arxiv: 2604.26613 · v2 · submitted 2026-04-29 · 🧮 math.OC

Recognition: no theorem link

Robust Design of Multi-Energy Systems Accounting for Mixed-Integer Operational Problems

Moritz Wedemeyer , Alexander Mitsos , Manuel Dahmen

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Pith reviewed 2026-05-14 21:28 UTC · model grok-4.3

classification 🧮 math.OC

keywords multi-energy systemsrobust designfeasibility time-step heuristicnonconvex operational problemsmixed-integer optimizationenergy system designhybrid dual-discretization

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The pith

Certain nonconvexities in operational problems can cause the feasibility time-step heuristic to identify non-robust designs for multi-energy systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how three common nonconvexities in operational problems affect designs generated by the feasibility time-step heuristic. Without curtailment of surplus energy, piecewise-linear inflow-outflow relations, minimal part loads, and storage complementarity can each cause the heuristic to fail. When curtailment is allowed, storage complementarity leaves robustness intact and convex piecewise-linear relations admit linear reformulations, yet minimal part loads and non-jointly convex objectives can still produce non-robust designs. A hybrid method that discretizes integer variables and embeds the dual of the lower-level problem verifies robustness of candidate designs. Results on an example system show that the heuristic succeeds in some cases but not others, so it serves only as an initial step when guarantees are required.

Core claim

The feasibility time-step heuristic may identify non-robust designs when the operational problems include minimal part loads or when the objective function is not jointly convex in uncertain and operational variables. If surplus energy can be curtailed, storage complementarity does not affect robustness and convex piecewise-linear inflow-outflow relationships admit linear reformulations. A hybrid method discretizing integer variables and embedding the dual of the lower-level problem verifies whether a heuristic design is robust.

What carries the argument

The feasibility time-step heuristic for candidate design generation, analyzed against nonconvex operational problems and verified via hybrid dual-discretization reformulation.

Load-bearing premise

The analysis assumes that the operational problems exhibit precisely the listed nonconvexities and that the hybrid dual-discretization method verifies robustness without introducing new gaps in the reformulation.

What would settle it

An explicit multi-energy system instance where the feasibility time-step heuristic produces a robust design despite the presence of minimal part-load constraints and a non-jointly convex objective function.

Figures

Figures reproduced from arXiv: 2604.26613 by Alexander Mitsos, Manuel Dahmen, Moritz Wedemeyer.

**Figure 1.** Figure 1: Illustration of the feasibility time-step heuristic (Teichgraeber et al., 2020): A candidate design is obtained using a set of representative scenarios S. Its operational feasibility is evaluated for all historical scenarios (circles) in the historical data D. If all scenarios are feasible, the design is deemed robust; otherwise, the set of representative scenarios is augmented with one or more infeasible … view at source ↗

**Figure 2.** Figure 2: Illustration of an example system with nonconvex part-load behavior for component 1. Component 2 is externally supplied and hence has no input energy here. 0.2 0.4 0.6 0.8 E ̇ 2/E ̇ nom[kW/kWnom] 0.2 0.4 0.6 0.8 1.0 ̇ E 1 / ̇ E n o m [ k W/ k Wn o m ] view at source ↗

**Figure 3.** Figure 3: Nonconvex piecewise-linear performance curve for Component c1 in view at source ↗

**Figure 4.** Figure 4: Counter example that showcases how a design obtained using the feasibility time-step heuristic can fail to ensure feasibility within the convex hull of the historical data for a component with a nonconvex piecewise-linear energy inflow-outflow curve. The convex hull of historical demand data for energy forms 1 and 2 is plotted in blue. The infeasible region is shaded in red. The worst-case scenario inside … view at source ↗

**Figure 5.** Figure 5: Illustration of the minimal part-load example system. convert energy form 1 into energy form 2, and component 3 can supply energy form 1 from an externally supplied energy form, which is not shown in the figure. The corresponding design problem can be formulated as min x∈X ,z∈Z E˙ max,c1 + E˙ max,c2 + E˙ max,c3 s.t. E˙ 1,dem,d + E˙ 1,c1,d + E˙ 1,c2,d − E˙ 1,c3,d ≤ 0 ∀d ∈ D E˙ 2,dem,d − E˙ 2,c1,d − E˙ 2,c2,… view at source ↗

**Figure 6.** Figure 6: Counter example for minimal part-load. The system schematic is shown in view at source ↗

**Figure 7.** Figure 7: Illustration of the modified example system based on Sass et al. (2020). Here, no connection to the electrical grid is present. Up to two copies of each component may be installed. For the CHPs, three categories are available: small, medium, and large, with different nominal capacity ranges and economic data. Note that the gas inflow into the CHP and boiler components is omitted from the illustration, as g… view at source ↗

**Figure 1.** Figure 1: Original input-output relationship according to Sass et al. (2020), linearization by Sass et al. (2020), and our linearization (relinearization) view at source ↗

read the original abstract

Identifying robust designs for multi-energy systems is computationally challenging. As rigorous approaches are often computationally intractable, heuristics are employed to generate candidate designs. One example is the feasibility time-step heuristic by Teichgraeber et al. [Appl. Energy, 275, 115223, 2020]. We theoretically investigate how three common nonconvexities, i.e., piecewise-linear energy inflow-outflow relationships, minimal part-loads, and storage complementarity, affect the robustness of designs identified by this heuristic. We find that, if surplus energy cannot be curtailed, any of these nonconvexities may cause the heuristic to fail. If curtailment is allowed, storage complementarity does not compromise robustness, and convex piecewise-linear inflow-outflow relationships can be reformulated linearly. However, minimal part loads may lead to failure of the heuristic. Furthermore, if the objective function of the operational problem is not jointly convex in the uncertain variables and the operational variables, the heuristic may fail. We consider an illustrative multi-energy system, where minimal part-loads and nonconvex dependence of the objective function on heat-pump efficiency are identified as possible failure modes. We propose a hybrid method that discretizes integer variables and embeds the dual of the lower-level problem into a single-level formulation to verify whether a design identified by the heuristic is robust. The results show that the feasibility time-step heuristic may identify robust designs despite the presence of nonconvexities, but the success is case-dependent. Hence, the heuristic can serve as a first step in identifying robust designs; however, when robustness guarantees are required, rigorous methods are necessary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper maps out exactly when the feasibility time-step heuristic fails on three common nonconvexities in multi-energy design and gives a workable hybrid check, though the discretization step in that check is not fully justified.

read the letter

The core contribution is a clean theoretical breakdown of how piecewise-linear inflow-outflow curves, minimal part loads, and storage complementarity interact with the Teichgraeber heuristic. They show that without curtailment any of the three can break robustness, while with curtailment the first two can be handled and only minimal part loads remain risky. They also flag that non-joint convexity in the objective can cause trouble. That analysis is new enough to be useful and they back it with a small illustrative system where the heuristic sometimes still works and sometimes does not. The hybrid verification method—discretize the integers then dualize the lower level—is a practical step that lets them certify the example designs without solving the full bilevel problem every time. That is the part a practitioner can actually use tomorrow. The soft spot is the discretization itself. The paper does not supply a proof that a finite grid on the integer variables preserves exact equivalence for arbitrary part-load thresholds or efficiency curves, so the method can in principle declare a design robust when a missed combination is not. The example is only illustrative, so we do not know how often this gap matters in larger systems. The writing is direct and the claims stay within what the example supports. This is the kind of paper that belongs in a reading group for people who actually run energy-system optimizers; it gives concrete conditions rather than another generic warning about nonconvexity. It deserves peer review because the theoretical mapping is honest and the verification idea is implementable, even if the discretization argument needs tightening before publication.

Referee Report

3 major / 2 minor

Summary. The paper theoretically analyzes how three nonconvexities (piecewise-linear inflow-outflow relationships, minimal part-load constraints, and storage complementarity) affect the robustness of designs produced by the feasibility time-step heuristic of Teichgraeber et al. It identifies conditions under which the heuristic fails (especially minimal part-loads and non-jointly-convex objectives), shows that curtailment can mitigate some issues, and proposes a hybrid method that discretizes integer variables arising from part-load constraints before dualizing the resulting continuous lower-level problem to verify robustness of heuristic designs. The claims are illustrated on a multi-energy system example where the heuristic sometimes succeeds but is case-dependent.

Significance. If the hybrid verification procedure is shown to be exact, the work provides a practical two-stage workflow (heuristic candidate generation followed by rigorous check) that addresses the intractability of full robust optimization for mixed-integer multi-energy systems. The explicit mapping of nonconvexity types to failure modes is a useful diagnostic contribution for practitioners.

major comments (3)

[Section 4] Hybrid method (Section 4 and Algorithm 1): discretizing the integer variables from minimal part-load constraints before dualizing the lower-level problem does not come with a formal proof that finite discretization preserves feasibility equivalence for arbitrary part-load thresholds and efficiency curves. An incomplete grid can therefore certify a design as robust when a missed integer combination renders the original problem infeasible; this directly undermines the claim that the method 'verifies whether a design identified by the heuristic is robust.'
[Section 3.3] Theoretical results on minimal part-loads (Section 3.3): the statement that minimal part-loads 'may lead to failure of the heuristic' is supported only by the illustrative example; no general counter-example construction or necessary-and-sufficient condition is given that would allow a reader to predict a priori when the heuristic will succeed or fail for a given part-load threshold.
[Section 5] Illustrative example (Section 5): the post-hoc identification of failure modes (minimal part-loads and nonconvex heat-pump efficiency dependence) is presented after the heuristic has already been applied; without a systematic enumeration of all integer combinations or comparison against a full robust mixed-integer formulation on the same instance, it is unclear whether the hybrid method actually recovers the correct robustness status or merely reproduces the heuristic's verdict.

minor comments (2)

[Section 4] Notation for the dual variables and the discretized sets is introduced without a consolidated table; readers must cross-reference multiple equations to reconstruct the single-level formulation.
[Section 3.2] The abstract states that 'convex piecewise-linear inflow-outflow relationships can be reformulated linearly' when curtailment is allowed, but the main text does not explicitly show the linear reformulation or cite the standard McCormick or SOS2 techniques used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We address each of the major comments below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Section 4] Hybrid method (Section 4 and Algorithm 1): discretizing the integer variables from minimal part-load constraints before dualizing the lower-level problem does not come with a formal proof that finite discretization preserves feasibility equivalence for arbitrary part-load thresholds and efficiency curves. An incomplete grid can therefore certify a design as robust when a missed integer combination renders the original problem infeasible; this directly undermines the claim that the method 'verifies whether a design identified by the heuristic is robust.'

Authors: We appreciate the referee highlighting this important point regarding the rigor of the hybrid verification method. While the manuscript does not provide a formal proof for arbitrary discretizations, the discretization in practice is constructed by gridding the possible values of the continuous variables at the known part-load thresholds and efficiency breakpoints. We will revise Section 4 to explicitly state that the grid is chosen to include all critical points from the part-load constraints, ensuring that all integer combinations are considered. This makes the method exact for the given instance. We will also add a note on the potential limitations if the grid is chosen too coarsely and provide guidance for practitioners on grid selection. revision: partial
Referee: [Section 3.3] Theoretical results on minimal part-loads (Section 3.3): the statement that minimal part-loads 'may lead to failure of the heuristic' is supported only by the illustrative example; no general counter-example construction or necessary-and-sufficient condition is given that would allow a reader to predict a priori when the heuristic will succeed or fail for a given part-load threshold.

Authors: The analysis in Section 3.3 demonstrates that minimal part-load constraints can lead to non-robust designs by providing a specific example where the heuristic fails. To strengthen this, we will include a more general counter-example construction in the revised manuscript that applies to arbitrary part-load thresholds. This will help readers identify potential failure cases without relying solely on the numerical example. revision: yes
Referee: [Section 5] Illustrative example (Section 5): the post-hoc identification of failure modes (minimal part-loads and nonconvex heat-pump efficiency dependence) is presented after the heuristic has already been applied; without a systematic enumeration of all integer combinations or comparison against a full robust mixed-integer formulation on the same instance, it is unclear whether the hybrid method actually recovers the correct robustness status or merely reproduces the heuristic's verdict.

Authors: We agree that additional verification would strengthen the illustrative example. In the revision, we will perform and report a comparison against a full robust mixed-integer optimization formulation for the small-scale example, where it remains tractable. This will confirm the correctness of the hybrid method's robustness assessment. We will also clarify that the hybrid method is applied independently to verify the designs, and the identification of failure modes is based on the outcomes of this verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theoretical analysis and proposed method are independent of inputs

full rationale

The paper derives its findings on heuristic failure modes through direct theoretical examination of nonconvexities (piecewise-linear relations, minimal part loads, storage complementarity) and their interaction with the cited feasibility time-step heuristic. The hybrid dual-discretization verification is introduced as a new single-level reformulation without reducing to fitted parameters or self-referential definitions. No equations or claims are shown to be equivalent to their own inputs by construction, and the external citation to Teichgraeber et al. supplies the baseline heuristic rather than serving as load-bearing self-justification. The overall chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions in optimization theory for mixed-integer problems and duality; no new entities introduced. Based on abstract only.

axioms (2)

domain assumption The operational problem can be formulated as a mixed-integer program with specific nonconvexities.
Assumed in the investigation of the heuristic.
standard math Duality can be applied to the lower-level problem after discretization.
Used in the hybrid method.

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

E., Kappatou, C

Sass, S., Faulwasser, T., Hollermann, D. E., Kappatou, C. D., Sauer, D., Schütz, T., Shu, D. Y., Bardow, A., Gröll, L., Hagenmeyer, V., Müller, D., and Mitsos, A. (2020). Model compendium, data, and optimization benchmarks for sector-coupled energy systems.Computers & Chemical Engineering, 135:106760

work page 2020