A unified framework for model-based multi-objective linear process and energy optimisation under uncertainty

Process and energy models provide an invaluable tool for design, analysis and optimisation. These models are usually based upon a number of assumptions, simplifications and approximations, thereby introducing uncertainty in the model predictions. Making model based optimal decisions under uncertainty is therefore a challenging task. This issue is further exacerbated when more than one objective is to be optimised simultaneously, resulting in a Multi-Objective Optimisation (MO2MO2) problem. Even though, some methods have been proposed for MO2MO2 problems under uncertainty, two separate optimisation techniques are employed; one to address the multi-objective aspect and another to take into account uncertainty. In the present work, we propose a unified optimisation framework for linear MO2MO2 problems, in which the uncertainty and the multiple objectives are modelled as varying parameters. The MO2MO2 under uncertainty problem (MO2U2)(MO2U2) is thus reformulated and solved as a multi-parametric programming problem. The solution of the multi-parametric programming problem provides the optimal solution as a set of parametric profiles

Game theoretic optimisation in process and energy systems engineering: A review

Game theory is a framework that has been used by various research fields in order to represent dynamic correlation among stakeholders. Traditionally, research within the process and energy systems engineering community has focused on the development of centralised decision making schemes. In the recent years, decentralised decision-making schemes have attracted increasing attention due to their ability to capture multi-stakeholder dynamics in a more accurate manner. In this article, we survey how centralised and decentralised decision making has been facilitated by game theoretic approaches. We focus on the deployment of such methods in process systems engineering problems and review applications related to supply chain optimisation problems, design and operations, and energy systems optimisation. Finally, we analyse different game structures based on the degree of cooperation and how fairness criteria can be employed to find fair payoff allocations

Multi-parametric mixed integer linear programming under global uncertainty

Major application areas of the process systems engineering, such as hybrid control, scheduling and synthesis can be formulated as mixed integer linear programming (MILP) problems and are naturally susceptible to uncertainty. Multi-parametric programming theory forms an active field of research and has proven to provide invaluable tools for decision making under uncertainty. While uncertainty in the right-hand side (RHS) and in the objective function's coefficients (OFC) have been thoroughly studied in the literature, the case of left-hand side (LHS) uncertainty has attracted significantly less attention mainly because of the computational implications that arise in such a problem. In the present work, we propose a novel algorithm for the analytical solution of multi-parametric MILP (mp-MILP) problems under global uncertainty, i.e. RHS, OFC and LHS. The exact explicit solutions and the corresponding regions of the parametric space are computed while a number of case studies illustrates the merits of the proposed algorithm

Multi-parametric linear programming under global uncertainty

Multi-parametric programming has proven to be an invaluable tool for optimisation under uncertainty. Despite the theoretical developments in this area, the ability to handle uncertain parameters on the left-hand side remains limited and as a result, hybrid, or approximate solution strategies have been proposed in the literature. In this work, a new algorithm is introduced for the exact solution of multi-parametric linear programming problems with simultaneous variations in the objective function's coefficients, the right-hand side and the left-hand side of the constraints. The proposed methodology is based on the analytical solution of the system of equations derived from the first order Karush–Kuhn–Tucker conditions for general linear programming problems using symbolic manipulation. Emphasis is given on the ability of the proposed methodology to handle efficiently the LHS uncertainty by computing exactly the corresponding nonconvex critical regions while numerical studies underline further the advantages of the proposed methodology, when compared to existing algorithms

Nonlinear Model-Based Process Operation under Uncertainty Using Exact Parametric Programming

In the present work, two new, (multi-)parametric programming (mp-P)-inspired algorithms for the solution of mixed-integer nonlinear programming (MINLP) problems are developed, with their main focus being on process synthesis problems. The algorithms are developed for the special case in which the nonlinearities arise because of logarithmic terms, with the first one being developed for the deterministic case, and the second for the parametric case (p-MINLP). The key idea is to formulate and solve the square system of the first-order Karush-Kuhn-Tucker (KKT) conditions in an analytical way, by treating the binary variables and/or uncertain parameters as symbolic parameters. To this effect, symbolic manipulation and solution techniques are employed. In order to demonstrate the applicability and validity of the proposed algorithms, two process synthesis case studies are examined. The corresponding solutions are then validated using state-of-the-art numerical MINLP solvers. For p-MINLP, the solution is given by an optimal solution as an explicit function of the uncertain parameters

Multi set-point explicit model predictive control for nonlinear process systems

In this article, we introduce a novel framework for the design of multi set-point nonlinear explicit controllers for process systems engineering problems where the set-points are treated as uncertain parameters simultaneously with the initial state of the dynamical system at each sampling instance. To this end, an algorithm for a special class of multi-parametric nonlinear programming problems with uncertain parameters on the right-hand side of the constraints and the cost coefficients of the objective function is presented. The algorithm is based on computed algebra methods for symbolic manipulation that enable an analytical solution of the optimality conditions of the underlying multi-parametric nonlinear program. A notable property of the presented algorithm is the computation of exact, in general nonconvex, critical regions that results in potentially great computational savings through a reduction in the number of convex approximate critical regions

A game-theoretic optimisation approach to fair customer allocation in oligopolies

Under the ever-increasing capital intensive environment that contemporary process industries face, oligopolies begin to form in mature markets where a small number of companies regulate and serve the customer base. Strategic and operational decisions are highly dependent on the firms’ customer portfolio and conventional modelling approaches neglect the rational behaviour of the decision makers, with regards to the problem of customer allocation, by assuming either static competition or a leader-follower structure. In this article, we address the fair customer allocation within oligopolies by employing the Nash bargaining approach. The overall problem is formulated as mixed integer program with linear constraints and a nonlinear objective function which is further linearised following a separable programming approach. Case studies from the industrial liquid market highlight the importance and benefits of the proposed game theoretic approach

Stochastic programming approach for optimal day-ahead market bidding curves of a microgrid

The deregulation of electricity markets has driven the need to optimise market bidding strategies, e.g. when and how much electricity to buy or sell, in order to gain an economic advantage in a competitive market environment. The present work aims to determine optimal day-ahead market bidding curves for a microgrid comprised of a battery, power generator, photovoltaic (PV) system and an electricity load from a commercial building. Existing day-ahead market bidding models heuristically fix price values for each allowed bidding curve point prior to the optimisation problem or relax limitations set by market rules on the number of price–quantity points per curve. In contrast, this work integrates the optimal selection of prices for the construction of day-ahead market bidding curves into the optimisation of the energy system schedule; aiming to further enhance the bidding curve accuracy while remaining feasible under present market rules. The examined optimisation problem is formulated as a mixed integer linear programming (MILP) model, embedded in a two-stage stochastic programming approach. Uncertainty is considered in the electricity price and the PV power. First stage decisions are day-ahead market bidding curves, while the overall objective is to minimise the expected operational cost of the microgrid. The bidding strategy derived is then examined through Monte Carlo simulations by comparing it against a deterministic approach and two alternative stochastic bidding approaches from literature

Traveling Salesman Problem-Based Integration of Planning, Scheduling, and Optimal Control for Continuous Processes

Advanced decision making in the process industries requires efficient use of information available at different decision levels. Traditionally, planning, scheduling, and optimal control problems are solved in a decoupled way, neglecting their strong interdependence. Integrated planning, scheduling and optimal control (iPSC) aims to address this issue. Formulating the iPSC, results in a large scale nonconvex mixed integer nonlinear programming problem. In the present work, we propose a new approach for the iPSC of continuous processes aiming to reduce model and computational complexity. For the planning and scheduling, a Traveling Salesman Problem-based formulation is employed, where the planning periods are modeled in discrete time while the scheduling within each week is in continuous time. Another feature of the proposed iPSC framework is that backlog, idle production time, and multiple customers are introduced. The resulting problem is a mixed integer programming problem and different solution strategies are employed and analyzed

Closed-loop integration of planning, scheduling and multi-parametric nonlinear control

In this article, motivated by the need for efficient closed-loop implementation of the control objectives set within the integrated planning, scheduling and control (iPSC) problem we introduce a novel framework that enables its online solution under dynamic disturbances. We introduce the concept of multi-setpoint explicit controllers through the use of a new multi-parametric nonlinear programming algorithm and develop a rigorous rescheduling mechanism that mitigates the impact of the dynamic disruptions on the operational decisions of planning and scheduling. The overall closed-loop problem is formulated as mixed integer linear program with the control problem integrated via an outer loop. The benefits of the proposed framework are highlighted through two case studies and the results indicate the necessity of considering dynamic disruptions within the scope of the integrated problem