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.

Following the last reply, a more detailed response is attached in the pdf

We appreciate the reviewer’s detailed clarifications and have revised the manuscript accordingly.

Regarding symbol usage, the authors have added a list of symbols as Appendix A to aid readers requiring a reference table. In addition, the manuscript has undergone further proofreading, and the readability and presentation of variables have been improved following the comments from both reviewers.

Regarding comments [59] and [494], we agree that the contribution of the present work cannot be adequately assessed without access to the referenced manuscript.

Valerio 2026b, is a conference paper accepted for publication at TORQUE 2026, which could have not been made publicly available until it was presented at the conference. For this reason, we contacted the editorial support team prior to the review process so that the non-public manuscript could  have been made available to reviewers upon request. (https://doi.org/10.5194/wes-2026-53-AC1) As of now have been published and available online https://iopscience.iop.org/article/10.1088/1742-6596/3224/5/052005

The conference manuscript presents the graph preparation stage for offshore wind developments and the core MILP routing formulation, focusing on maximum turbines per string and maximum substation connections. These components are incorporated into the present WES manuscript and further extended as follows:

•  Graph preparation: A distinction is introduced between offshore and onshore distances due to the offshore connection from the sea surface to the seabed, which depends on bathymetry. In addition, a distinction is made between weighted path length and conductor length for each path, differentiating the quantities used for routing and cable cost calculations.
•  Routing algorithm: The core formulation is extended with additional user-defined strategies, including minimum turbines per string and unbalanced substation capacities.

Additionally, as referenced in comment [494], the conference manuscript presents a benchmark of the core MILP routing formulation against OptiWindNet from Souza de Alencar et al. (2025). The benchmark compares problem size in terms of the number of variables and constraints, as well as solution time. The authors did not consider it appropriate to repeat these results in the WES manuscript, as the present work extends beyond the core MILP routing formulation by incorporating an independent conductor size selection methodology and, most importantly, a joint routing and conductor size optimisation framework, which cannot be directly benchmarked against OptiWindNet. As such, the case studies presented in the WES manuscript focus on comparing the two joint optimisation methodologies, namely the integrated and sequential approaches, as well as two additional case studies specific to the sequential framework with non-linear conductor size selection.



A more detailed response can be found in the attached pdf.