  To account for volatile renewable energy supply, energy systems optimization
problems require high temporal resolution. Many models use time-series
clustering to find representative periods to reduce the amount of time-series
input data and make the optimization problem computationally tractable.
However, clustering methods remove peaks and other extreme events, which are
important to achieve robust system designs. We present a general decision
framework to include extreme events in a set of representative periods. We
introduce a method to find extreme periods based on the slack variables of the
optimization problem itself. Our method is evaluated and benchmarked with other
extreme period inclusion methods from the literature for a design and
operations optimization problem: a residential energy supply system. Our method
ensures feasibility over the full input data of the residential energy supply
system although the design optimization is performed on the reduced data set.
  We show that using extreme periods as part of representative periods improves
the accuracy of the optimization results by 3% to more than 75% depending on
system constraints compared to results with clustering only, and thus reduces
system cost and enhances system reliability.
