RiSES3: Rigorous Synthesis of Energy Supply and Storage Systems via time-series relaxation and aggregation
Introduction
Synthesis of energy systems is a complex two-stage optimization task. On the design stage, decisions on the type and sizing of units are made. On the operational stage, the on/off status and the load allocation are determined for each energy supply and storage unit in each time step. The synthesis problem requires the simultaneous optimization of the design stage and the operation stage (Lin et al., 2016).
In general, synthesis of energy systems would require a formulation via mixed-integer non-linear programming problems (Bruno, Fernandez, Castells, Grossmann, 1998, Goderbauer, Bahl, Voll, Lübbecke, Bardow, Koster, 2016). Discrete decisions are required to consider existence of units on the design stage and their on/off status on the operation stage. Typically, non-linear part-load performance and investment-cost curves have to be considered for each unit in the synthesis of energy systems (Bruno et al., 1998). Often, these non-linearities can be linearized by piecewise linearization (Floudas, 1995, Misener, Gounaris, Floudas, 2009, Voll, Klaffke, Hennen, Bardow, 2013). Thus, synthesis problems most often correspond to mixed-integer linear programming (MILP) problems (Elia and Floudas, 2014).
Synthesis problems of energy systems are generally strongly NP-hard (Goderbauer et al., 2019), and thus, cannot be solved efficiently (unless P = NP). In particular, this implies that the computational effort increases exponentially with the problem size. Practical application has shown that advanced solution software is able to solve small-scale MILP instances of synthesis problems in acceptable time (Grossmann, 2012). However, synthesis problems of energy systems consider time scales ranging from minutes up to several years of planning (Floudas et al., 2016). As a result, many time steps have to be considered, leading to large time series in synthesis problems. The operation stage usually depends on multiple of these large time series, e.g. demand profiles, electricity prices and renewable resources (Pfenninger et al., 2014). Consequently, the resulting synthesis problems are large-scale MILPs which are computationally challenging even for the most advanced solution software and often not solvable within reasonable computational time or memory limits.
To still solve large-scale synthesis problems, the problem structure can often be exploited (Grossmann, 2012). In particularly, the two-stage character of the synthesis problem forms the basis of two-stage algorithms as developed by Fazlollahi et al. (2014) and Elsido et al. (2017): Evolutionary algorithms are used for the first-stage decisions on the investment of units, while the second-stage decisions for the operation are determined by rigorous optimization methods. However, an evolutionary algorithm does not provide rigorous information on the quality of the solution. Yokoyama et al. (2015) therefore propose a decomposition method utilizing the two-stage character of synthesis problems, which allows to evaluate the quality of the solution. In the proposed decomposition method of Yokoyama et al. (2015), the first-stage decisions are solved with relaxed second-stage decisions. Following, the second-stage decisions are solved with fixed first-stage decisions. However, as Yokoyama et al. (2015) conclude, the implementation of this method is complicated as the input data needs to be manually prepared.
Besides exploiting the problem structure, the size of the synthesis problem can be reduced by model aggregation to allow solution of large-scale problems (Grossmann, 2012). Model aggregation methods are discussed in Mancarella (2014). Mancarella (2014) distinguishes between the aggregation of spatial distribution, of the number of energy flows, and of multiple energy conversion units into 1 component. Besides these aggregations methods, often, time-series aggregation is applied in synthesis problems for the design of chemical processes and energy systems, as reviewed by Maravelias and Sung (2009) and categorized by Nahmmacher et al. (2016). However, the solution of an aggregated synthesis problem is not directly related to the solution of the original synthesis problem with the full time series. Thus, the quality of the solution is unknown. Moreover, the resulting design of the aggregated problem might even be infeasible for the full time series.
To obtain a feasible solution of large-scale problems with known quality, exact solution strategies are needed (Grossmann, 2012). In our previous work, we proposed an exact decomposition method based on time-series aggregation with known quality of the solution (Bahl et al., 2018a). The method exploits both the problem structure and model aggregation and was shown to outperform state-of-the-art commercial solver. However, the proposed decomposition method by Bahl et al. (2018a) still has the following shortcomings:
- 1.
It is not applicable for energy systems with time-coupling constraints as ocurring in storage systems.
- 2.
It is only applicable for time series of energy demands.
- 3.
It requires significant computation time to proof optimality of the solution.
Modeling storage systems or ramping constraints (shortcoming 1) requires the consideration of chronological time steps. The decomposition method proposed in our previous work only addressed independent time steps in the time-series aggregation. Moreover, the time-series aggregation underestimated the energy demands to obtain a lower bound (shortcoming 2). To rigorously solve synthesis problems, not only energy demands need to be considered but also other time-dependent input parameters such as prices or renewable resources. Although our previous method outperformed a commercial state-of-the-art solver, the optimality proof requires significant more computation time than identifying feasible solutions with excellent quality. To solve even larger and more complex synthesis problems, the optimality proof needs to be improved (shortcoming 3).
In this paper, we propose the Rigorous Synthesis method of Energy Supply and Storage Systems RiSES3 as a generalization of the decomposition method by Bahl et al. (2018a) to overcome the above-mentioned shortcomings (1–3). In RiSES3, we address the chronology of times steps by using typical periods (shortcoming 1). We simultaneously under- and overestimate all time-dependent input parameters (shortcoming 2). RiSES3 employs 2 competitive lower bounds (optimality proofs), and parallel computing to further improve the computational time (shortcoming 3). A preliminary version of RiSES3 has been presented in a conference paper (Baumgärtner et al., 2018). Here, the method is extended by the simultaneous over- and underestimation of all time series, improved convergence of the first competitive lower bound, the second competitive lower bound, and parallel computing. Moreover, we provide a second case study for validation.
In Section 2, we state a generic synthesis problem for energy systems, accounting for volatile energy prices and time-coupling constraints, as typically introduced by storage systems and peak power prices. In particular, the time-coupling constraints due to the peak power price and the storage systems are known to increase the complexity of the synthesis problem (Rong et al., 2008). In Section 3, we present the RiSES3 algorithm in detail. In Section 4, we apply RiSES3 to 2 real-world industrial synthesis problems and validate the results by large computational studies. In Section 5, we present our conclusions.
Section snippets
Generic synthesis problem with time-coupling constraints
A generic synthesis problem of an energy system is stated in Eqs. (1)–(7), as a MILP. This problem class is extensively studied in literature, as analyzed by Elia and Floudas (2014) and Baumgärtner et al. (2019a). The objective function, Eq. (1), typically consists of 2 parts representing the two-stage character of the synthesis problem. The capital expenditures () and operational expenditures () (Pintarič and Kravanja, 2015). Here, we employ the total annualized costs as objective
The RiSES3 method
The proposed method RiSES3 aims to solve MILP synthesis problems that depend on large-scale time series. RiSES3 consists of 3 parallel branches, Fig. 1, to calculate upper bounds and 2 competitive lower bounds. In the master branch, upper bounds are obtained from feasible solutions. These feasible solutions are computed using time-series aggregation and restriction. These upper bounds are tightened by iteratively increasing the time resolution. The final time resolution is reached once a
Real-world case studies
To validate the proposed rigorous synthesis method, we apply RiSES3 to 2 real-world synthesis problems. In Section 4.1, we study an utility system providing electricity, low-temperature heat, steam, and cooling for an industrial site. In Section 4.2, we study a pump system providing cooling water to a chemical site.
Conclusions
The synthesis of energy systems typically includes multiple large time series representing, e.g., energy demands and volatile prices. As a consequence, the synthesis results in large-scale MILP optimization problems, which are computationally challenging and often not solvable within reasonable computational time or memory limits. Time-coupling constraints, e.g., due to storage systems, further increase the complexity of the synthesis problem.
To obtain a feasible solution with known quality, we
Acknowledgments
The research leading to this contribution was funded. Nils Baumgärtner thanks the German Federal Ministry of Education and Research (BMBF) for funding under grant number 03SFK3L1 and Björn Bahl thanks the German Federal Ministry of Economic Affairs and Energy for funding under grant number 03ET1259A.
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2022, Applied EnergyCitation Excerpt :In the literature, four predominant categories of time series aggregation (TSA) approaches are distinguished: downsampling (decreasing of temporal resolution), heuristic (selecting (contiguous) days based on defined criteria), clustering (selecting representative (contiguous) days), and optimization (selecting (contiguous) days by minimizing an error indicator), see, for example, [4,5,18]. In addition, more complex approaches are presented such as directly including TSA in energy system models by decomposition [19]. However, a comprehensive comparison conducted by Pfenninger [5] shows that there is no consistently best TSA approach.