A multi-stage methodology for wind park inter-array cabling: graph preparation, layout, and sizing

Castro Valerio, Bernardo; Gebraad, Pieter M. O.; Cheah-Mane, Marc; A. Lacerda, Vinicius; Gomis-Bellmunt, Oriol

doi:10.5194/wes-2026-53

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https://doi.org/10.5194/wes-2026-53

Preprints

16 Mar 2026

| 16 Mar 2026

Status: this preprint is currently under review for the journal WES.

A multi-stage methodology for wind park inter-array cabling: graph preparation, layout, and sizing

Bernardo Castro Valerio, Pieter M. O. Gebraad, Marc Cheah-Mane, Vinicius A. Lacerda, and Oriol Gomis-Bellmunt

Abstract. Inter-array grid optimisation plays a critical role in the technical and economic performance of wind parks and is typically divided into two primary tasks: layout length minimisation and conductor size selection. This work proposes the consideration of three primary tasks, beginning with graph preparation to account for terrain profiles, soft and hard exclusion zones, and export cables. Once a graph with viable connection pathways is established, a pathfinding mixed-integer linear formulation for radial arrays, subject to crossing constraints and a maximum number of turbines per string, is presented. This formulation can be adapted to user requirements to include a minimum number of turbines per string, as well as a methodology to determine the minimum number of substation connections required under irregular turbine distributions arising from terrain constraints. Conductor size selection is addressed through a linear approximation of the power flow, enabling the limitation of the number of conductor types used within the system. Pathfinding and conductor size selection may be combined within an integrated approach or implemented as a sequential algorithm to explore trade-offs between trenching and cabling costs. The proposed approaches are evaluated using current turbine ratings and the layouts of both onshore and offshore existing projects.

Received: 10 Mar 2026 – Discussion started: 16 Mar 2026

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Bernardo Castro Valerio, Pieter M. O. Gebraad, Marc Cheah-Mane, Vinicius A. Lacerda, and Oriol Gomis-Bellmunt

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AC1: 'Comment on wes-2026-53', Bernardo Castro Valerio, 16 Mar 2026

This manuscript is an extension of a paper presented at TORQUE 2026, which is not currently publicly available. To facilitate the review process, the non-publicly available article can be requested by the reviewers through Copernicus Editorial Support.

Citation: https://doi.org/10.5194/wes-2026-53-AC1
RC1:
'Comment on wes-2026-53', Anonymous Referee #1, 06 Apr 2026

General comments: The submitted manuscript addresses an interesting and relevant problem: Optimization of inter-array cabling in wind farms. The chosen method, mixed integer linear programming (MILP) is standard.
Specific comments: It is hard to find contributions to the existing literature. Section 1 gives a brief overview of the literature in the field, but fails to identify any research gaps that the submissions claims to close successfully. Neither the 'graph preparation' (Section 2), the model that 'minimises the connections' (Section 3), the model that optimizes the selection of cable types (Section 4), the heuristic that addresses the computational challenge of the model integrating all features (Section 5), represent scientific novelty beyond decent engineering approaches.
Technical comments:
1. The writing quality is not up to the standards of a scientific journal, especially in the more mathematical sections. Nearly none of the mathematical formulas are smoothly integrated with the flow of text, as exemplified already in (1): The trivial statement that $p_i=1$ for all $i\in\mathcal{N}_t$ is given as a displayed equation separated by two periods, and given a forward reference in the text. Such forward references, where the formula id occurs earlier than the formula itself occur throughout the manuscript, whereas they should be totally avoided.
2. Several symbols are not carefully introduced before they are used (see $a_{ik}$ in (4)), whereas others are introduced more than once (see the calligraphic letter that probably is an 'M' introduced on lines 167 and 196). Other symbols that are not defined include $S_{a,\iota_{ik}}$ (see (16a)), $\theta_i$ and $\theta_k$ (see (19)), $\gamma_t$, $t_i$, and $s_i$ (see (20)), etc. Also, while the authors should be perfectly free to choose their own notational style, it is likely that more readers than this reviewer would appreciate less excessive use of calligraphic letters (what do we call the symbol denoting the 'maximum number of string connections' on line 175?)
3. The distinction between non-linear and linear CSS (see Sections 4.1 and 4.2, respectively) is at best unclear. What non-linearities are in fact addressed in Section 4.1? Apparently, the only non-linear relations that occur are products between one binary and one continuous variables, which by standard, simple techniques can be expressed in terms of linear inequalities (see inequalities (23)-(24)). The value of Section 4.1 seems dubious.
Conclusion: The manuscript is not recommended for publication.

Citation: https://doi.org/10.5194/wes-2026-53-RC1
- AC2:
  'Reply on RC1', Bernardo Castro Valerio, 07 Apr 2026
  We appreciate the reviewer’s comments and provide the following clarifications.
  
  Regarding the general and specific comments, the paper presents a multi-stage methodology addressing three key steps in the optimisation of wind farm array layouts. In the existing literature (Fischetti and Pisinger 2018a,2018b; Pérez-Rúa et al. 2020; Souza de Alencar et al., 2025), it is standard practice to approximate cable lengths using two-dimensional distances between turbines. In contrast, we propose a process that incorporates three-dimensional terrain distances, accounts for inclination thresholds, and includes both hard and soft exclusion zones. Soft exclusion zones, for instance, may represent varying terrain types with distinct trenching costs, which are explicitly considered during pathfinding. This results in a more representative graph for the interconnection optimisation problem.
  
  For the second and third steps, we agree that mixed-integer linear programming (MILP) is widely used, as demonstrated in prior studies (Pillai et al., 2015; Wędzik et al., 2016; Fischetti and Pisinger, 2018a, 2018b; Ulku and Alabas-Uslu, 2020; Pérez-Rúa et al., 2020; Souza de Alencar et al., 2025). However, in most of these works, the MILP formulation does not explicitly account for radial array structures and typically introduces two binary variables per arc (one per direction). In contrast, our formulation employs a single binary variable per arc, while allowing the associated flow variable to take both positive and negative values.
  
  As presented in our earlier conference paper (Torque), referenced in the introduction, this approach reduces the number of variables compared to recent MILP formulations such as Souza de Alencar et al. (2025), for which a benchmark comparison is provided. This benchmark is not included in the present manuscript, as the conference paper focuses solely on layout optimisation, where benchmarking is directly applicable. Whereas the current work extends the framework to include conductor size selection and its interaction with the layout process. The conference paper has been accepted but is not yet publicly available. To facilitate the review process, the manuscript can be requested by the reviewers through Copernicus Editorial Support.
  
  In the third step, the conductor size selection (CSS) process is typically treated in the literature as a post-processing task. In this work, we instead present CSS either in a unified framework with the layout optimisation or as a sequential but integrated step. This allows CSS to play an active role in the layout decision-making process. For example, the user may define a set of candidate conductor sizes (e.g. ten options) and restrict the final selection to a subset (e.g. three). Treating CSS purely as post-processing can limit optimality, as the shortest connection is not necessarily the most cost-effective, as demonstrated in our case studies. We are currently revising the introduction to more clearly highlight these contributions. In the following bullet points after outlining out the literature criteria;
  
  “The main contributions of this works consist of addressing these criteria as follows:
  A graph construction methodology that incorporates three-dimensional terrain, inclination constraints, and both hard and soft exclusion zones into the candidate edge generation process.
  
  A reduced MILP formulation for inter-array routing using a single binary variable per arc, thereby decreasing the number of decision variables relative to standard formulations.
  
  A joint treatment of routing and conductor size selection, allowing CSS to influence layout decisions rather than being treated as post-processing.
  
  The development and comparison of integrated and sequential approaches for joint routing and CSS, highlighting trade-offs between solution quality and computational complexity. “
  
  Regarding the technical comments:
  
  We acknowledge the concerns regarding writing quality and the integration of mathematical expressions. We are revising the manuscript to improve the flow of the text, ensure proper placement of equations, and eliminate forward references to equations that have not yet been introduced.
  We agree that several symbols were not adequately introduced and that some notation was defined more than once. We will revise the manuscript to ensure that all symbols are clearly and consistently defined prior to use. For example, for candidate edges we will explicitly define $a_{ik} = (i,k)$. We also acknowledge the duplicate definition of $\mathfrak{M}$ (lines 167 and 195) we intended these repetitions only to aid readability by reminding the reader of previously defined symbols; this has now been streamlined. Additionally, symbols such as $S_{a,\iota_{ik}}$, $\theta_i$, $\theta_k$, and $\gamma_t$ were insufficiently introduced, as they originate from Valerio et al. (2026a), where the non-linear ACOPF-based CSS formulation is described in detail. We will include explicit definitions in this manuscript: $\gamma_t$ represents the curtailment of turbine $t$, $S_{a,\iota_{ik}}$ denotes the apparent power on edge $a$ for conductor size $\iota$, and $\theta$ denotes the voltage angle at nodes. The indices $t_i$ and $s_i$ are used within a unified formulation for node $i$, which may be associated with either turbines or a substation, rather than defining separate expressions for each node type.
  We acknowledge that the distinction between non-linear and linear CSS was not sufficiently clear. While products of binary and continuous variables can indeed be linearised using standard techniques (e.g. McCormick envelopes, as applied in our MILP CSS formulation), the non-linear ACOPF formulation involves additional non-linearities arising from power flow equations. In particular, the apparent power $S$ is defined in terms of real and reactive power components, which depend non-linearly on voltage magnitudes and phase angles:
  
  S_{ik}& = P_{ik} + \text{j} Q_{ik} \\\
  P_{ik} &= |V_i|^2 G_{ff} + |V_i||V_k| \left[ G_{ft} \cos(\theta_i - \theta_k) + B_{ft} \sin(\theta_i - \theta_k) \right] \\\
  Q_{ik}&=-|V_i|^2 B_{ff}+|V_i||V_k|[G_{ft} \sin(\theta_i-\theta_k)-B_{ft} \cos(\theta_i-\theta_{k})
  
  Here, $S$ denotes apparent power, composed of real power $P$ and reactive power $Q$; $V$ is the voltage magnitude; and $\theta$ is the voltage angle at nodes $i$ and $k$. The conductance $G$ and susceptance $B$ correspond to the real and imaginary parts of the admittance matrix $Y_{\text{bus-branch}}$. We will revise Section 4.1 to clarify these distinctions and better justify its inclusion.
  
  Citation: https://doi.org/10.5194/wes-2026-53-AC2
RC2: 'Comment on wes-2026-53', Anonymous Referee #2, 19 May 2026

Overall comments
The work addresses a relevant aspect of wind power plant optimization and offers a contribution regarding the inclusion of realistic details and constraints within the problem modeling and the optimization framework. The set of example cases presented as experiments employing the proposed method and its variants is valuable and illustrates some of the alternatives discussed.
That said, the manuscript is hard to follow. Symbol use precedes symbol definition in several cases. The manuscript would benefit from a structured symbol description summary early in the text. Moreover, the names for variables and procedures could be chosen so as to help the reader understand what they represent. A few specific examples are given in the detailed comments, but one that is used throughout the document is conflating “path-finding” with network design.
Reference Valerio et al. (2026b) is not published and seems essential to support part of the work presented in this manuscript. This reference seems to present one aspect that is missing from the manuscript: a comparison to the modeling choices and frameworks already described in the literature.
Detailed comments and technical corrections follow.

Abstract
[11~12] The abstract could mention the main outcomes of the evaluation of the approaches.
1 Introduction
[19~20] Prefer citing the original sources instead of citing works that just refer to the original sources.
[22] “Layout optimisation” in the context of wind power usually refers to choosing the positions of the wind turbines. Although it is technically correct to use “cable layout optimisation” for the inter-array cabling problem, using a more exclusive term such as “collection system” or “electrical network” or “internal grid” or “array cable routing” optimization would draw a clearer distinction between the disciplines.
[25~26] MST and TSP are not algorithms
[51-52] Sentence is missing a verb
[59] Reference Valerio et al. (2026b) is not published, but the current work builds on it. At least a pre-print should be made available to enable the assessment of the contributions of the current work.
[71] When you use “comprises”, what follows should cover the entire solution, not only the first step. Maybe you meant “includes”?
2 Graph preparation
Is the time spent on graph preparation negligible?
[93-95] This description of the additional vertices would really benefit from a figure specifically illustrating the densification approach.
[106] Introduce what you mean by “every segment”, you were just talking about vertices in the previous sentence.
3 Minimum path connection
Reconsider the section title. The section seems to cover the minimization of the length of the entire network. “Path” is a simple sequence of nodes.
[134] “minimises the connections” sounds like an edge count minimization, but you are minimizing total length or total weight (penalized length).
[136~137] Consider using “cycles” instead of “loops”, as the former is better aligned with graph terminology. “Cable strings” can also be improved by specifying that turbines can have a maximum degree of 2. Do you mean “simple path” instead of “single path”?
[139] Instead of “categorized”, use “partitioned” (more precise). Or state that N_{ac} = Union(N_t, N_s).
[Eq. 5] Any particular reason to not use the simpler |N_t| as the right-hand-side?
[161] Avoid symbol that is juxtaposition of two symbols (CP).
[215] spatially instead of “spacially”. If the development area is split in disjoint polygon, why not treat each as a separate problem?
[222] It is not clear what you mean by “which serves as a feasibility requirement rather than an optimisation constraint”. The optimization already includes many constraints that are connected to feasibility and that use user-specified parameters. What is different here?
4 Conductor size selection
[230~231] Avoid creating mathematical symbols by simply concatenating letters (CT), it makes expressions harder to read. Does installation cost include acquisition cost? Is the latter not relevant?
[Eq.18] Discount rate is typically adimensional, so expression 17 has energy dimension and expression 15 has currency dimension. Why not bring both to a common dimension instead of introducing this abstract alpha parameter?
[Eq.20] γ_t is not defined
[Eq.25] This seems in conflict with the previous use of Xi. Shouldn’t the right-hand-side be Xi_a? Or maybe you could formally define A_{ac}, given that “Aac is defined as the subset of E previously selected by a path algorithm” is not very precise.
[296] State what v_i is supposed to indicate.
5 Joint routing and conductor size optimisation
[310] Feasibility is only determined by the maximum capacity cable type available. In terms of feasibility, the CSS and routing problems are decoupled.
[331] Isn’t the number of active branches at least the same as the number of turbines?
[350] When you say it does not increase linearly, is the relationship superlinear or sublinear? How does the conclusion follow from the premise?
[360] It seems like the first formal definition of A_{ac} is in fig.3. This definition would be useful much earlier in the manuscript. It seems like a poor naming choice to call it “permissible edges”. Isn’t A_{ac} the set of active edges of a routing solution?
[Fig.3] Maybe unifying “initialize CT” and “CT <- {…}” at the top of the diagram would make it clearer what the loop is actually iterating over.
6 Layout modelling
[397] By “those obtained without bifurcations” do you mean those obtained with bifurcations? You should show some evidence (empirical or theoretical) that radial configurations help in approaching optimality.
[Table 1] The “Type” column should use some short handle that helps to understand the different methods. Using subsection numbers is too obscure.
[405] The linear costs used for each cable type should be presented as a table.
[420-425] What is the voltage level used for that example?
[436] Hornsea 1 has 174 turbines.
[439] Please clarify “pathfinding mechanism can identify hook-shaped strings to optimise interconnections”
[470] The role of the use of the assumed LCoE in the comparison is not clear. What was the value for alpha used in those non-linear examples?
[Table 5] How are the costs associated with losses computed?
7 Discussion
[491] Again, a reference to the unavailable work with great relevance for the assessment of the current work.
[494] Using the time to “last feasible solution” as a reference seems odd. Either you reference the first feasible solution (i.e. time to feasibility) or you use the gap (time to a reach a pre-defined gap).

Citation: https://doi.org/10.5194/wes-2026-53-RC2

Bernardo Castro Valerio, Pieter M. O. Gebraad, Marc Cheah-Mane, Vinicius A. Lacerda, and Oriol Gomis-Bellmunt

Data sets

Code and data for Case Studies of: A multi-stage methodology for wind park inter-array cabling: graph preparation, layout, and sizing Bernardo Castro Valerio https://doi.org/10.5281/zenodo.18937429

Model code and software

pyflow acdc v0.5.1 Bernardo Castro Valerio https://github.com/CITCEA-UPC/pyflow_acdc

Bernardo Castro Valerio, Pieter M. O. Gebraad, Marc Cheah-Mane, Vinicius A. Lacerda, and Oriol Gomis-Bellmunt

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Short summary

Designing inter-array cable networks is a key challenge in wind farm development. This work proposes a methodology that combines graph-based routing with mixed-integer optimisation to determine cable layouts and conductor sizes while accounting for spatial constraints and electrical limits. Two strategies are investigated: an integrated optimisation and a sequential approach. The method is tested on layouts from existing offshore and onshore wind farms.


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