control system structure. The
hierarchical architecture has been used in process control with satisfactory results optimising economic profits when
disturbances are slowly varying (Tatjewski, 2007). In DWNs, these disturbances follow a pattern in an daily basis and
can be well predicted for an hourly sampling time, which makes the hierarchical structure suitable to optimise targets
for the policies of the direct control level.
The proposed controller is based on a three-layer structure (see Fig. 1). First, a Learning and Planning layer
(LPL) executes strategic and on-line determination of the dynamic safety constraints, economic references and demand forecasting by training artificial neural networks (ANNs). Secondly, a Supervision and Adaptation layer (SAL)
implements a fuzzy rule-based inference system (FIS) to continually adjust the parameters of the MPC, which computes, through the receding horizon approach, the optimal control policy according to the current status of the plant.
Finally, a Basic Feedback Control layer (BFCL), commonly based on PID controllers or direct MPCs, is responsible
for real-time operation of the system and has direct access to manipulated variables. The BFCL is not addressed
in this paper because it is assumed that the design problem of each local regulatory controller is already solved by
the operators of the DWN. Hence, as a common practice in the design of hierarchical controllers, perfect reference
tracking of the control loops at the lower layer is assumed in this paper.
3.2. Learning and Planning Layer
Classical hierarchical MPC structures exhibit a high computational burden when set-point optimisation, governing
of constraints and forecasting of disturbances are required to be executed at the same frequency that the control
problem is solved. Sequential optimisations are often required for the planning of actions in top layers but many
parameters and non-linear operations cause these tasks to be intractable for on-line tuning of controllers, specially in
large-scale systems as DWNs. Therefore, this paper proposes a planning layer that uses ANNs in order to reduce the
computational burden of optimisation-based approaches.
ANNs have a remarkable ability to derive meaning from complicated or imprecise data and can be employed to
extract patterns and detect trends that are too complex to be noticed by other computer techniques. Specifically, this
5
General Management Policies
Prices
S erviceLevel
Constraints
Priorities
J
Weather conditions
(Temperature, humidity)
Learning and Planning
(Neural-Networks based)
d̃, x̃e , x̃ s
x, J
Knowledge base
(rules, fuzzy-sets)
Supervision & Adaptation
(Fuzzy-tuned MPC)
u∗
x
Basic Feedback Control
u
Disturbance
(Real demand)
y
d
Plant
Fig. 1: Multilayer hierarchical intelligent control architecture
Input
wh11
z1
Output
Hidden
nh1
Σ
b
fh
yh1
o
w
b 11
bh1
b
b
1
z2
1
b
nhj
zi
b
zN
b
Σ
fh
yhj
b
b
1
1
whNS
nhs
Σ
b
1
h
fh
yhs
b
woS L
bhs
nok
yo1
fo
yok
bok
noL
Σ
b
fo
bo1
Σ
bhj
b
no1
Σ
fo
yoL
boL
1
yh = f (Wh + bh )
yo = f o (Wo yh + bo )
Fig. 2: Feed-forward Neural Network diagram
paper uses multi-layer perceptron (MLP) neural models, which as proved in Cybenko (1989) and Funahashi (1989),
can represent continuous functions to any degree of accuracy, with at least one hidden layer, provided that the number
of neural units is sufficiently large. MLP is considered a universal approximator and has been efficiently used on-line
predictive control due to its natural capability for storing and generalising experience-based knowledge, (Fukuda and
Shibata, 1992; Ławryńczuk, 2007; Parisini and Zoppoli, 1995). A basic MLP consists in one input layer, one or more
hidden layers with several active neurons, and one output layer (see Fig. 2). Learning is a process through which free
parameters (i.e., synaptic weights whj , wok and bias levels bhj , bok ) of an ANN are adapted through a continuous process
of stimulation by the environment in which the network is embedded.
In this paper, three MLPs supervised training structures (see Fig. 3) with one hidden layer are devised in order
to forecast water demands and to plan future base-stocks and optimal economic references. Tan-sigmoid and linear
activation functions are used for the hidden and the output layers, respectively, following the results presented in
Yonaba et al. (2010). The training patterns (i.e., 8640) used in each ANN is divided randomly with a ratio of 60%,
20% and 20%, to form training, validation and testing sets, respectively. Networks with different number of active
neurons have been trained off-line in order to find the one with the smallest test data cost. The training of the ANNs is
done with the Gauss-Newton approximation to Bayesian regularisation (GNBR) algorithm developed in Dan Foresee
6
d(k)
T (k)
+
d̃(k) −
NND
RH(k)
ed (k)
BP
JE
t(k)
d(k)
NNE
b
x̃e (k) −
+ xe (k)
DWN
αk X, U
Economic
MPC
u(k)
ee (k)
BP
k+H p
d̃k
b
γ
x̃ s (k) −
NNS
BP
b
d̃(k)
−
+
ed (k)
+ x s (k)
τ
JE
D
Base
Stocks
u(k)
αk X, U
Economic
MPC
e s (k)
DB
e0:k
d
k+H p
d̃k
d(k)
Fig. 3: ANNs training diagrams for Demand Forecasting (top), Economic trajectory (middle) and Base-stocks setting
(bottom)
and Hagan (1997), which avoids overfitting and improve the generalisation ability of the neural models. This training
method performs the Bayesian regularisation employing the Levenberg-Marquardt back-propagation algorithm; it
ensures the ANNs provide accurate output values for inputs not represented in the calibration set of the neural network
and imposes constraints to get smaller weights and a smoother network.
3.2.1. ANN for Demand Forecasting
This module focuses on the problem of water demand forecasting for real-time operation of the DWN. An hourly
consumption data analysis is proposed here for training an MLP (see Fig. 3 top) with 100 neurons in the hidden
layer. The inputs to the neural model are chosen based on literature review (Babel and Shinde, 2011) and correlation
analysis, considering consumption data and meteorological variables such as temperature T and air relative humidity
RH. Principal component analysis (PCA) preprocessing is applied to the training patterns in order to reduce the
dimension of the input vectors and to obtain uncorrelated values that facilitate the learning process.
3.2.2. ANN for Optimal Economic Trajectory
The optimal economic trajectory of water volume in each tank is the one obtained considering only the objective
(2a) in Problem 1 related to a DWN, solving an LP constrained optimisation problem on-line. With the intention to
reduce computational effort, an MLP with 50 neurons in the hidden layer is trained off-line to emulate the economic
MPC controller. A non-linear explicit model of the optimal operation of the network is obtained to compute the
economic trajectory x̃e , taking into account the hourly electric tariff α2 and the measured demands d as inputs (see
Fig. 3 middle).
3.2.3. ANN for Dynamical Safety Volumes
An MLP with 50 neurons in the hidden layer is trained with input-output patterns that are generated using historical
records of demand forecasting errors as inputs, while using safety inventories as outputs. These targets have been
calculated for a service level γ = 95% and a lead-time τ = 4h, by the stock management methodology shown in
Subsection 2.2. The goal with this neural modelling is to avoid on-line optimisations which are required to virtually
decouple the tanks for the calculation of the safety stock vector in (18), and to use the non-linear explicit model of the
trained MLP to dynamically set the safe base-stock vector x̃ s (see Fig. 3 bottom).
7
Knowledge Base
Fuzzification
Rules-based Inference
Defuzzification
R1 (ēE , ēS )
R2 (ēE , ēS )
Rr (ēE , ēS )
ēE , e¯S
Linking
Heuristic
H p , We , W x , Wu
Fig. 4: Fuzzy rules-based MPC tuner diagram
Remark 3. The number of hidden layers and neuron units depend upon the complexity of the problem and the available
computational resources, i.e., both are design parameters.
♦
Remark 4. Neural models may not adapt to all characteristics of the environment, specially when there are uncertainties and no feedback correction mechanisms. The performance of ANNs in any application will be as good as the
scenarios and the quality of the training data set are. This lack of guarantees is the reason why this paper enhances a
constrained MPC with neuro-learning for tuning purposes instead of implementing a pure neuro-control.
♦
3.3. Supervision and Adaptation Layer
Self-tuning on-line algorithms for MPC of large-scale flow networks is not a widely reported topic in literature.
Most of the tuning strategies for the inherent multi-objective optimisation problems take into account the exploration
of the complete Pareto front to choose a non-dominated solution in line with the management objectives. The aim
behind the Pareto frontier applied to MPC of a DWN is to find a direct relation between the weights of the solution
points and the water demands taking d periodic (Toro et al., 2011).
To reduce computational complexity of common approaches, this paper presents an adaptation scheme, similar to
the one in Ali (2003), where fuzzy logic interacts with MPC by automatically adjusting the tuning parameters of the
controller based on the supervision of output feedback and measured disturbances.
3.3.1. Reasoning Mechanism
A knowledge-based tuner for the MPC controller is proposed here after a phase of experimentation and understanding of the effect that tuning parameters have on the controlled DWN. The tuning mechanism consists in a fuzzy
inference system (FIS) that involves: (i) a fuzzyfication interface, (ii) an a priori rule basis, (iii) a defuzzyfication
interface, and (iv) a linking heuristic (see Fig. 4).
In this paper, the fuzzyfication phase converts the normalised crisp values of the economic state error, which is
defined as ēE , |xk − x̃e |/x̃e ∈ Rn , and the safety volume state error, defined as ēS , (xk − x̃ s )/x̃ s ∈ Rn , into fuzzy
values. Indeed, to each input of the supervisor it is associated ni f s input fuzzy sets labelled with ni f s linguistic input
variables and described by specific Gaussian membership functions with values in the interval [0, 1]. The universe of
discourse for the input ēE is: small (S ), medium (M), large (L); and for the input ēS is: large-negative (LN), mediumnegative (MN), small-negative (S M), very-small-negative (VS N), very-small-positive (VS P), small-positive (S P),
medium-positive (MP), large-positive (LP).
The fuzzy-logic supervisor involves expert knowledge using the Mamdani’s fuzzy inference system, which applies
the set of linguistic rules presented in Table 1 to the fuzzy inputs in order to evaluate the supervisor fuzzy outputs. The
inference table associates no f s output fuzzy sets (no f s linguistic output variables), described by specific trapezoidal
membership functions, to the supervisor crisp output variables H̃ p , W̃e , W̃ x , W̃u ∈ Rn . The universe of discourse for
output H̃ p is: small (S ), medium (M), large (L); and for the outputs W̃e , W̃ x , W̃u is: very small (VS ), small (S ),
medium (M), large (L). The min-max inference method is used for the evaluation of the fuzzy rules contribution, and
the gravity center method is used in the defuzzyfication phase. For a detailed explanation on fuzzy inference reasoning,
the reader could refer to Wata et al. (2000).
8
Table 1: Fuzzy-logic rules for LB-MPC.
Rule
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Algorithm 1
Inputs
Outputs
eE
eS
H̃ p
W̃e
W̃x
W̃u
L
L
L
L
M
M
M
M
S
S
S
S
any
any
any
any
LP
MP
SP
VSP
LP
MP
SP
VSP
LP
MP
SP
VSP
LN
MN
SN
VSN
L
L
L
L
M
M
M
M
S
S
S
S
L
L
L
L
L
L
M
S
L
L
M
S
L
M
M
S
VS
VS
VS
VS
VS
VS
VS
S
VS
VS
VS
S
VS
VS
VS
S
L
L
L
L
VS
S
M
L
VS
S
M
L
VS
S
M
L
VS
VS
VS
VS
Linking Heuristic
1: procedure LinkingHeuristic(W̃e , W̃x , W̃u , H̃ p )
2:
m := number of actuators
3:
n := number of tanks
4:
we ← 0m×1
5:
wu ← 0m×1
6:
for {i = 1 → m} do
7:
if {u(i) connects node-node or tank-demand} then
8:
we (i) ← 0
9:
wu (i) ← 0
10:
else
11:
if {u(i) connects tank-tank} then
12:
we (i) ← max{W̃e (tank1 ), W̃e (tank2 )}
13:
wu (i) ← max{W̃u (tank1 ), W̃u (tank2 )}
14:
else
15:
we (i) ← W̃e
16:
wu (i) ← W̃u
17:
end if
18:
end if
19:
end for
20:
We ← Rm×m diagonal matrix whose diagonal is we
21:
Wu ← Rm×m diagonal matrix whose diagonal is wu
22:
Wx ← Rn×n diagonal matrix whose diagonal is W̃x
23:
H p ← max{H̃ p }
24:
return H p , We , Wx , Wu
25: end procedure
Remark 5. The fuzzy sets and the logic rules of the fuzzy inference mechanism depend on the application and the
desired response of the plant. They are assigned a priori based on the experience the engineer has acquired from the
system operation and also on the performance requirements. The practicing engineer may improve the performance
of the controller by updating the knowledge base (i.e., logic rules and fuzzy sets).
♦
Remark 6. Outputs from the FIS are in Rn since they are computed for every tank in the network. Therefore, it is
necessary to transform the results of weights W̃e and W̃u into an actuator base in Rm×m and the results for H̃ p into an
integer value in N. The weight W̃ x is already related to states but have to be transformed into a diagonal matrix in
Rn×n .
♦
3.3.2. Linking Heuristic
To obtain the quantitative value of the tuned parameters in the corresponding domain for the MPC problem stated
in Section 2, an associative heuristic is proposed in Algorithm 1 based on the interconnection of elements in the DWN.
9
Fig. 5: Case Study: Aggregate model of the Barcelona DWN
Remark 7. Most of the tuning guidelines for MPC controllers reported in literature assume weighting matrices with
equal elements in the diagonal. The learning-based tuning approach presented in this paper allows to adapt on-line
individual elements of the cost function weighting matrices giving more degrees of freedom to the managers of the
network.
♦
4. Simulation and Results
In this section, simulation results are presented. The selected case study is, without loss of generality, an aggregate
and representative version of the entire Barcelona DWN. In this aggregate model, some consumer demand sectors of
the network are concentrated in a single point. Similarly, some tanks are aggregated in a single element and the
respective actuators are considered as a single pumping station or valve (Ocampo-Martinez et al., 2009, 2010). The
model consists of 17 tanks, 61 actuators, 25 measured demands and 11 nodes (see Fig. 5). All the simulations have
considered a time period of four days (96 hours). The selected sampling time is one hour. Simulations have been
TM
carried out using the CPLEX solver of the TOMLABR 7.6 optimisation package, Fuzzy Logic Toolbox and Neural
TM
TM
Network Toolbox of MatlabR R2010b (64 bits). The computer used to run the simulations is a PC IntelR Core
E8600 running both cores at 3.33GHz with 8GB of RAM.
4.1. Demand Forecasting and States Planning with ANNs
Table 2 shows the mean absolute percentage error (MAE%) of the ANNs described in Subsection 3.2, that were
TM
trained using the function trainrb included in the Neural Network Toolbox of MatlabR . The final architecture of
10
Table 2: Performance of the ANNs (MAE%)
Neurons
10
20
30
40
50
60
70
80
90
100
110
ANNd
8.8299
7.1257
10.9843
7.8918
8.6294
11.2109
11.3223
7.4649
14.0614
6.6945
14.8369
ANNe
629.4652
87.2827
89.3765
1.4455
1.4098
1.4263
1.4788
−
−
−
−
ANNs
3.5512
2.0423
1.5513
1.4124
1.0376
1.0323
1.0206
−
−
−
−
each ANN was selected by comparing the performance of experiments varying the number of neurons in the hidden
layer. The selected number of neurons were 100, 50 and 50 for the ANNd , ANNe , and ANN s , respectively. Forecasting
of demand with the ANNd is based on water consumption and meteorological (temperature and air relative humidity)
records available from AGBAR Company and from the Servei de Meteorologia de Catalunya, respectively. The results
show the prediction of water demand is reliable and the magnitude of the forecasting error is not a reason to reject the
obtained model. In fact, these minimal discrepancies are reflected in an increase of the safety volume. In contrast to
ANNd , the ANNe for the economic optimal trajectory and the ANN s for the safety volume trajectory perform with
higher accuracy as it could be expected, because both MLPs are trained with the solution of quadratic programming
(QP) problems instead of experimental driven data as in the forecasting demand case. The advantage of all of the
neural models used in this paper is that the time invested in processing data and training the ANNs will be gained in
the on-line solving process once they are accurately validated and tested.
4.2. Fuzzy Tuning of MPC Parameters
The tuning parameters of Problem 1 are the prediction horizon H p and the weights We , W x , Wu . These parameters
TM
were computed and adapted by the FIS described in Subsection 3.3 using the Fuzzy Logic Toolbox of MatlabR .
Histograms of their values are shown in Fig. 6. These results were obtained for an unmeasured random disturbance
of at most 20% of the nominal demand pattern.
Remark 8. In most of the results presented in literature for control of DWNs, tuning is focused on the weighting
matrices and most of the times with no adaptation schemes. Nevertheless, for large-scale systems, an efficient selection
of the prediction horizon is demanded because the size and complexity of the optimisation problem is based mainly on
this parameter. Therefore, this paper adapts the prediction horizon according to a trade-off between risk and economic
cost.
♦
4.3. LB-MPC Controller for DWN
The proposed learning-based approach has been implemented for the tuning of a constrained MPC to operate the
aggregate model of the Barcelona DWN. Results have been compared with two previous strategies. The controllers
are the following ones:
• MPCo: the baseline MPC approach presented in Pascual et al. (2011), which uses fixed prediction and control
horizons (24h), constant safety water stocks and constant tuning weights for the prioritisation of management
objectives.
• MPCss: a bi-level MPC which implements analytically the dynamic optimisation of safety stocks following
Section 2.2. It considers fixed horizons (24h) and fixed weights as well.
11
Hp Histogram
We Histogram
80
20
Frequency
25
Frequency
100
60
40
15
10
20
0
5
0
5
10
15
Hp: Prediction Horizon
0
50
20
25
25
20
20
15
10
15
10
5
5
0
90
Wu Histogram
30
Frequency
Frequency
Wx Histogram
60
70
80
We: Economic Weight
5
10
15
Wx: Safety Weight
0
20
0
10
20
30
Wu: Smoothness Weight
40
Fig. 6: Histograms of tuning parameters for the LB-MPC
Table 3: Key performance indicators for the different approaches
Controller
MPCo
MPCss
LB-MPC
KPIE (103 )
183.74
176.77
178.99
KPIS
28.8022
5.0295
5.2138
KPIU
0.1318
0.1340
0.1172
Time [s]
142.01
286.17
132.91
• LB-MPC: the learning-based MPC addressed in Section 3, with adaptive horizons, tuning weights and safety
stocks.
Table 3 shows the specific key performance indicators (KPI) used to assess the aforementioned controllers over
the simulation period (96h). The indicators are defined as follows:
KPIE ,
96
X
k(α1 + α2 (k))T u(k)k1 ∆t,
(19)
k=1
KPIS ,
96 X
17
X
εn (k),
(20)
k∆u(k)k2 ,
(21)
k=1 n=1
KPIU ,
96
X
k=1
where KPIE is the total economic cost of the DWN operation, KPIS is the addition of all safety level violations and
KPIU is the accumulated RMS of control variations.
Simulations show that static MPC design parameters (safety stocks, tuning weights and prediction horizon) are a
drawback for the management of DWNs, because under uncertain disturbances, the fixed value of these parameters
12
Tank # 12
4
x 10
10
x
x
8
x
3
LB−MPC
s,MPCo
xs,MPCss
7
Volume [m ]
MPCo
xMPCss
9
x
6
s,LB−MPC
dnet
5
xmax
x
4
min
3
2
1
0
0
24
48
72
96
Time [h]
Fig. 7: Dynamic variation of tanks volumes for the different approaches
Economic units [e.u.]
x 10
Daily Electric Costs
2.15
2.1
MPCo
MPCss
LB−MPC
2.05
1
2
Daily Water Costs
4
3
Economic units [e.u.]
4
2.2
3
x 10
2.5
2
1
1
4
Time [days]
MPCo
MPCss
LB−MPC
1.5
2
3
4
Time [days]
Fig. 8: Comparison of the daily electric and water costs for the different approaches
might cause an undesired restriction of the solution space that degrades the economic performance, and/or increment
the risk of constraint violation. In Fig. 7 it is shown the excursion of water in a tank of an important sector of
the Barcelona DWN, which behaviour is representative of most storages in the DWN. It can be seen that all of the
compared MPC controllers keep the volume in tank within the hard and soft constraints satisfying also the net demand
along the simulation horizon but with differences in the computational burden (see Table 3) and the management of
safety stocks which impacts the aforementioned KPIs. As expected, the safety stocks of the LB-MPC controller tend
to reproduce the ones computed with the MPCss controller because this latter was used to train the ANN that predicts
the safety in the LB-MPC.
The MPCo controller presents the highest economic cost due to the conservative and static safety volumes that
limits the economic optimisation. In general, this approach do not guarantee optimal results for any condition because
the safety is fixed heuristically without taking into account demand variations. Instead, the MPCss controller is robust
to disturbances by optimising the dynamic safety stocks (see Fig. 7) in accordance to the deviation of the forecasting
error. The MPCss presents the best economic performance but the highest computational effort (see Table 3) since it
involves more on-line optimisation problems to set those safety stocks.
Results show that the LB-MPC controller outperforms the previous strategies. It presents similar results to the
MPCss for the economic, safety and smoothness indicators but reduces the computational burden (see Table 3). In
addition, Fig. 7 and Fig. 8 show that, despite the similar safety stocks of the MPCss and the LB-MPC, this latter makes
13
better use of hydraulic and economic resources due to its flexibility to self-adapt the parameters of the controller if
the operational conditions change (see Fig. 6). This capability helps managers to deal with demand uncertainty and
prediction errors in an optimal and economic way, guaranteeing the desired safety and service level with less volume
of water and less electric energy. Furthermore, Fig. 8 details that even when the MPCss approach have a lower value
than the LB-MPC for the economic performance indicator (see Table 3), which integrates water costs and electric
costs, the LB-MPC presents a lower cost in the electric component due to the adaptation of weights by means of the
AI techniques used in this application. From an operational point of view, this difference in costs implies that the
LB-MPC decides to take water from more expensive sources in order to further reduce the electrical costs by pumping
more, with respect to the other approaches, when the electric tariff is cheaper.
5. Concluding Remarks
This paper has presented a Multilayer Model Predictive Controller with self-tuning capabilities for the efficient
management of water transport systems based on soft-computing techniques. The resultant controller architecture
has been applied to an aggregate model of the Barcelona DWN obtaining important improvements in the computation
time towards on-line implementation for large scale systems. The selected parameters to be tuned in the MPC problem
were the prediction horizon and the weighting matrices of the multi-objective cost function. The main advantage of
the fuzzy tuner is that it is able to tune every element independently, which is a difficult task in analytical approaches
due their lack of intuitiveness for multivariable large-scale systems. The proposed scheme is a quasi-explicit MPC
because most of the heavy computational tasks are converted into non-linear explicit modules using neural networks.
The controller also tunes the set-points based on inventory management theory, enriching the controller design with
reliability aspects to assure a customer service level under disturbances uncertainty. Further research will be conducted
in using other ANNs structures and in reinforcement learning algorithms with two main directions: (i) to implement
intelligent distributed MPC of DWNs where shared variables are negotiated using learning techniques, and (ii) to
adapt and improve the presented fuzzy inference system.
6. Acknowledgements
This work has been partially supported by the Spanish research project WATMAN (CICYT DPI2009-13744) of
the Science and Technology Ministry, the EU Project EFFINET (FP7-ICT-2011-8-31855) and the DGR of Generalitat
de Catalunya (SAC group Ref. 2009/SGR/1491).
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15
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