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Impact of different time series aggregation methods on optimal energy system design

Leander Kotzur, Peter Markewitz, Martin Robinius, Detlef Stolten

arXiv:1708.00420v1math.OCcs.CE

TL;DR

Renewable energy-system design is computationally demanding because fluctuating supply and demand create large optimization problems, while existing aggregation methods have been applied to different systems and are difficult to compare. The paper extends these methods to multidimensional time series and evaluates them across energy-system configurations. Aggregation substantially reduces computational resources, but representative real periods outperform averaged values, and system characteristics matter more than the algorithm choice.

  • Problem

    Fluctuating supply and demand make renewable energy-system design computationally demanding, while aggregation methods applied to different systems are difficult to compare.

  • Method

    The paper extends time-series aggregation methods for multidimensional inputs and compares them across three energy-system design configurations using accuracy and computational-load analyses.

  • Results

    Time-series aggregation significantly reduces model complexity and computation; medoids generally outperform averaged centroids, while the aggregation algorithm itself has minor influence.

  • Takeaways & Limitations

    Centralized-resource systems can achieve cost-prediction errors below 2% with eight to twelve typical days and 50-fold shorter solution times than full-year series.

  • Takeaways & Limitations

    Current typical-period models do not properly represent seasonal storage because typical periods are treated independently without inter-period energy exchange.

Abstract

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Modelling renewable energy systems is a computationally-demanding task due to the high fluctuation of supply and demand time series. To reduce the scale of these, this paper discusses different methods for their aggregation into typical periods. Each aggregation method is applied to a different type of energy system model, making the methods fairly incomparable. To overcome this, the different aggregation methods are first extended so that they can be applied to all types of multidimensional time series and then compared by applying them to different energy system configurations and analyzing their impact on the cost optimal design. It was found that regardless of the method, time series aggregation allows for significantly reduced computational resources. Nevertheless, averaged values lead to underestimation of the real system cost in comparison to the use of representative periods from the original time series. The aggregation method itself, e.g. k means clustering, plays a minor role. More significant is the system considered: Energy systems utilizing centralized resources require fewer typical periods for a feasible system design in comparison to systems with a higher share of renewable feed-in. Furthermore, for energy systems based on seasonal storage, currently existing models integration of typical periods is not suitable.

1. Introduction

Energy-system design optimization is computationally demanding because supply and demand vary across space and time, motivating temporal aggregation into representative periods. The paper compares aggregation methods across several energy-system configurations and evaluates their accuracy and computational load.

  • Motivation: Optimization inputs directly determine problem size and processing-resource requirements, making advance simplification necessary.
  • Motivation: Temporal aggregation reduces redundant data by creating typical periods that represent recurring hourly, daily, and seasonal patterns.
  • Research gap: Existing aggregation methods include time-chronological averaging, k-means, hierarchical, and k-medoids clustering, but differing system applications hinder direct comparison.
  • Research questions: The paper asks which aggregation method, number of aggregated steps, and period length best suit different energy-system design applications.
  • Approach: The study applies deterministic aggregation methods and extreme-period variants to typical-day selection, then evaluates three heat-and-electricity system designs against full-time-series optima.

2. Time series aggregation methods

The paper preprocesses multidimensional time series into candidate periods, groups them into clusters, and derives representative periods using several aggregation methods.

  • General procedure: Time series aggregation groups similar original periods and represents each group with one typical period.The raw input is preprocessed before grouping, generally using distance measures between candidate periods.
  • Pre-processing: Normalized multidimensional time series are arranged into candidate periods with equal length and multiple attributes.Each candidate period becomes a row in a matrix whose columns combine period steps and attributes.
  • Pre-processing: For two hourly yearly series, a typical-day approach produces 365 candidate days and a 365-by-48 matrix.This example uses 24 time steps and two attributes.
  • Averaging periods: Averaging assigns periods by their original order, producing ordered typical profiles without grouping by similarity.Its main advantages are straightforward application and a clear order of resulting periods.
  • k-means clustering: k-means forms clusters by minimizing squared error between each candidate and its cluster mean, using an iterative greedy algorithm.The algorithm alternates candidate assignment and cluster-center recomputation and can converge to a local minimum.
  • Alternative clustering methods: k-medoids selects an existing representative candidate as each cluster center, whereas hierarchical clustering repeatedly merges the closest clusters until Nk remain.Exact k-medoids can reach a global optimum but requires high computational load correlated with the number of candidate periods.

2.3. Integration of extreme periods

Aggregation can omit extreme periods that contain design-relevant demand or supply values. The paper therefore compares variants for incorporating these peak periods into the representative set.

  • Motivation: Peak periods may be excluded because they are not representative of an entire group, despite containing design-relevant extrema such as peak heat demand.These extrema matter because the energy system design should be able to meet them.
  • Integration variants: The compared integration variants are none, append, additional cluster center, and replacement of a representative period.The variants differ in whether extreme periods are omitted, added, used to reassign candidates, or substituted within their original cluster.
  • Design implication: The choice of aggregation method determines the robustness of the system design.This conclusion is stated in the context of integrating extreme periods into aggregated profiles.

2.4. Scaling of aggregated time-series

The aggregated profiles are scaled after aggregation so their averages match the original time series while respecting the original extrema.

  • Average-value correction: Each time series is scaled a posteriori so the aggregated time series retain the original average values.The scaling is performed separately for each time series.
  • Extreme-value constraint: Values exceeding 1 are capped at 1, and the remaining values are rescaled to recover the correct average.This prevents scaling from exceeding the original time series' extreme values.
  • Rescaling: The corrected profiles are finally scaled back to their original scale.

3. Exemplary time series aggregation

The study compares four aggregation methods across five hourly annual profiles, showing that profile type strongly affects aggregation quality. Clustering generally preserves time-series structure better than averaging, but peaks and fluctuations can be lost, limiting indicator-based predictions of design performance.

  • Experimental setup: Four aggregation methods reduce five hourly annual profiles to four typical days for comparison across solar, temperature, wind, and electrical-load time series.The profiles include GHI, temperature, wind speed, household load, and regional load.
  • Qualitative comparison: Clustering can represent nonconsecutive candidate periods, whereas averaging groups consecutive periods and smooths much of the original fluctuation.This flexibility gives clustering-based methods a more accurate representation of the original time series.
  • Profile dependence: Temperature aggregation captures major patterns but may omit minimum-temperature days that are relevant for energy-system design.The result motivates adding extreme periods to aggregated time series.
  • Profile dependence: Solar and regional-load profiles are comparatively well represented, while wind and household-load profiles are difficult to aggregate because their fluctuations or peaks are poorly captured.Regional load has strong daily patterns; wind has weak intraday structure, and household load contains high fluctuation and lagged patterns.
  • Indicator-based evaluation: The regional electrical-load profile has the lowest average RMSE at 6.02 % for k-mean clustering, while wind has the highest at 9.99 %.Household load follows with 7.75 % error, while temperature and GHI fall between these values.
  • Indicator-based evaluation: K-mean clustering performs best on profile RMSE, but duration-curve RMSE favors different methods across profiles, including hierarchical clustering for GHI and k-medoids for wind.Duration-curve errors include 1.71 % for hierarchical GHI aggregation, 4.09 % for k-medoids on wind, and 2.65 % for k-means on temperature.
  • Indicator-based evaluation: The indicators provide limited guidance for predicting effects on optimal system design, so the methods are evaluated in subsequent energy-system optimization problems.This motivates comparing aggregated results with optimization using the full time series.

4. Application of optimal energy system design

The aggregation methods are compared across CHP, residential, and island energy systems, showing that representative-period choices affect cost estimates and designs differently. Centralized systems converge with few typical days, whereas highly renewable systems with storage require longer periods and remain difficult to model accurately.

  • Application scope: Three energy supply systems—CHP, residential heat pump and photovoltaics, and a renewable island system—are compared using varied aggregation methods and typical-period counts.The study also examines extreme-period integration in the residential system and period length in the island system.
  • CHP system: Small numbers of typical periods underestimate CHP system costs because smoother representative profiles improve apparent heat–electricity matching and reduce capacity exceedances.This increases the modeled CHP supply rate and overestimates CHP coverage.
  • Aggregation methods: Medoid-based methods generally outperform k-means and averaging because real representative days retain more variation than smoothed profiles.No consistent dominance is found between hierarchical and k-medoid clustering.
  • Residential system: Residential extreme-period treatment changes cost estimates: excluding extremes underestimates costs by roughly 2%, manual peak inclusion reduces this to 1%, and replacing clusters with peaks overestimates costs by 3%.The latter approach produces a more conservative system design.
  • Design effects: Typical-period smoothing overestimates optimal photovoltaic capacity while underestimating storage capacity by making feed-in, demand, and heat demand appear steadier.The resulting profiles show greater overlap and require less balancing storage.
  • Island system: The island system shows no robust convergence even with 72 typical days; groups of three days improve nonmonotonously, while typical weeks converge more predictably but remain above 2% error at 24 weeks.Typical weeks also solve more slowly because each period contains more time steps.
  • Island system: Independent typical periods prevent energy exchange between periods, so storage-heavy island designs cannot represent seasonal storage properly.Storage technologies account for almost 30% of system costs, including fuel cell, electrolyzer, hydrogen storage, and battery.

5. Summary and discussion

The paper evaluates typical-period aggregation across energy systems to determine how method and system structure affect optimal designs. It finds that medoid periods generally outperform averaged profiles, while centralized systems need few periods and seasonal-storage systems are not adequately represented by current models.

  • Summary: Averaging time series by their occurrence in the year produces inaccurate energy system designs.
  • Summary: Medoid periods generally perform better than centroids or averaged time series, while the choice between k-medoid and hierarchical clustering has relatively little influence.
  • Time-series characteristics: Solar irradiation, temperature, and regional electricity load are represented with small errors, whereas single-building load and wind data are more difficult to aggregate.Wind aggregation smooths much of the original intraperiod fluctuation.
  • System dependence: CHP and residential systems reach cost-prediction errors below 2% with eight to twelve typical days while reducing optimization time by a factor of 50.These systems depend on centralized supply infrastructures such as fuel imports or the electricity grid.
  • System dependence: Island-system aggregation produces high estimation errors because independent typical periods cannot exchange energy, preventing proper treatment of storage and seasonal storage.Extending typical days to weeks provides only a small improvement.

6. Conclusion

Time series aggregation substantially reduces model complexity and computational time, but its usefulness and accuracy depend more on the system configuration and representative-period choice than on the aggregation algorithm itself.

  • Clustering-based time series aggregation significantly reduces model complexity and required computational time.
  • The aggregation algorithm has only a minor impact on optimal system design, whereas using real cluster medoids produces more accurate results than averaged k-means centroids.
  • Regional time series are easier to aggregate than single-location series, requiring fewer typical periods for sufficient representation.
  • The trade-off between time-series reduction and modeling error depends strongly on the energy system configuration.
  • Centralized-supply systems can be represented with a few typical days, but storage-heavy systems cannot be properly represented by independent typical days or weeks.
  • Aggregation methods must be evaluated separately for each energy system model because performance indicators alone can significantly deviate from the optimal design.

7. Outlook

The outlook calls for models that preserve the sequence of aggregated periods and for clustering features that better retain original time-series variability.

  • New modeling methods should account for the sequence of aggregated typical periods when representing long-term storage operation.
  • Future clustering research should identify parameters such as gradients and variance because current clustering smooths typical periods and underestimates original variability.
  • The introduced aggregation methods are available in the Python package tsam and can be applied and extended.

Appendix A. System modeling

The system model represents an energy network through connected technologies and energy-flow variables, while typical-period aggregation replaces global time steps with within-period indices.

  • The validation model is a simple mixed-integer linear program consisting of connected technologies and energy-flow variables.
  • Each energy connection is defined by an output component, an input component, and membership in a connection set.
  • For a typical period, the original time index t is replaced by the within-period index g for period k.

Appendix A.1. Objective function

The objective function minimizes annualized supply-system cost by combining annualized device costs, capital expenditures, installed capacities, binary installation decisions, and operating energy-flow costs.

  • The objective function describes the annualized cost of the considered energy supply system.
  • Annualized device costs use a capital recovery factor based on weighted average cost of capital and device lifetime.
  • Each component uses a binary existence variable and a continuous installed-capacity variable to represent fixed and scale-dependent capital expenditures.
  • Variable operating costs are linked to energy flows, while device scaling contributes to the objective function.

Appendix A.2. Constraints

The system constraints are represented through five device classes covering flows, energy aggregation, transformation, storage dynamics, and scaling restrictions. The formulation also incorporates storage capacity, self-discharge, timestep length, and BigM-based device-variable limits.

  • Device models establish system constraints through five classes: Source/Sinks, Collectors, Transformers, Storages, and associated scaling restrictions.The classes represent the main device-model components used to formulate the system constraints.
  • Source/Sinks represent system input and output flows, including photovoltaic feed-in and electricity demand.Lower and upper bounds can represent minimum demand satisfaction or maximum photovoltaic feed-in per installed capacity.
  • Collectors enforce equivalence between all input energy flows and all output energy flows at a hub.
  • Transformers convert energy between forms using energy-type-specific efficiencies, with fuel cells and heat pumps as examples.Each energy flow has an energy type, and the transformation efficiency depends on the input and output types.
  • Storages track state of charge over time using an Euler formulation, while self-discharge, timestep length, capacity, and boundary relations constrain operation.The storage capacity limits state of charge, and the initial and final states are related across the considered timeframe.
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