# Temporal Aggregation

The main task of temporal aggregation consists of the complexity reduction of temporally resolved input data in order to keep energy system models mathematically tractable and to reduce the computing time while preserving the overall optimization result as good as possible. In contrast to spatial aggregation, temporal aggregation focuses on summarizing redundant repetitions along the time axis.

Temporal aggregation focuses on summarizing environmental input data consisting of time series such as wind speeds or solar radiant power in certain places, for example from spatially aggregated mean time series of regions. Besides, the time series of already existing components e.g. demand profiles of households or industry can be aggregated as well. In this context, time series are lists of discrete values that represent a certain physical property for a defined time step, e.g. an annual time series with hourly resolution for wind speed in a certain place is a list of 8760 values.

The major challenge of temporal aggregation is to maintain the most relevant information for the energy system model while reducing the input time series. Those are especially peak values, which have a great impact on the design of surplus capacities to achieve operational feasibility during exceptional periods, as well as the mean values which are essential for the overall cost optimization.

The most common methods used for temporal aggregation are [1]:

- Averaging, also referred to as down sampling
- Clustering, i.e. grouping time slices of the series in a multi-dimensional hyperspace consisting of the time steps each time slice consists of and representing each group by a single representative time slice, called “typical period”
- Heuristics which take extreme periods separately into account

The most basic method of temporal aggregation is averaging. This method simply reduces the number of time steps by representing a number of time steps by their average. This can be done for an arbitrary interval length, e.g. hourly to daily intervals, but also on even coarser levels such as seasonal averages. Since the law of large numbers leads to a negligence of extreme periods which are highly important for the operational feasibility, this method is outperformed by more sophisticated methods and heuristics for the determination of extreme periods, which will be presented in the following.

Clustering focuses on the fact that, in contrast to spatially resolved data, certain phenomena in time series appear with certain regularity, for example the solar irradiance oscillates on a daily basis. This is exploited by clustering algorithms, which group time series into clusters of similar time periods, mostly typical days. Each group (i.e. “cluster”) of typical periods is then represented by the mean values of the cluster or their medoid. Figure 1 and Figure 2 show the effect of clustering a time series for the capacity factors of Photovoltaic. Here, each day is one of 365 columns across the heat plots and all days from Figure 1 are represented by eight typical days in Figure 3, i.e. by only eight different color slices. Since the k-means algorithm was chosen, each typical day is the mean of its group. This means that the effect of averaging is still present and the extreme values are not represented sufficiently, but in general the daily pattern is well-represented.

As shown above, the clustering still results in a negligence of the extreme values which leads to an underestimation of surplus capacities in energy system optimizations. Therefore, a large number of heuristics exist that try to capture these periods as well. The most common method is to exclude days with extreme values from the clustering algorithm and to include them afterwards to try to achieve a robust optimization, which is a highliy non-trivial task in systems with a large number of components.

All things considered, the most crucial tasks of temporal aggregation methods developed in METIS remain to further improve both, the representation of frequently appearing constellations in multi-dimensional time series and to determine the most critical periods for the energy system before the aggregation and the optimization itself. This aims at maintaining operational feasibility and achieving a cost-optimal solution while staying computationally tractable [2].

The developed temporal aggregation methods will be implemented into the Python package tsam.