1. Introduction
One of the main benefits and motivations for introducing advanced control systems is a more economic operation of plants and processes. The most widespread solution for achieving this goal is to use a two-layer hierarchy architecture for economic optimal process management [
1]. The first layer, often referred to as real-time optimization (RTO), determines the economically optimal operation point by solving a steady-state economic optimization of the system variables. This operation point is typically updated on a time scale of hours or days. The RTO sends the results of its optimization as a set-point to the second layer, usually referred to as the
(advanced) control layer. This control layer is designed to steer the plant’s state to the set-point while ensuring the satisfaction of the operation is suitable according to the management policies in the presence of model mismatches and disturbances. This process control layer often exploits
Model Predictive Control (MPC) because of its flexibility, performance, robustness, and its ability to directly handle hard constraints on both inputs and states [
2]. MPC has been extensively studied and successfully applied in many real-life industrial applications; nowadays, it is a widespread and established advanced control strategy [
3,
4]. The objective of the MPC-based advanced control layer is usually to achieve asymptotic tracking of set-point changes, minimizing the effect of the disturbances over the closed-loop system performance [
5].
While this two-layer approach has been demonstrated as a successful control technique in many industrial applications, the hierarchical separation of economic analysis and control is either inefficient or inappropriate due to the slow reaction to disturbances and the mismatches between the models used in each layer. This fact has motivated the question of improving the economic performance of the controlled systems by integrating the economic optimization into the dynamic control layer [
6], leading to what is known as
Economic Model Predictive Control (EMPC). EMPC is a variant of MPC that directly optimizes an economic performance index instead of a tracking error [
1]. This line of work is of particular interest for critical infrastructure systems because the operation of the system is guided by the cost of energy that varies along the day and the user demands that present periodic behavior. For example, in [
7,
8], an EMPC has been used to reduce the energy consumed in drinking water distribution networks, while in [
9,
10], the benefits of real-time optimization in the management of energy grids have been demonstrated. Theoretical analysis of EMPC with Lyapunov-based stability proofs can be found in, e.g., [
1,
11]. Other control approaches also seek to improve performance, taking into account the economic perspective but using alternative control strategies (see, e.g., [
12]).
Different solutions have been proposed to enhance the economic performance of the system, e.g.,
- (i)
by adding a steady-state target optimization layer between the RTO and the MPC [
6],
- (ii)
by considering the dynamics of the system in the real-time optimization stage by replacing the RTO by a Dynamic RTO (DRTO) [
13], or
- (iii)
by moving economic information into the control layer, where the control problem is posed as an optimization problem, similar to MPC [
14,
15].
As a result, the controller directly and dynamically optimizes the economic operating cost of the process without reference to any steady state. However, the resulting optimization problem typically involves multiple objectives that are typically combined in a
weighted sum, without considering this multi-objective nature. However, the tuning of these controllers, i.e., selecting appropriate weights, is often non-trivial. This is especially the case when the different objectives are incommensurable or price information is only inaccurately known or fluctuates [
16,
17]. In the related literature, several MPC-tuning approaches have been proposed for linear (see, e.g., [
18,
19,
20]) and non-linear model predictive controllers (see, e.g., [
17,
21,
22]) in the case of standard tracking formulation. For the case of economic formulation, recently, some methods have started to appear, e.g., [
23], that propose the use of evolutionary game theory in order to complement an MPC approach for finding a management region where the weights are determined by using fitness functions. In [
24], it is proposed a formal procedure that tunes a tracking MPC scheme so that it is first-order equivalent to a scheme based on EMPC.
In the current paper, several tuning methods for EMPC based on multi-objective optimization tools are proposed and applied to a Drinking Water Transport Network (DWTN). The proposed EMPC formulation seeks for the complementarity among the proposed control objectives (terms into the multi-objective cost function) in such a way that the operation of the critical infrastructure was not exclusively handled by the cost of the electric energy but by other particular interests given by the managers of the related DWTN. The proposed tuning methods are based on an offline learning and an online operation phase. During the offline training phase, Pareto sets with trade-off solutions are computed and preferred solutions and the corresponding Weighted Sum weights are selected. In order to avoid the calculation of Pareto sets during the online operation phase of the controller, two approaches are evaluated. The first and simplest is based on a histogram that helps to find out the most selected weight combinations. The second is based on a regression model between the measured disturbances and the weight combination. The proposed methods are suitable for dealing with disturbances which are not only time-varying but also periodic. Consequently, they allow to obtain sequences of tuning factors according to measured disturbances. The first main objective is to explore the Pareto optimal solutions for the EMPC strategy with its multiple objectives, and to choose a solution in line with the management objectives of the control problem. The second main objective behind the Pareto front calculation is to look for a direct relation between the weights of the solution points and the measured disturbances of the control problem in order to derive an adaptive tuning strategy for the online EMPC implementation.
The main contributions of this work are twofold: (i) to highlight that the Pareto front is not static as disturbances change the EMPC problem constantly (hence, it is necessary to adjust the controller continuously) and (ii) to note that the tuning of the controller is explicitly related to the disturbances.
The remainder of this paper is organized as follows. In
Section 2, the general problem statement of the EMPC for DWTN is presented and formulated. In
Section 3, methods to calculate the Pareto front of an EMPC controller in view of its different objectives are presented. In
Section 4, strategies to tune the EMPC’s weighting factors are discussed. In
Section 5, the case study is described and the main simulation results are presented and discussed. Finally, in
Section 6, the most relevant conclusions as well as further paths for future research are drawn.
3. Pareto Front Calculation of Multi-Objective Optimization Problems
As mentioned, the EMPC described in the previous section can be regarded as a multi-objective optimization (MOO) problem of the form
subject to
Here,
y represent the decision variables, which, for the MOEMPC, are the sequences
. Each
denotes an individual
objective function, which are all grouped into the
cost vector. In the MOEMPC, these
correspond to the functions
. The vector
and vector
,…,
represent the
inequality and
equality constraints, respectively. In the MOEMPC setting, these relate to the constraints (
2b)–(2d). The
feasible decision space is
and its mapping into the cost space yields the
feasible cost space . In MOO, typically no single optimal solution exists except a set of optimal solutions following the Pareto optimality concept [
27].
Definition 1. A point is Pareto optimal if no other point exists , such that for all i and for at least one objective function j.
Moreover, the following additional concepts are introduced considering a minimization framework: the minimizer of the i-th cost function , the utopia point, which contains the minima of the individual objective functions , and the individual minima cost vectors , which is the cost evaluated for the individual minimizer . The approximated nadir point contains the worst value for each objective obtained from the individual minima cost vectors with . Using the individual minimizers as anchor points, the pay-off matrix contains, in its i-th column, the vector . Alternatively, when using the pseudo-anchor points , the pay-off matrix has, in its i-th column, the vector .
Finally, in order to obtain an approximation of the Pareto set, a scalarization approach can be used [
27]. The original MOO problem (2) is converted into a parametric single optimization problem. By solving this problem for different values of the scalarization parameters, a part of the Pareto front is obtained. Several scalarization methods exist in the literature.
3.1. Normalization
The first step when scalarizing the MOO problem is to normalize the different objective functions in order to avoid scaling deficiencies. Then, the optimization is performed in the normalized space. Normalization can be achieved by first shifting the objectives such that the utopia point coincides with the origin and afterwards pre-multiplying them with a matrix
, i.e.,
When considering only the shifting and scaling of the individual objectives, the matrix
is diagonal with elements
where
and the
are the approximated nadir point and the utopia point, respectively. Alternatively, the objectives can be mapped to the corners of a unit hypercube by using a matrix
as follows:
with
a matrix containing zeros on the diagonal and ones on the off-diagonal.
3.2. (Normalized) Weighted Sum (WS)
The most widely scalarization method is based on formulating a Weighted Sum of different terms as
where
w is the vector of
scalarization parameters or often called
weights with
and
. In this paper, a Normalized Weighted Sum (NWS) approach is obtained when the weighted sum scalarization approach is applied to a normalized multi-objective optimization problem based on the pay-off matrix with pseudo-anchor points. To obtain an approximation of the Pareto set, the weight parameters can be varied.
3.3. (Enhanced) Normalized Normal Constraint ((E)NNC)
(E)NNC reformulates the MOO problem in an alternative way, as [
28,
29]
subject to
with
and
as scalarization parameters. Here, indicate normalized variables. The rationale is to minimize the single most important objective
(
14a), while reducing the feasible cost space by adding
hyperplanes (
14b) that are orthogonal to the plane through the (normalized) individual minima. The normalization can be achieved using (
10), resulting in the traditional NNC, or using the linear transformation (
12) with either the individual minimizers or the pseudo anchor-points as anchors points, yielding Enhanced Normalized Normal Constraint (ENNC) and Enhanced Normalized Normal Constraint with Pseudo-Anchor points (ENNCP), respectively. Again varying the weights leads to an approximation of the Pareto set.