The graph nodes (Nac) are initially partitioned into two sets: turbine nodes Nt⊆Nac and substation nodes Ns⊆Nac, such that Nac=Nt∪Ns. To maintain power flow tracking within the system, each turbine node injects one unit of power, modelled as a fixed parameter such that pi=1 for all i∈Nt.

For substation nodes, the net injection is modelled as an integer decision variable representing power withdrawal, such that pi∈Z≤0 for all i∈Ns.

Power balance is enforced by requiring that the total substation withdrawal matches the total turbine injection: 1∑i∈Nspi=-|Nt|. Units of power are transferred through each edge and an integer variable fa∈Z is associated with each candidate edge a∈E, representing the number of turbine units transmitted between nodes i and k. Although the graph is undirected, each edge a=(i,k) is assigned an arbitrary reference orientation for flow accounting purposes. A negative value of fa therefore indicates flow opposite the chosen orientation. The following constraint ensures that the power generated by all turbines is transmitted through the grid and that no turbine remains unconnected: 2pi=∑a=(i,k)∈Ek∈Nac,k≠ifa∀i∈Nac. To determine whether an edge is selected within the system, a binary variable ξa∈{0,1} is assigned to each candidate edge, indicating whether the corresponding edge is activated. To guide the optimisation towards selecting an appropriate number of edges, the following constraint is introduced: 3∑a∈Eξa=|Nt|. This constraint states that the minimum number of edges required to interconnect all nodes in a single graph is one less than the number of nodes. Accordingly, the number of edges necessary to connect all turbines to a given number of disjoint substations can be determined, resulting in the formation of separate graphs within the wind farm structure.

Furthermore, the binary variable enables the restriction of crossing pair activations, ensuring that at most one edge from each crossing pair is active. Crossing pairs are defined by the set X which denotes the set of unordered edge pairs (a,b) such that edges a and b geometrically intersect, as constrained by 4ξa+ξb≤1∀(a,b)∈X. To limit the maximum number of turbines connected via a single string, the power flow on each edge is constrained such that 5-ξa⋅M≤fa≤ξa⋅M. Here, M is a parameter that represents the maximum allowable integer power flow per edge, equivalent to the maximum number of turbines per string, and ξa ensures that no power is permitted through inactive edges. To regulate the number of permissible connections at each turbine node, the following constraint is imposed, limiting the number of active edges connected to each turbine node. 61≤∑a∈Eiξa≤2∀i∈Nt. This is achieved by summing the binary activation values over the subset Ei⊂E of edges connected to node i. In radial arrays, this threshold is set to two for turbine nodes, ensuring that each turbine is connected to at most two active edges and thereby preventing bifurcations. Relaxing this constraint permits bifurcations and enables radial-star configurations when required.

A key consideration in the design of array systems is the number of permissible string connections to a substation. This is enforced by the following constraints: 7ass≤∑a∈Eiξa≤Ss∀i∈Ns,7bss=|Nt||Ns|⋅M. Here, Ss denotes the maximum number of string connections permitted at each substation node i∈Ns and is treated as a fixed integer within each MILP solve. To reduce the search space, the minimum number of string connections per substation, ss, determines the distribution according to the maximum number of turbines permitted in each string.

The objective of the optimisation is to minimise the total weight of all active edges, defined as 8min⁡∑a∈Eξa⋅Wa. Here, Wa denotes the pre-computed weight of edge a obtained during the graph preparation step (e).

To improve the practicality and balance of the solutions produced by the core MILP formulation, additional modelling strategies are introduced. These refinements aim to guide the optimisation process towards routings that are not only feasible but also well-structured in terms of string distribution and substation connectivity.

One possible formulation for cable size selection employs a non-linear model of AC power flow (ACOPF). This formulation considers the voltage magnitude V and angle θ at each node and calculates the active P and reactive Q power flowing in each branch based on the branch admittance representation derived from Y=G+jB. The apparent power Sik from node i to node k is then defined as 14aSik=Pik+jQik,14bPik=|Vi|2Gff+|Vi||Vk|Gftcos⁡(θi-θk)+Bftsin⁡(θi-θk),14cQik=-|Vi|2Bff+|Vi||Vk|Gftsin⁡(θi-θk)-Bftcos⁡(θi-θk). Each node i∈Nac is assumed to correspond to either a turbine or a substation, such that at most one of Pti or Psi is non-zero. Accordingly, the nodal power balance is given by 15aγtiPti-Psi=∑k∈Nack≠iPaik-eq∀i∈Nac,15bQti-Qsi=∑k∈Nack≠iQaik-eq∀i∈Nac. Here, variables Pti and Qti denote the active and reactive power generated at node i by a connected turbine, while Psi and Qsi represent the corresponding substation consumption. The term γtiPti represents curtailed generation, where γt=1-curtailmentpu.

This allows consideration of the trade-off between system losses and cable investment costs. Non-linear modelling captures the complete π model of cables, including inductance and, crucially for submarine cables, capacitance, which can increase reactive power and thereby lead to additional losses. The formulation of a non-linear CSS power flow is presented in , where each array section a includes multiple Ybus-branch elements corresponding to the various conductor types that may connect the node pair, as illustrated in Fig. .

The use of equivalent bus admittance matrices establishes a relationship in which the power at each end of the equivalent branch is determined by multiple binary factors ξa,ι∈{0,1}, which relate the apparent power flow associated with each conductor type, Sa,ι, to the equivalent branch flows Saik-eq and Saki-eq, such that 16aSaik-eq=∑ι∈Iξa,ι⋅Sa,ιik,16bSaki-eq=∑ι∈Iξa,ι⋅Sa,ιki. Here, Sa,ιik denotes the apparent power flow associated with conductor type ι on branch a=(i,k), directed from node i to node k, where ξa,ι is indexed by both the branch position a and the conductor type ι∈I. The referenced paper presents a non-linear approach that considers internal system losses, allowing for the intentional over-sizing of conductor sections to reduce losses. These losses are incorporated into the main CSS objective within a multi-objective framework as follows: 17min⁡1-1+r-yr⋅Hy⋅ρ⋅∑i∈NtPti-∑i∈NsPsi. Here, the losses are calculated as the total available power to be injected by the turbines, ∑i∈NtPti, minus the total power extracted by the substations, ∑i∈NsPsi, in the array system. The economic parameter Hy denotes the full-load hours of system operation over y years, while ρ denotes the electricity cost, for example based on an assumed LCoE. The discount rate r enables a net present value (NPV) comparison with conductor investment costs. Both individual objectives are combined into a multi-objective function using a weighting parameter α∈R≥0, which determines the relative importance of the loss-related term, 18min⁡Eq. (13)+αEq. (17).

While non-linear modelling provides the most comprehensive representation of losses and power flow, it can be computationally intensive. If efficiency is a priority, a strategy based on a linear simplification of power flows, commonly referred to as the DCOPF approximation, can be adopted. This subsection presents the corresponding equality and inequality constraints associated with this formulation.

In the integrated formulation, conductor size selection is incorporated directly into the routing optimisation. Each candidate edge may therefore be associated with multiple conductor types, and the optimisation determines both whether the edge is activated and which conductor type is assigned.

Each conductor type ι∈I is characterised by a flow capacity Mι. To represent the assignment of conductor types to candidate edges, the binary edge-activation variable is extended with an additional dimension ξa,ι associated with each branch. This variable links the activation of an edge to the selection of a specific cable type, such that 27∑ι∈Iξa,ι=ξa∀a∈E, Here, the sum of assigned conductor types equals one if the branch is active and zero otherwise. As the flow limit now depends on the selected conductor type, Eq. () is modified to reflect type-specific capacities: 28-∑ι∈Iξa,ι⋅Mι≤fa≤∑ι∈Iξa,ι⋅Mι. In this formulation, the parameter M used in Eq. () is defined as the maximum value over all conductor capacities, i.e. M=max⁡ι∈IMι, ensuring that each branch satisfies criterion C3. To constrain the permissible number of active conductor types from the set and comply with criterion C5, the previously defined binary variable νι is employed. Leveraging Eq. (), which defines the total number of active branches based on the number of turbines and substations, the term (|Nt|) provides an upper bound on the number of branches. Accordingly, Eq. () is reformulated as follows: 29∑a∈Eξa,ι-|Nt|⋅νι≤0∀ι∈I. The objective of the optimisation is to minimise the total cost of all active edges as follows: 30min⁡∑a∈Eξa⋅Wa⋅ψa+∑ι∈Iξa,ι⋅La⋅ψι. Here, ψι and ψa represent the cost per unit of length of each cable type and the installation cost per unit of length respectively. It is important to note that installation costs are based on the weighted length Wa, which accounts for soft exclusions from graph preparation (d, e), while conductor costs are based on the actual length La of each segment. The objective thus captures a combined cost of installation and cable type selection.

However, the inclusion of cable size selection is mutually exclusive with the iterative determination of substation connections, as the computation of the initial connection set-point to the substation depends on the largest conductor used. If this is decided in advance, it constrains the space of possible solutions.

To address this limitation, a sequential approach is proposed. This formulation allows the routing stage to incorporate more detailed constraints that would otherwise overly restrict the integrated optimisation problem. However, it also highlights a limitation of single-pass routing approaches. When routing is performed independently of CSS, the routing stage primarily minimises geometric length, while conductor costs are determined only subsequently during cable size selection. As illustrated in Fig. a, the relationship between conductor cross-sectional area and power capacity is sublinear, such that progressively larger increases in conductor cross-sectional area are required to accommodate additional power. Furthermore, Fig. b shows that higher-capacity conductors incur disproportionately greater costs. Therefore, the shortest trench or cable routing typically concentrates power into fewer, larger conductors and does not necessarily correspond to the minimum total investment cost, as the routing stage optimises geometric cable length independently of conductor costs.

This mismatch is further influenced by the definition of the maximum number of turbines per string, M, which is based on the capacity of the largest available conductor. Consequently, routing solutions that minimise cable length tend to favour configurations with a larger number of turbines per feeder, implicitly encouraging the use of higher-capacity cables and potentially increasing the overall investment cost.

The sequential algorithm, illustrated in Fig. , addresses this issue by iterating between routing and conductor sizing decisions. The procedure begins with the graph prepared using the method described in Sect. and considers a set of possible conductor types I. In the first iteration, the minimum path connection problem (Sect. ) is solved in conjunction with the substation connection strategy described in Sect. to determine the trenching investment cost, denoted by ΨMP. The resulting subset of edges determined by the routing optimisation, A, is then evaluated using either a non-linear CSS approach (Sect. ), which accounts for conductor losses and investment, or a linear CSS model (Sect. ), which focuses solely on conductor investment and is computationally more efficient.

Following the CSS assignment, the total cost Ψi for iteration i is calculated by summing the trenching and conductor investment costs. The largest conductor type used in that iteration, denoted ιmax⁡(i), is then identified. The solution is subsequently added to the feasible solution set F.

In the subsequent iteration, the candidate conductor set I is restricted to conductor types with capacities strictly smaller than ιmax⁡(i), and the procedure is repeated. This progressively limits the search to solutions relying on smaller conductor types. The algorithm terminates under three conditions: (i) if the routing problem becomes infeasible, for example due to path limitations, a restricted number of allowable substation connections, or an empty conductor set or (ii) if the user-defined iteration limit is reached, or (iii) if the reduction in conductor costs achieved by using smaller cable types is outweighed by the increase in trenching costs due to longer routing distances, resulting in a higher total cost than in the previous iteration.

The final array configuration is selected as the solution with the minimum total cost from the feasible solution set F.

To provide a visual example of the routing formulation and its refinements, the turbine distributions of Baltic Eagle [DE] and Moray East [GB-SCT] are used. Based on the turbine and substation locations, several tests are conducted. The evaluated formulations, summarised in Table , are identified using the following short descriptive handles.

Core: core formulation for determining minimum paths (Sect. )

The corresponding cable routing solutions are shown in Fig. for Baltic Eagle [DE] and Fig. for Moray East [GB-SCT].

As shown in Fig. , the core formulation yields short strings that are balanced according to the minimum turbines-per-string constraint. However, for the Moray East [GB-SCT] case, the algorithm fails to produce a feasible solution under the minimum turbines-per-string criterion within the specified parameters, owing to limitations on inter-substation connectivity. In the subsequent formulation, where the minimum number of substation connections is determined iteratively, the solution time is reduced in both cases.

As illustrated in Table , relaxing the constraint (Eq. ) to allow radial-star configurations results in shorter trench lengths in both cases, although at higher computational cost. Since radial configurations are a subset of radial-star configurations, the resulting system routings cannot be shorter than those obtained with bifurcations; however, the reduced solution space can aid the solver in approaching optimality.

To enable broad application of the formulation, a cable database was created. For offshore cases, the database is based on copper cables from and includes 22, 33, and 66 kV voltage levels, covering 10 conductor cross-sectional areas ranging from 95 to 1000 mm². For onshore cases, the database is based on copper cables from at the 33 kV voltage level, covering 15 cross-sectional areas from 95 to 2000 mm², assuming buried cables under trefoil conditions.

In all simulations, the cost per metre of each conductor type is estimated using the procedure described in , which relates cost to the conductors cross-sectional area. Based on the same reference, a trenching cost of 304 k EUR km⁻¹ is used in offshore simulations. For onshore simulations, the trenching cost is assumed to be half this value, as no reliable public data were found.

In each case study, the number of conductor types is limited to N=3, meaning that at most three conductor types from the cable database may be selected in the optimisation. In addition, no explicit general constraint is imposed on substation connections. Both the cable database and the implementation of the methodology are available in the repository . Simulations were performed on a desktop computer equipped with a 12th Gen Intel^® Core™, i5-12400 processor (2.50 GHz) and 64 GB of RAM. The Gurobi solver was used to solve the mixed-integer linear problems.

For the integrated approach, a time limit of 6 h (21 600 s) was imposed. In contrast, the sequential approach used a time limit of 1.5 h (5400 s) per sub-problem; therefore, the total runtime may exceed this limit depending on the number of iterations required. In the sequential approach, all CSS evaluations are performed using the linear formulation presented in Sect. .

To illustrate the joint routing and conductor optimisation, the case of Westermost Rough [GB-ENG] is first presented. This example uses the same graph shown in Fig. 1f. For the sequential approach, the intermediate results of the algorithm iterations are depicted in Fig. 7a–e.

In the first iteration, with all 10 conductor capacities available, the minimum-route solution connects seven turbines per string, requiring a conductor size of 800 mm². In the second iteration, the 800 and 1000 mm² conductors are excluded from the available options. The process continues until iteration five, where the total investment exceeds that of iteration four. The resulting optimal configuration includes cable types of 95, 120, and 185 mm².

For comparison, the integrated CSS result is shown in Fig. 7f. This result uses the same conductor selection as iteration three of the sequential algorithm (see Fig. 7c). In iteration three, the connections to the substation are determined within the optimisation. In contrast, the integrated CSS does not perform this evaluation, allowing the algorithm to generate shorter strings, assign the 300 mm² conductor to only three strings, and increase the total number of strings, thereby reducing the overall cost. A detailed cost breakdown is provided in Table , where increases in cable length, trench length, and installation cost can be observed as the iterations progress; however, the total cost continues to decrease until the reduction in conductor costs no longer compensates for the additional trenching.

Further case studies were implemented, covering wind parks ranging from 12 to 174 turbines, with power capacities between 2 and 14 MW, and including arrays with one to three substations. Detailed information on the number of permissible edges, number of turbines, identified crossings, and turbine power and voltage is given in Table . The table also reports the graph preparation time for each case.

Hornsea I [GB-ENG] (Fig. 8a, b) serves as the largest test case, with 174 turbines and three substations. Anholt [DK] (Fig. 8c, d) highlights how inter-turbine edges are prevented from crossing export cables, in this case due to the connection from Anholt Island to Denmark. Horn Rev 2 [DK] (Fig. 8e, f) illustrates the effect of the sequential balancing constraints, which favour a more even distribution of turbines across strings than in the integrated solution. Moray West [GB-SCT] (Fig. 8g, h) exemplifies configurations with large-capacity turbines. Seagreen [GB-SCT] (Fig. 8i, j) demonstrates the use of irregular development areas, where densification of points generates non-standard edges that comply with spatial constraints.

Table summarises all offshore test cases, reporting the objective value, solution time, and the final optimality gap if the time limit was reached. The optimality gap measures the difference between the best feasible solution and the solver's lower bound, which comes from LP relaxations and represents a provable bound on the optimal objective . Simulation times marked with an asterisk (^*) indicate when the last feasible solution was obtained. Detailed results are given in Tables and of the individual objective costs.

In the onshore cases, the z plane incorporates elevation, with inclination playing a more significant role in further limiting the connectivity of the densified points. The test cases cover wind parks ranging from 10 to 106 turbines, with power capacities between 225 kW and 3.6 MW. Detailed information on the number of permissible edges, number of turbines, identified crossings, and turbine power and voltage is given in Table .

For example, Bhlaraidh [GB-SCT] (Fig. 9a, b) illustrates the role of exclusion zones in routing around waterbodies such as lakes, while Coromuel [MX] (Fig. 9c, d) demonstrates that the algorithm can still achieve feasible solutions in highly constrained development areas. Finally, Table summarises all onshore test cases, reporting the objective value, solution time, and the final optimality gap if the time limit is reached. Simulation times marked with an asterisk (^*) indicate when the last feasible solution was obtained. The results are given in more detail in Tables and of the individual objective costs.

To bridge the difference to comparable project data, the inter-array documentation for Moray East [GB-SCT] and Moray West [GB-SCT] are used as a reference. In the simulations, cable selection is restricted to two aluminium types, with a maximum of eight and six strings per substation for Moray East [GB-SCT] and Moray West [GB-SCT] respectively.

The simulation utilises four aluminium conductors, defined based on the cable capacities reported in for conductor sizes of 240, 300, 650, and 800 mm², as well as technical data and cost approximations from and . The project documentation specifies the cable length used for each conductor size, and for comparability, costs are calculated based on the same costs per length as those used in the simulations. The reported routing for Moray East [GB-SCT] is reproduced in Fig. 10a for comparison with Fig. 10b, c, and d, while the reported routing for Moray West [GB-SCT] is presented in Fig. 10e and compared with Fig. 10f, g, and h, which show the resulting routings from the integrated and sequential methodologies, which respectively show the simulation of the integrated, linear sequential, and non-linear sequential approaches.

To account for losses, a non-linear approach is applied. Assuming an LCoE of EUR 91 per MWh and 5170 full-load hours annually over a 25-year project lifetime , a net present value analysis of losses relative to investment is conducted. Electrical losses are obtained from the ACOPF solution at the equivalent full-load operating point and are converted to lifetime cost using the assumed full-load hours. To enable a direct comparison between losses and investment costs, the value of α in Eq. () is set to 1. For the non-linear simulation, the sequential algorithm is executed in a WSL environment, using OR-Tools for routing and Bonmin for the CSS.

In both cases, the non-linear CSS shown in Fig. 10d and h differs in conductor size selection from the linear solutions, while more closely resembling the reported project data. Comparing the linear and non-linear CSS for Moray East, the use of smaller conductor capacities is reduced, whereas higher-capacity conductors are utilised across a greater number of connections. In the case of Moray West, two strings connect six turbines instead of five, requiring a higher-capacity cable.

For a comparable assessment, losses are also calculated using a Newton–Raphson power flow applied to the grids obtained from the linear approaches. These results are presented in Table .

Several state-of-the-art cable-routing formulations represent each candidate connection through two directed arcs, allowing the optimiser to select the direction associated with power flow towards the substation . This provides a useful structural reference for the routing component of the proposed formulation. For the quantitative benchmark, however, OptiWindNet  is selected because it is available as an open-source framework and directly addresses the spatial embedding of selected connections. Its search space is built from a sparsified candidate graph based on constrained Delaunay triangulation, diagonals, and feeder links. In particular, it goes beyond purely static crossing constraints by separating topology selection from route embedding, enabling non-straight cable routes and reducing the number of crossing constraints considered during optimisation. Gurobi was selected as the common solver for both approaches with a 6 h time limit. For cases reaching the imposed time limit, the reported value corresponds to the best incumbent solution.

The comparison is restricted to the routing component and uses total cable length and solution time as common metrics. Three test types were used to enable comparison across the case studies. The first was a non-restrictive test, in which no substation-connection limit was enforced and the maximum string length was set equal to the number of turbines in the wind park, M=|Nt|. In the second test, the maximum number of turbines per string was set according to the highest-capacity conductor available for the corresponding voltage level, as reported in Sect. . In the third test, substation connections were limited to obtain a more balanced string configuration. The resulting Ss and M values are shown in Table .

The first aspect evaluated is the problem size. Table presents the number of variables and constraints considered by the solver. In the proposed formulation, the number of variables is reduced primarily because each candidate edge is represented by a single activation variable. However, the number of constraints increases substantially owing to the use of static crossing constraints.

The second evaluation concerns computational performance, shown in Table . The proposed methodology is slower in most cases. The explicit modelling of cable crossing restrictions increases the number of constraints and introduces a combinatorial structure that is more difficult for the branch-and-bound algorithm to resolve. Although feasible solutions of comparable quality are identified early in the search, the LP relaxation bound is weaker than that of the planar-embedding approach, making the optimality gap more challenging to close within a given time limit. The advantage of the reduced variable representation is therefore not necessarily a reduction in solution time but a simpler representation of each candidate connection that facilitates the incorporation of additional routing and CSS constraints. This simplification is, however, counterbalanced by the static crossing constraints.

Both methodologies produce comparable routing lengths across the nine wind farms, with larger differences appearing mainly in the more constrained balanced cases. It is worth noting that OptiWindNet's topology-optimisation objective can differ from the final embedded route length when post-processing detours are required. For example, in the third Horn Sea 2 test, the topology-optimisation length before re-routing is 325.87 km; however, post-processing re-routing to avoid physical cable crossings introduces additional cable length, increasing the final embedded route length to 334.56 km.

In contrast, the proposed methodology enforces crossing avoidance directly within the MILP solution. While OptiWindNet resolves many crossing issues through topology-embedding separation and post-optimisation detours, the proposed methodology considers crossing candidate edges explicitly within the optimisation through static crossing constraints.

The case study information used for benchmarking is available in the Zenodo repository .

The comparison in Table evaluates the routing formulation quantitatively, while several modelling aspects require qualitative positioning. Existing routing formulations already include many of the individual components used here: Delaunay-based pathfinding and GIS obstacle avoidance , Steiner-node obstacle detours and practical routing variants , penalty-based obstacle and boundary treatments , and topology-embedding separation with post-optimisation detours . These approaches differ mainly in where spatial feasibility is imposed: during path generation, through additional graph nodes, as penalty terms, or after topology optimisation. The proposed formulation instead applies routing constraints to candidate routes generated by the graph-preparation stage, so hard exclusions, soft-exclusion weights, clearance constraints, and static crossing constraints are reflected before or within the MILP solve.

The core routing MILP follows standard network-design principles, including edge activation, flow conservation, capacity limits, and crossing constraints. Similar routing structures have been used in wind farm collection-system formulations based on directed arc-pairs representations . As shown in the quantitative benchmark, the distinction of the present core routing formulation is the use of a single activation and flow variable per candidate edge, together with the enforcement of a strict radial topology. The formulation is further extended through constraints on the minimum number of turbines per string, the determination of substation connections under unfavourable GIS constraints, and the possibility of unbalanced substation capacities, thereby expanding the range of user-defined design requirements that can be represented.

For CSS and loss-aware design, and provide the closest comparison with precomputed loss treatments. Both approaches estimate loss-related costs from the cable type and the number of turbines carried by a segment, while further include wind-scenario probabilities in the expected loss contribution. This keeps the optimisation tractable, but the loss assigned to each segment is based on local loading assumptions rather than on the full electrical state of the selected network. Consequently, upstream voltage conditions, reactive power flows, cable capacitance effects, and cumulative losses along preceding segments are not explicitly represented. In this paper, the ACOPF-based CSS formulation captures losses within the optimisation by representing voltage magnitudes and angles, as well as the full π model of the cable sections. However, this formulation is computationally demanding. For this reason, the linear CSS formulation is used to enable joint routing–CSS optimisation in both the integrated and sequential approaches, although losses are not included in the linear CSS objective.

The strongest joint routing–CSS benchmark is , which integrates network routing optimisation with CSS and includes restrictions on the number of cable cross-sections used. However, coupling CSS with balanced radial-routing constraints introduces an additional difficulty: the routing balance depends on the maximum number of turbines that can be carried by the largest conductor selected, but this conductor is itself an optimisation outcome. If this maximum capacity is fixed in advance, the routing problem can enforce balanced strings more easily, but the conductor-selection problem is partially predetermined. Conversely, if the conductor set is left fully open, the routing formulation cannot directly know which turbine-per-string limits should be imposed to obtain a balanced configuration. The sequential strategy is therefore used to explore this dependency step by step, restricting the available conductor set across iterations and re-solving the routing problem with the corresponding balance and substation-connection constraints. This allows the effect of conductor availability on routing, trenching cost, and CSS cost to be evaluated more explicitly than in a single integrated solution.

In all cases, the routing algorithm is constrained by the graph preparation process. The inclination threshold determines the viability of candidate connections, while soft-exclusion penalties modify their weights and therefore influence their selection. The level of densification further determines the spatial resolution of the graph, where higher densification can provide better-informed candidate paths at the expense of increased computational cost. Connections that may appear feasible upon visual inspection can be excluded due to cable crossing constraints, an inherent limitation of using static constraints. As illustrated in Fig. 12a, a visually preferable routing might involve connecting the leftmost turbine (highlighted in blue) of the orange string directly to the substation, thereby reducing total cable length. However, this configuration is excluded by the algorithm because it intersects with other inter-turbine edges.

In another example, shown in Fig. 12c, the connection set in orange originates from the original project documentation but would be excluded under the current method, as it crosses other active connections. Another restriction concerns the connectivity of possible turbine-to-turbine links. In the methodology proposed in Sect. , such connections are defined using Delaunay and diagonal edges. In Fig. 12b, the orange link, also derived from the project documentation, would not be considered under a Delaunay-diagonal configuration. This could be extended, optionally, by incorporating k-nearest neighbours into the search; however, doing so increases the graph preparation time and complicates the routing problem.

Moreover, under the graph preparation procedure, connections originating from the substation and overlapping with other substation links are not classified as crossings by the algorithm. Consequently, multiple such connections are permitted. This is also visible in Fig. 12a, where area constraints result in parallel segments running from the substation to different turbines. These examples underscore the importance not only of careful graph preparation but also of incorporating human oversight to mitigate limitations inherent in automated routing strategies.

Some of the limitations discussed above could be alleviated by refining the candidate-edge construction. Additional Steiner points, as used in related routing studies, could be introduced to increase the number of feasible detours and reduce the restrictive effect of static crossing constraints. In the limiting case, the routing optimisation could be performed directly on the densified graph produced during graph preparation step (d), rather than on the reduced candidate graph. This would provide greater routing flexibility but would substantially increase the number of candidate edges and crossing constraints. A practical compromise is therefore to adjust the density of the graph-preparation stage according to the size and complexity of the test case. Higher densification improves the spatial resolution of the three-dimensional path assessment but also increases the computational cost of pathfinding and subsequent routing optimisation.

The MILP scalability is mainly limited by the explicit crossing constraints. These constraints increase with the number of intersecting candidate-edge pairs and can make it difficult for the solver to close the optimality gap, particularly in larger or more constrained cases. The optimality gap is the difference between the best feasible solution and the solver's lower bound, which is obtained from LP relaxations and provides a provable bound on the optimal objective .

In the linear CSS formulations, losses are not included in the objective in order to keep the routing–CSS problem linear and computationally tractable. Loss-related terms could be incorporated through precomputed coefficients, as in previous loss-aware routing formulations, but such an extension would require assumptions on segment loading and would not represent the full AC state of the selected network. For this reason, losses are instead evaluated through the separate non-linear CSS formulation.

The non-linear formulation is evaluated using an equivalent full-load-hours analysis, which represents the annual operation of the wind farm by a single operating point. Consequently, the optimisation does not capture the variation in active and reactive power flows under different operating conditions. In particular, it does not account for the increased relative contribution of cable charging losses at lower loading conditions, especially for oversized conductors with higher capacitance . A more detailed representation could be obtained through a multi-scenario approach, as presented in , in which the wind speed Weibull distribution and turbine power curve are used to generate representative operating scenarios, each associated with a probability. Similarly, time series data can be clustered into representative operating conditions to capture spatial and temporal variations in turbine production. The main limitation of these approaches is the substantially increased computational burden, as the optimisation must be solved simultaneously for multiple operating scenarios rather than a single equivalent operating point.