# (118g) Advanced Computational Methods for the Solution of Nonlinear Scheduling Problems with Time-Variable Electricity Prices

Authors:
RWTH Aachen University
RWTH Aachen University
During the past decade, an increasing amount of literature has shown that the exploitation of time-variable electricity prices represents a promising measure to decrease the electricity costs of energy-intense processes, such as air separation, steel production or aluminum electrolysis [1]. Typically, time-discrete formulations are used, which subdivide the considered planning horizon into intervals of known length (usually in accordance with the discretization of the electricity prices) and aim at assigning optimal process loads as well as modes of operation to the individual intervals [2,3]. The vast majority of literature formulates these optimization problems as mixed-integer linear programs (MILPs) in order to allow for the use of sophisticated algorithms/solvers for the solution of MILPs such as CPLEX (IBM Corporation) or Gurobi (Gurobi Optimization, LLC.). In particular, authors have proposed MILP reformulations for process features like logical constraints on mode switching [4,5] or nonlinear relations between process variables [6]. For instance, the method introduced by Zhang et al. approximates the feasible region by a set of polytopes with piecewise linear objective function [7]. However, MILP formulations like these possibly require a large number of disjunctions to be sufficiently accurate, resulting in a high number of binary variables in each time interval of the optimization problem.

Despite this drawback, only few works rely on the use of nonlinear process models that lead to mixed-integer nonlinear programs (MINLPs) [8]. The nonlinear process models in general introduce nonconvexities to the optimization problem, involving the potential existence of multiple suboptimal local minima. Thus, global solution methods are desired in this case. The size of the MINLP however scales linearly with the number of considered intervals, making the solution of the optimization problem with state-of-the-art algorithms for global optimization of MINLPs like BARON [9] tractable, only if short horizons, i.e., few variables, are considered. Consequently, a reduction of the temporal dimensionality of the scheduling problem is required for the consideration of longer planning horizons. Particularly in the field of energy systems engineering, various methods have been proposed for time-series aggregation, which essentially correspond to selecting a small amount of short horizons to represent a substantially larger horizon [10]. However, the focus of these methods rather lies on synthesis than on operational optimization, i.e., the resolution of the time series is selected to ensure that design variables are chosen reasonably [11]. Furthermore, the consideration of storage relations between time steps that are common when modeling process systems remains a challenging problem when using aggregated time series.

We herein present an alternative approach for reducing the temporal dimensionality of scheduling problems. For this purpose, we first present a reduced space formulation for scheduling problems that allows for optimizing only in the space of degrees of freedom (DOFs) instead of the full space of all model variables. In fact, we thereby utilize a concept that has recently been shown promising for deterministic global optimization of process flowsheets [12] and is de facto the state-of-the-art for local flowsheet optimization methods implemented in commercial software packages [13]. Furthermore, the reduced space scheduling formulation shows a similarity with sequential solution approaches for dynamic optimization (see, e.g., [13]). This similarity motivates the application of sophisticated computational methods for using problem specific coarsened discretizations of the DOFs whilst considering sufficiently fine discretizations for feasibility of the solution. Regarding recent work from Pattison et al. concerning the use of low-dimensional dynamic models for scheduling purposes [14], the motivation for applying computational methods from the field of dynamic optimization to scheduling problems is even more evident.

We show the tailoring of a wavelet-based algorithm [15] for adaptive refinement of the discretization of the DOFs to scheduling problems with time-variable electricity prices [16]. We thoroughly discuss the properties of the algorithm that include the guarantee for feasible solutions in the finest, i.e., the original temporal discretization, a systematic approach to decide on grid refinements based on previous solutions, and the maintenance of the chronology of time steps (i.e., the possibility to include storage relations, etc.). We apply the algorithm to the operational optimization of a compressed air energy storage subject to historic German electricity price data. Here, substantial nonconvexities arise from the efficiency characteristics of the turbomachines, which are expressed using artificial neural networks [17]. We solve the optimization problem using our in-house optimization software MAiNGO [18] based on McCormick relaxations [19]. We demonstrate that by using a suitable assignment of the DOFs to intervals of the scheduling problem, a substantial reduction of the number of DOFs and hence of the computational time can be achieved with only minor influence on the quality of the obtained solution point. Finally, we briefly outline possible further developments in the algorithm. These include treatments for additional kinds of constraints that arise in scheduling problems, e.g., ramping and transitional constraints, and further improvements in the algorithm itself, such as tailored refinement strategies for binary variables.

Acknowledgements:

The authors gratefully acknowledge the financial support of the Kopernikus project SynErgie by the Federal Ministry of Education and Research (BMBF) and the project supervision by the project management organization Projektträger Jülich. Furthermore, the authors thank Artur M. Schweidtmann and Susanne Sass valuable discussions about the solution approach as well as Dominik Bongartz and Jaromil Najman for helpful advice concerning the use of MAiNGO.

References:

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[14] Pattison RC, Touretzky CR, Johansson T, Harjunkoski I, Baldea M. Optimal process operations in fast-changing electricity markets: Framework for scheduling with low-order dynamic models and an air separation application. Indus Eng Chem Res. 2016;55:4562-4,584.

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[16] Schäfer P, Schweidtmann AM, Lenz PHA, Markgraf H, Mitsos A. A wavelet-based adaptive grid algorithm for the solution of nonlinear scheduling problems with time-variable electricity prices. In preparation. 2019.

[17] Schweidtmann AM, Mitsos A. Deterministic global optimization with artificial neural networks embedded. J Optim Theory Appl. 2019;180(3):925-948.

[18] Bongartz D, Najman J, Sass S, Mitsos A. MAiNGO - McCormick-based algorithm for mixed-integer nonlinear global optimization. 2019.

[19] McCormick GP. Computability of global solutions to factorable nonconvex programs: Part I - Convex underestimating problems. Math Program. 1976;10(1): 147-175.

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