Typical Periods for Two-Stage Synthesis by Time-Series Aggregation with Bounded Error in Objective Function

Abstract

Two-stage synthesis problems simultaneously consider here-and-now decisions (e.g., optimal investment) and wait-and-see decisions (e.g., optimal operation). The optimal synthesis of energy systems reveals such a two-stage character. The synthesis of energy systems involves multiple large time series such as energy demands and energy prices. Since problem size increases with the size of the time series, synthesis of energy systems leads to complex optimization problems. To reduce the problem size without loosing solution quality, we propose a method for time-series aggregation to identify typical periods. Typical periods retain the chronology of time steps, which enables modeling of energy systems, e.g., with storage units or start-up cost. The aim of the proposed method is to obtain few typical periods with few time steps per period, while accurately representing the objective function of the full time series, e.g., cost. Thus, we determine the error of time-series aggregation as the cost difference between operating the optimal design for the aggregated time series and for the full time series. Thereby, we rigorously bound the maximum performance loss of the optimal energy system design. In an initial step, the proposed method identifies the best length of typical periods by autocorrelation analysis. Subsequently, an adaptive procedure determines aggregated typical periods employing the clustering algorithm k-medoids, which groups similar periods into clusters and selects one representative period per cluster. Moreover, the number of time steps per period is aggregated by a novel clustering algorithm maintaining chronology of the time steps in the periods. The method is iteratively repeated until the error falls below a threshold value. A case study based on a real-world synthesis problem of an energy system shows that time-series aggregation from 8,760 time steps to 2 typical periods with each 2 time steps results in an error smaller than the optimality gap of the synthesis problem (2%). This corresponds to a reduction of the number time steps and thus a reduction of the size of the synthesis problem by a factor of 1,000 with excellent accuracy in cost estimation. Thus, the proposed method enables an efficient and accurate synthesis of energy systems.

A Review on Time Series Aggregation Methods for Energy System Models

“…This was also pointed out in comparison to a typical periods approach by Pineda et al [72], who used the centroid-based hierarchical Ward's algorithm [73] with the side condition to only merge adjacent time steps. Bahl et al [74], meanwhile, introduced a similar algorithm as Fazlollahi et al [69] inspired by Lloyd's algorithm and the partitioning around medoids algorithm [75,76] with multiple initializations. This approach was also utilized in succeeding publications [77,78].…”

“…This approach was also utilized in succeeding publications [77,78]. In contrast to the approach of Bahl et al [74], Stein et al [79] did not use a hierarchical approach, but formulated an MILP in which not only extreme periods could be excluded beforehand, but also so that the grouping of too many adjacent time steps with a relatively small but monotone gradient could be avoided. The objective function relies on the minimization of the gradient error, similar to the method of Mavrotas et al [54].…”

“…One of the most common partitional clustering algorithms used in energy system optimization is the k-means algorithm, which has been used in a variety of studies [14,15,24,37,57,58,63,69,74,78,[83][84][85][86][87]97,[137][138][139]141,142,[145][146][147][148][153][154][155][156][157][158][159][160][161]. The objective of the k-means algorithm is to minimize the sum of the squared distances between all cluster members of all clusters and the corresponding cluster centers, i.e., min…”

“…This clustering algorithm was used by numerous authors [14,19,41,55,57,58,74,78,86,139,141,149,150,159,[163][164][165][166][167], either by using the partitioning around medoids (PAM) introduced by Kaufman et al [168] or by using an MILP formulation introduced by Vinod et al [169] and used in several studies [14,41,55,57,139,159,164]. The MILP can be formulated as follows:…”

Reducing Computational Load for Mixed Integer Linear Programming: An Example for a District and an Island Energy System

“…In this way, the problem size is further decreased. However, it can only account for storages within each typical period, which means that seasonal storage is not accounted for, as mentioned by Bahl et al [32]. The lower bound of the problem is determined in two ways: Either by aggregating the input time series to typical periods with typical sequences and simultaneously over-and underestimating supply and demand for each segment while stepwise increasing the number of periods, or by using a common branch-and-bound procedure, which is used as a benchmark.…”