the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 16 Mar 2026
A multi-stage methodology for wind park inter-array cabling: graph preparation, layout, and sizing
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.
- Preprint
(7334 KB) - Metadata XML
- BibTeX
- EndNote
Status: open (until 13 Apr 2026)
- AC1: 'Comment on wes-2026-53', Bernardo Castro Valerio, 16 Mar 2026 reply
-
RC1: 'Comment on wes-2026-53', Anonymous Referee #1, 06 Apr 2026
reply
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
reply
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
-
AC2: 'Reply on RC1', Bernardo Castro Valerio, 07 Apr 2026
reply
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
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 128 | 72 | 14 | 214 | 11 | 12 |
- HTML: 128
- PDF: 72
- XML: 14
- Total: 214
- BibTeX: 11
- EndNote: 12
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
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.