1. Introduction
The real-time control and supervision of drinking water networks (DWNs) is a field of increased interest given the environmental, economic and social impact [
1]. DWNs are critical infrastructures in urban environments. These networks provide important services in modern society and maintaining the service availability is an important requirement. Therefore, reliability and resilience are important properties to be guaranteed in DWNs while being subject to constraints and continuously varying conditions of probabilistic nature [
2]. DWNs are multivariate dynamic constrained systems that are described by the interconnection of several subsystems (tanks, actuators, sources, nodes and consumer sectors). Moreover, DWN optimal management is a complex challenge for water utilities that can be addressed as a multi-objective optimization problem. This problem can be solved online using a Model Predictive Control (MPC) scheme [
3].
Generally, the structure of the MPC approach follows a moving horizon strategy. The control action is obtained solving an optimal control problem that provides a control action sequence in a prediction horizon that minimizes the considered control objectives and satisfies the set of constraints including the system model and physical/operational limitations. Therefore, MPC can provide suitable strategies to achieve the DWN operational control improving their performance, as it allows computing optimal control approaches ahead of time for all the pressure and flow control elements [
4]. Revising the literature, different approaches can be found that show the benefits of the optimal DWN management. In [
5,
6,
7], by optimizing a mathematical function that considers operational goals in a specific time horizon and using a model of the network dynamics and demand forecasts, optimal strategies are computed. These references also assumed predicted disturbances as defined in the model, but involve a soft constraint to penalize evacuation of water volume below a heuristic safety threshold without forcing any target regulation. Regarding optimised control strategies for managing water systems, MPC is not implemented in a classical way, as there is no reference volume to be tracked [
8]. The standard MPC forces the system to follow the set point, but it does not guarantee that the system evolution toward the set points is obtained in an economic efficient way. The general aim in the operation of several process industries, as, e.g., DWNs, is the reduction of costs associated to the consumption of energy, which is not the main goal of standard MPC. For this purpose, Economic MPC (EMPC) provides a systematic method for the optimization of economic system operation [
9]. The problem of optimization associated to the EMPC strategy aims at obtaining a family of optimal set points considering economic efficiency rather than aiming that the controlled system reach a certain set point [
9].
The use of control strategies that take into account the system and component reliability that guarantee the quality of service is necessary. The health monitoring of the actuator and system should be considered for increasing the system reliability, minimising the fault appearance and reducing the operational costs. In the later stages, system reliability in the process of control system has been considered using a Prognosis and Health Management (PHM) framework. This is because reliability is a standard method for evaluating how long the system will achieve its function without malfunctions. Moreover, it can be used to predict future damages in the system according to the health state of its components [
10].
In the past few years, the problems of system reliability and actuator lifetime in service has received considerable interest for the researcher community. In [
11], to decrease the maintenance cost, the actuator lifetime is regarded as a controlled parameter that is considered as additional goal when using a linear quadratic optimal controller. On the other side, MPC predicts the suitable control actions to obtain optimal performance according to multi-objective cost functions and physical constraints, and therefore it can be considered as a suitable approach for developing health-aware control schemes. An MPC strategy based on distributing the loads among redundant actuators is introduced in [
12], while forcing constraints to guarantee that the accumulated actuator degradation will not arrive at the unsafe level at the end of the prediction horizon. In [
13], the authors proposed a health-aware MPC controller that incorporates a fatigue-based prognosis into MPC to minimize the component damage. Most of the other methods that consider component health and system reliability management stand within the structure of fault-tolerant control or in the area of preservation scheduling see, e.g., Gallestey et al. [
14], Khelassi et al. [
15], Salazar et al. [
16] and references therein. However, none of these methods consider uncertainty.
The reliability is the system’s ability (or component) to carry out its expected functions. The reliability of DWN is influenced by different conditions such as the capacity and the quality of the water accessible at the sources and the pump/pipe failure rates [
17,
18]. In most of the works, the actuator reliability is assumed that follows an exponential distribution that varies with the control action [
19]. The system reliability is characterised according to the interdependence topology based on the combining of each actuator reliability. Subsequently, the system reliability has a demonstrative relationship with the control input that leads to a nonlinear mathematical model. In several studies, this is achieved by including a damage index in the optimization problem and establishing a trade-off by weight tuning [
20] or by imposing constraints with respect to the actuator reliability [
17]. However, considering the reliability at the actuator level not at the system level is the main drawback of the previous methods; otherwise, it leads to the use of nonlinear MPC according to nonlinearity of the resulting constraints. Generally, Economic Nonlinear MPC (ENMPC) implies a high computational cost and, the existing gradient-based numerical algorithms do not certify that the obtained solution corresponds to the global one because of the non-convexity of the associated optimization problem. Transforming the nonlinear optimization problem into a quadratic problem through a linearisation method is one way of addressing the non-convexity problem and guaranteeing a unique optimum. In this way, the system is modelled by an incremental model because the model has to be linearised at each iteration. This approach has been improved by means of of the use Linear Parameter Varying (LPV) models that do not require linearisation [
21]. The LPV models can describe both nonlinear phenomena and time-varying that can be estimated/measured online.
Another weakness of previous approaches combining reliability analysis and MPC is the conservatism of the resulting control strategies, which affects negatively the efficient DWN operation. Furthermore, in real applications, the assumption of bounded disturbances in real applications is not always satisfied. Thus, constraint violations can not be avoided because of the appearance of faults, unexpected events, etc. A more realistic representation of uncertainty is based on using the stochastic approach that leads to less conservative control methods by incorporating explicit disturbance models in the control design and by converting hard constraints into probabilistic constraints. The stochastic approach is a sophisticated theory in the field of optimization, but a revived consideration has been provided to the stochastic programming methods as powerful tools for the design of controllers, leading to the stochastic MPC, which has a particular alternative called chance-constrained MPC (CC-MPC) [
22,
23]. The stochastic control approach that represents robustness in terms of probabilistic (chance) constraints, which need that the probability of violation of any operational condition or physical constraint is under a designated value. By placing this value suitably, the user/operator can obtained the desired trade-off between robustness and performance. For related works that proposed the CC-MPC approach in water networks the reader is referred to [
24,
25]. Some economic-oriented controller that consider the reliability issue has been proposed [
20], but without considering reliability at the system level and probabilistic constraints based on the reliability of the system.
The aim of this paper is to include in an EMPC strategy for DWN an additional objective that takes into account PHM information obtained by the online evaluation of the system reliability. The system reliability is incorporated into the control algorithm by using an augmented model that includes both the reliability and DWN models. As the reliability model of the whole DWN is nonlinear, its model is expressed as an LPV model such that at each time instant the varying parameters are updated according to the value of the scheduling variables. This allows to solve the optimization MPC problem associated to the health-aware approach using quadratic programming instead of nonlinear programming. Considering the probabilistic nature of system reliability, it is included in the MPC optimization problem in the form probabilistic constraints as the demands (disturbances) using the chance constraints programming paradigm. The resulting control inputs obtained by the proposed health-aware MPC approach are able to achieve the economic control objectives and simultaneous to increase the lifespan and reliability of the system components.
Chance-constraints programming allow to determine an optimal strategy by establishing the desired level of infeasibility and system reliability. Moreover, it allows considering the system reliability, which is assessed online using an LPV-MPC strategy; representing the main contribution of this paper. The second contribution is to propose an advanced health-aware LPV-MPC approach that formulates a quadratic optimization problem taking into account the functional dependency of scheduling variables and state vector. This approach avoids the use of nonlinear optimization. Moreover, it uses chance constraints programming to manage dynamically designate safety stocks in flow-based networks to satisfy nonstationary flow demands and system reliability.
The structure of the paper is as follows. The control-oriented model considered for DWN when considering the transportation layer is introduced in
Section 2.
Section 3 presents the chance-constraints programming and the way to use it into the MPC controller. The system reliability modeling and the relationship between reliability and chance constricted are described in
Section 4. In
Section 5, the economic reliability-aware MPC-LPV including chance-constraints programming is provided. The results of the application of the proposed control strategy to the DWN network using the proposed case study are analyzed and summarized in
Section 6. Finally, the conclusions and research future paths are presented in
Section 7.
Notation: Throughout this paper indicate the field of real numbers, the set of non-negative real numbers, the set of column real vectors of length n, and the set of m by n real matrices, respectively. Equivalently, presents the set of non-negative integer numbers including zero. Define the set for some and for some . The operator ⊕ is direct sum of matrices (block diagonal concatenation). Furthermore, denotes the spectral norm for matrices and is the squared 2-norm symbol. The superscript ⊤ represents the transpose and operators indicate element-wise relations of vectors.
3. Chance-Constrained Model Predictive Control
If the stochastic nature of disturbances (demands) and reliability of components of the system is not explicitly considered, an optimal solution of (10) satisfying all constraints can not be found in real scenarios. Therefore, to guarantee feasibility of the optimization problem (10), it is appropriate to relax the original constraints that involve stochastic elements with probabilistic statements in the form of chance constraints. In this manner, the constraints are needed to be satisfied with predefined risk levels to manage the uncertainty and component reliability of the system. Chance-constrained programming is a technique of stochastic programming dealing with constraints of the general form as
where
indicates the probability operator,
is the decision vector,
a random variable, and
a constraint mapping. The level
is user given and defines the preference for safety of the decision
v. The constraint (
11) means that we wish to take a decision
v that satisfies the
-dimensional random inequality system
with high enough probability. As demonstrated in [
29], if
is jointly convex in
and
is quasi-concave, then the feasible set
is convex for all
. All chance-constrained models need prior knowledge of the acceptable risk
connected with the constraints. A lower risk acceptability proposes a harder constraint. In general, joint chance constraints lack from analytic expressions because of the involving multivariate probability distribution [
30]. In this paper, by following the results in [
30,
31], a uniform distribution of the joint risk is approximated by upper bounding the joint constraint and assuming a similar distribution of the joint risk among a set of individual chance constraints are transformed inside equivalent deterministic constraints.
Consider the general joint chance constraint (
11), and define
with
. Therefore, the additive stochastic element is separable and the following chance constraint is achieved,
Then, by rewiring
, for any duple
, it follows that
Describing the events
(as e.g., faults in the actuators or unexpected changes in the demand), it follows that
Indicating the complements of the events
by
, and it is obvious from probability theory that
and consequently
By using the union bound, the Boole inequality let to bound the result in (
17a), declaring that for a countable set of events, the probability that at least one event occurs is not higher than the sum of the individual probabilities [
32], such that
and, by applying (
18) to (
17a), it yields to
Then, a set of constraints rises from previous results as sufficient conditions to enforce the joint chance constraint (
13), by allotting the joint risk
in
separate risks
These constraints are described as follows,
where (
20a) produces the set of
effective individual chance constraints, which bounds the probability that each inequality of the receding horizon problem could not be satisfied. Moreover, (
20b) and (
20c) are conditions forced to bound the new single risks in such a way that the joint risk bound is not breached. Each solution that satisfies the aforesaid constraints is guaranteed to provide (
13).
According to the satisfaction of each individual constraint is an event
. A joint chance constraint needs that the connection of all the individual constraints is satisfied with the wanted probability level, such as
Considering that each individual constraint is probabilistically dependent, the level of conservatism can be derived by using the inclusion–exclusion principle for the union of finite events,
, which proves the following equality,
Note that by considering as an event a fault in an actuator, it can be observed that Equation (
22) has a similar as formulation as the one used for evaluating the system reliability based on the component reliability.
In a DWN, the constraints come from models (10b) and (10c) that can be formulated as chance constraints statements taking into account the probabilities associated to the component reliability. Considering only faults in actuators, the reliability of the system is related to the system inputs
. Therefore, (
11) can be formulated in case of the actuators as follows,
where
is a stochastic variable which considers if the actuator is one of two states
(or
) defined as follows,
where
is the actuator reliability. In the case that
, the input
associated to the
i-th actuator is bounded by (2b); otherwise, an additional constraint setting
should be included. Furthermore, to determine the reliability associated to the system that associates a probability to the system model constraint (1), the joint-chance constraint probability calculation (
22) should be used leading to the following probabilistic formulation for the MPC optimization problem (10),
subject to
The main difficulty in solving this stochastic problem using chance constraints is that at each time iteration, the probabilities associated to the system reliability should be updated taking into account the value of the optimal control actions . In the following section, a solution procedure is proposed to solve this problem.