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A modeler's guide to handle complexity in energy systems optimization
Leander Kotzur, Lars Nolting, Maximilian Hoffmann, Theresa Groß, Andreas Smolenko, Jan Priesmann, Henrik Büsing, Robin Beer, Felix Kullmann, Bismark Singh, Aaron Praktiknjo, Detlef Stolten, Martin Robinius
TL;DR
Energy system optimization models face increasing computational difficulty from renewable integration, higher spatiotemporal detail, and sector coupling, while solver and hardware advances remain insufficient. The paper reviews complexity determinants and reduction methods, concluding that modelers need tailored designs and systematic error evaluation, while noting unresolved choices and aggregation limitations.
Problem
Renewable integration, increased spatiotemporal resolution, sector coupling, and growing model data and nonlinearity make energy system optimization increasingly difficult to solve within practical computational limits.
Method
The paper reviews and qualitatively compares complexity-reduction methods ranging from linearization and aggregation to decomposition and mixed multi-level approaches.
Results
The review identifies tailored model design, systematic model-size reduction, and quantified simplification errors as approaches for addressing computational limitations.
Takeaways & Limitations
Modelers should choose complexity-management methods according to the research question, beginning with a coarse model and adding detail where necessary.
Takeaways & Limitations
No clustering indicator has proved superior for selecting the number of typical periods, and spatial aggregation can lose information and underestimate variability in connected systems.
Abstract
from arXiv · showhide
The determination of environmentally- and economically-optimal energy system designs and operations is complex. In particular, the integration of weather-dependent renewable energy technologies into energy system optimization models presents new challenges to computational tractability that cannot only be solved by advancements in computational resources. In consequence, energy system modelers must tackle the complexity of their models daily and introduce various methods to manipulate the underlying data and model structure, with the ultimate goal of finding optimal solutions. As which complexity reduction method is suitable for which research question is often unclear, herein we review some approaches to handling complexity. Thus, we first analyze the determinants of complexity and note that many drivers of complexity could be avoided a priori with a tailored model design. Second, we conduct a review of systematic complexity reduction methods for energy system optimization models, which can range from simple linearization performed by modelers to sophisticated multi-level approaches combining aggregation and decomposition methods. Based on this overview, we develop a guide for modelers who encounter computational limitations.
1 Introduction
Energy system optimization models have become harder to formulate and solve as renewable integration, sector coupling, spatial-temporal detail, and model connectivity increase. Because available computing resources and solver parallelization do not reliably overcome this complexity, the paper reviews complexity-management methods and aims to guide modelers facing computational limits.
- Increasing data requirements, nonlinear objectives or constraints, system variables, and uncertainties enlarge optimization problems and can threaten feasible-solution identification.
- Renewable technologies increase required spatiotemporal resolution and introduce nonlinear structures, while sector coupling further increases system intricacy.
- Advances in processor cores have not translated reliably into faster large-scale optimization because many solvers cannot exploit parallelized resources beyond several threads.
- Large-core supercomputers therefore cannot efficiently tackle the mathematical complexity of energy system optimization models, limiting access to computational-resource advances.
- The paper reviews and qualitatively compares complexity-management methods, from qualitative model evaluation to systematic data-reduction approaches, to guide modelers and identify research gaps.
2 Determinants of complexity
Energy-system complexity arises from interacting elements, growing variety, connectivity, and dynamics, while model complexity depends on how those systems are represented. Modelers therefore balance computational tractability against accuracy through system boundaries, modeling depth, and abstraction.
- Models represent selected aspects of reality for a purpose, simplifying properties that the modeler deems relevant.
- Energy-system models seek to depict emergent behavior through outputs and their dependence on external inputs.
- Complexity can be reduced, controlled, or avoided by changing parts, variables, interdependencies, or anticipation of dynamics.
- Energy systems are complex systems whose variety, connectivity, and dynamics are increasing.
- Algorithmic complexity is hardware-independent but provides only upper scalability limits and cannot validly compare complexity classes.
- Greater abstraction lowers computational complexity but can reduce fidelity, creating an accuracy–complexity trade-off shaped by boundaries, modeling depth, and interdependencies.
3 Methods for complexity reduction
Computational complexity in energy system optimization models can be systematically altered and quantified. The paper organizes complexity-reduction dimensions and subsequently discusses specific reduction and decomposition approaches.
- Computational complexity can be systematically altered and quantified, unlike the underlying system’s complexity.
- The paper illustrates the dimensions for reducing energy-system optimization-model complexity in Section 3.1.
- Sections 3.2–3.5 describe possibilities for reducing energy-system optimization-model complexity.
3.1 Dimensions to reduce complexity
Computational complexity in energy system optimization models is shaped by model size, problem class, and connectivity. These dimensions link modeling choices such as resolution, variable types, constraints, and system coupling to computational requirements.
- Three factors directly affect computational complexity: model size, optimization problem class, and connectivity within the model.Model size concerns variables and constraints; problem class concerns variable and constraint types; connectivity concerns their linkage.
- Model size grows with scope, spatial resolution, and temporal resolution, including the number of modeled time steps and observation period.Time steps may range from sub-minute to hourly resolution, while observation periods can span typical days to decades.
- Large-scale energy system models primarily use linear programs because continuous variables and linear constraints support convexity and efficient polynomial-time solving algorithms.LPs are especially relevant for large bottom-up systems supplied by renewable power.
- Connectivity rises with dense spatio-temporal links, including transmission networks, storage states, investment dynamics, and hierarchical time grids.Strong linkage can make models difficult to decouple even when their overall size is small.
- The model’s computational representation connects energy-system complexity drivers to optimization complexity and motivates systematic complexity reduction.The paper organizes these drivers in a holistic list and relates them to computational representation.
3.2 Temporal aggregation
Temporal aggregation reduces the number of modeled time steps either by lowering resolution or by replacing similar periods with representative periods. These reductions improve tractability but can distort variance, chronology, extremes, and other dynamics relevant to energy-system design and operation.
- Temporal aggregation: Temporal aggregation represents long time series with fewer time steps through direct resolution reduction or representative typical periods.The two approaches are summarized as direct temporal reduction and clustering of similar periods.
- Decreasing the temporal resolution: Down-sampling averages predefined adjacent time steps, whereas segmentation merges adjacent steps according to similarity and produces irregular time-step lengths.Down-sampling is regular; segmentation is irregular and can use clustering or MILP formulations.
- Decreasing the temporal resolution: Down-sampling underestimates input variance, potentially underestimating required capacities and overestimating self-consumption and renewable feed-in.These effects are particularly relevant for systems with high renewable shares.
- Decreasing the number of periods: Typical-period methods cluster comparable periods after normalizing and aligning time series in a dissimilarity matrix, then use cluster representatives as typical periods.Clustering seeks high within-cluster similarity and high between-cluster dissimilarity.
- Decreasing the number of periods: No clustering-indicator set is universally superior for selecting the number of typical periods, partly because continuous phenomena produce poorly separated sample points.The clustering error may decrease monotonically as the number of periods increases.
- Decreasing the number of periods: Clustered periods lose chronology, requiring alternative representations for inter-period dynamics such as seasonal storage and validation because results can change.Extreme periods and cumulative extremes also matter for feasibility, surplus capacity, and operational costs.
- Robust temporal reduction: Robustness strategies include preserving important time-series characteristics, optimizing across multiple scenarios, and bounding fully resolved objectives with aggregated inputs.These directions address variance and feasibility concerns through heuristic, scenario-based, and systematic approaches.
3.3 Spatial aggregation
Spatial aggregation reduces energy-system detail by grouping regions and aggregating technologies within the resulting regions. Its computational benefits depend on whether connections are represented, while information loss can distort variability, balancing, and costs.
- Spatial aggregation: Spatial complexity depends on the number of model regions and the technologies or agents represented within each region.Regions contain energy-system components and connect through networks or grids.
- Spatial aggregation: Spatial aggregation groups regions with similar properties and aggregates regional information, including time series, within newly created regions.Grouping aggregates the network; representation aggregates technologies inside the grouped regions.
- Spatial aggregation: Existing spatial grouping approaches use administrative boundaries, grid characteristics, geographic distance, power-transfer factors, demand patterns, and energy or socioeconomic indicators.The literature also considers highly resolved renewable-potential and demand data, buildings, industrial sites, and representative municipalities.
- Computational effects: Spatial aggregation affects both computational indicators and aggregated costs, motivating methods that select the number and composition of regions according to optimization complexity.Aggregating independent entities yields only linear computational reduction relative to the aggregation rate but avoids modeling connections between candidates.
- Information loss: For connected systems, aggregation causes information loss through within-region representation and balancing effects that can underestimate data variability.Copper-plate assumptions can externalize costs, while renewable technologies may require richer capacity-factor and configuration representations.
3.4 Reduction of the level of detail in modeling system behavior
Reducing the level of detail in system-behavior models often means approximating nonlinear, non-continuous, multidimensional, or mixed-variable relationships. These choices trade mathematical simplicity against representation accuracy and complexity.
- Accurate operation modeling generally produces an MINLP, whereas simplifying technical characteristics can yield an MILP or LP.
- General simplifications: Continuous nonlinear constraints can be approximated through linearization, while non-continuous constraints can be implemented with the Big-M method.The Big-M method adds a binary variable and a sufficiently large value M, producing an MILP.
- General simplifications: Multidimensional functions can be handled by precomputing results after fixing all but one variable to predefined values.
- General simplifications: Products of continuous and integer variables can be replaced by a continuous variable and post-processed to recover the original variables.
- Operation modeling: For part-load-dependent efficiency, constant values produce an LP, while binary-step and piecewise-linear approximations increase both accuracy and model complexity.More sections in the approximation generally provide greater accuracy and complexity; piecewise quadratic approaches produce MIQCPs but were not proven superior to conventional MILPs.
- Operation modeling: Minimum-load operation and start-up or shut-down costs require binary variables, while ramping restrictions use dynamic constraints and may add continuous cost variables.Conversion units commonly operate above minimum loads of 20% to 50% of rated power.
3.5 Simplification of system dynamics and connectivity
System dynamics and connectivity can be simplified by separating technology choice, sizing, and operation, reducing operational resolution, or using limited-foresight approaches. These methods reduce computational burden but can sacrifice global optimality or require validation.
- Decision-layer separation: Technology choice, sizing, and operation can be separated into iterative layers, with simulation or separate operation optimization evaluating the resulting design.
- Decision-layer separation: Design-layer operation can be simplified by fixing selected transmission parameters or aggregating temporal resolution before full-resolution validation and refinement.
- Foresight approaches: Perfect-foresight models use complete information across expansion phases, whereas myopic models assume limited knowledge of future requirements.Perfect foresight can identify a cost-minimal transformation pathway across all expansion phases.
- Foresight approaches: Rolling horizons overlap smaller intervals, combining myopic links between subsets with perfect foresight within each subset.This reduces computational burden compared with perfect foresight while raising the question of suitable interval lengths for balancing complexity and accuracy.
- Validation: Manual decomposition can converge for an underlying energy-system model without producing a globally optimal solution, so subsequent validation is required.Exact decomposition methods that quantify the error could therefore be advantageous.
4 Solving and Decomposition methods
Decomposition methods address large or difficult optimization models by splitting them into coupled subproblems. Their effectiveness depends on exploiting structure, balancing work packages, and preserving links between subproblems.
- Problem classes: Non-convex optimization is harder than convex optimization because local optima are not guaranteed to be globally optimal.
- Convex decomposition: Large linear programs can be decomposed into smaller parts coupled through linking variables and constraints.Common convex decomposition methods include Lagrangian relaxation, Benders decomposition, and the alternating direction method of multipliers.
- Automatic decomposition: Efficient automatic decomposition seeks approximately equally sized work packages, but optimal decomposition is generally not computable in polynomial time.Heuristics and approximations such as graph partitioning can identify arrowhead and bordered structures for structure-exploiting solvers.
- Temporal decomposition: Temporal decomposition creates smaller problems for time segments and links them by matching end states to the beginning states of adjacent segments.
- Mixed-integer methods: Decomposition is also used within branch-and-bound algorithms through generalized Benders decomposition, outer approximation, and cutting planes.
- Energy-system applications: Nested Benders decomposition has been applied to stochastic scenario analyses and large-scale, multi-period energy-system optimization problems.
5 Conclusions
The guide recommends designing ESOMs around the research question, then reducing resolution, nonlinearities, and model structure systematically while quantifying resulting errors. It also identifies open needs for cross-method evaluation, abstract-model aggregation, and parallel computing.
- Model design: Model design should begin with a coarse superstructure aligned to the research question, then add detail only where necessary.The authors caution against letting data availability determine model design.
- Aggregation: Higher renewable shares increase the importance of spatiotemporal resolution, which directly enlarges ESOM size and calculation time.Temporal down-sampling, typical-period clustering, and spatial aggregation can reduce model size, but may require chronology adaptations or evaluation of copper-plate assumptions.
- Formulation: Avoiding binary variables and linearizing relationships where possible preserves tractable convex formulations, whereas MILPs and non-convex nonlinear programs are NP-hard.The resulting exponential solving time limits applications combining renewable generation with storage and transmission.
- Validation: Errors from aggregation and simplification should be quantified using benchmark models, error bounds, or multi-stage approaches before relying on the resulting designs.Conservative upper bounds can guarantee feasibility of the original problem, but are challenging and often simplification-specific.
- Decomposition: Exact decomposition must be tailored to the model because solver support for automatic large-scale decomposition remains unavailable, while heuristic decompositions may sacrifice global optimality.Computational gains also depend strongly on model connectivity and parallelization.
- Research gaps: Open research gaps include holistic cross-impact analysis, aggregation based on abstract mathematical models, and improved exploitation of parallel computing infrastructure.The review specifically calls for quantifying the effects of avoiding binaries and nonlinearities in larger energy system models.
7 Authors Contribution
The authors distributed the work across conceptualization, validation, investigation, writing, funding, resources, and supervision.
- Author roles: The author team shared investigation and original-draft writing, while responsibilities also covered conceptualization, validation, review, funding, resources, and supervision.The contribution statement assigns these roles across the listed authors.
8 Appendix
The appendix surveys bottom-up energy-system optimization frameworks by formulation, application, geographic scope, and temporal representation. The listed tools span LP, MILP, dispatch, planning, capacity expansion, and power-flow applications.
- Framework overview: The appendix identifies frameworks including EFOM, BESOM, MARKAL, MESSAGE, IKARUS, PERSEUS, TIMES, DESOD, DER-CAM, CALLIOPE, OEMOF, URBS, and PYPSA.The overview table is described as covering bottom-up energy-system optimization frameworks.
- Geographic scope: Applications range from single sites, districts, and microgrids to national, multinational, continental, and global energy systems.The appendix also lists systems spanning community, state, national, and global scales.
- Formulations and tasks: The listed frameworks use LP or MILP formulations for tasks such as capacity expansion, dispatch, planning, unit commitment, and power-flow simulation.Examples include GAMS-based LP models, C# or GAMS-based MILP models, and Python-based LP or MILP frameworks.
- Temporal representation: Temporal representations include single time steps, typical days, representative time slices, and dispatch or planning over energy-system networks.Several frameworks combine planning with dispatch using reduced temporal representations.
- Energy-system scope: The frameworks cover electricity and non-electric demands, including heat and electricity planning for residential and commercial districts.The appendix includes technology networks supplying consumer demand and models of coupled heat-and-electricity systems.