Cabling Optimization in Wind Power Plants, Enhancing the Cable Type-Based Formulation

Abstract

The collection grid represents a relevant share of the total initial investments into a wind power plant. Planning/optimizing collection grids is a task that grows severely in complexity according to the size of the analyzed plant, i.e., its number of wind turbines. This paper enhances a well-known mixed-integer linear programming formulation based on the types of cables available for installation and meant for radial grids. More specifically, this work proposes valid constraints that tighten the search space and enable faster convergence. Results indicate that small wind power plants do not benefit from the novel constraints, whereas the time to solve medium and large plants can significantly decrease. However, comparisons against an alternative algorithm based on the flowing power reveal that the proposed enhancement to the cable type-based formulation does not make it the most computationally efficient. In studies regarding Thanet, a wind power plant with 100 wind turbines, the mean convergence time has decreased from 18% up to 85% for different cases when applying the proposed constraints to the cable type-based formulation. Nonetheless, such durations are 2 to 15 times more extensive than what is required by the power-based algorithm. Thus, this paper seeks to raise awareness regarding the assessed algorithms and aid in more efficient inter-array cabling optimization studies.

1. Introduction

Despite recent declines in the development of wind energy assets, intermittent renewable sources have been subject to significant investment following the energy sector transition [1]. The year 2023 has shown the second-highest offshore wind power capacity growth in history, with 117 GW of wind power capacity integrated into electrical grids, according to the Global Wind Energy Council [2]. Although most of the investments into a wind power plant (WPP) relate to wind turbines (WTs) [3], the collection grid (CG) represents a relevant share of the initial costs. For instance, Nieradzinska et al. [4] claims that the cost of Dogger Bank’s CG is around 12.9% of the total. Thus, CG studies have a relevant impact on the planning and development of WPPs.

1.1. Literature Review

The literature presents different approaches for cabling optimization problems (COP), such as methods based on metaheuristics, e.g., genetic or swarm algorithms [5,6,7,8,9,10], mathematical heuristics such as Clark and Wright’s saving algorithm [11,12], or optimization solvers addressing mixed-integer linear programming (MILP) formulations [13,14,15,16].

Regarding MILP approaches, Cerveira et al. [13] improved a power-based MILP cabling formulation (PBF) and demonstrated the method through small to medium-sized onshore WPPs. Pérez-Rúa et al. [14] presented a heuristic to reduce the COP solution space and globally optimize CGs of large WPPs. Pérez-Rúa et al. [15] studied ring and radial CGs and addressed reliability via stochastic programming. Different scenarios were defined according to cable failure probabilities. Abritta et al. [16] proposed a new reliability indicator for MILP COP formulations and demonstrated its strong correlation with the expected energy not supplied via Monte Carlo simulations.

1.2. Motivation

In general, MILP problems can be computationally intensive depending on the number of integer variables. COP’s computational complexity is strongly correlated with the number of WTs in the WPP, as this number affects the number of binary variables in the problem. Following the energy transition toward more sustainable energy systems, more and larger WPPs tend to be developed, implying more computationally expensive COP problems to be solved. Studies such as [14] report cases regarding large WPPs in which more than 10 h were required to conduct a single optimization of the problem. Therefore, there is a need to improve the efficiency of the current methods to address COP.

1.3. Contributions

This paper enhances the convergence of a well-known cable type-based COP MILP formulation (CTBF), which aims to yield optimal radial grids. References [7,15,17,18] are examples that utilize the addressed formulation as their foundation. Despite presenting particularities of their own in terms of additional characteristics, such as Steiner nodes [18] or expansion to ring CGs [15], these works have the cable type-based MILP formulation as their core. Then, this work compares its results with those of the COP MILP formulation presented in [13], which is based on network flowing power and has been implemented in [14,16,19]. Hence, this paper reports the results obtained by implementing novel constraints to tighten the optimization search space in CTBF. In other words, similarly to how [13] enhanced PBF, this research aims to improve CTBF by enabling faster convergence without compromising optimality. This paper’s approach does not regard heuristic strategies, which can potentially eliminate high-quality solutions. In contrast, it augments the problem’s structure in a way that reduces the number of possible solutions while the global optimum is guaranteed to be preserved. To the best of the authors’ knowledge, no such effort to enhance CTBF’s computational efficiency has been reported in the literature. However, this work highlights that such enhancements were insufficient to outperform PBF, as suggested by the obtained results.

1.4. Paper Organization

Beyond this introduction, Section 2 describes the COP formulations and proposed constraints. Section 3 presents the methodology for optimization assessment. Section 4 discusses the results. Section 5 concludes the work.

2. Problem Formulation

This paper addresses COP MILP formulations to yield radial CGs and assesses two different formulations. One is based on the cable types available for installation [7,15] and is referred to as CTBF. The other relates to the power that flows in each cable [13] and is referred to as PBF. Although this paper is built upon offshore WPP examples, note that the addressed concepts also apply to onshore plants subjected to similar formulations.

2.1. Cable Type-Based Formulation

In this mathematical model, the CG is a directed graph that converges to the substation(s), as illustrated in Figure 1. Binary variables $u_{i,j,t}$ indicate if the type $t\in \mathcal{C}$ cable linking $i\in \mathcal{N}$ to $j\in {\mathcal{N}}_0$ exists, where $\mathcal{N}$, ${\mathcal{N}}_0$, and $\mathcal{C}$ are the sets of WTs, WTs plus substations, and cable types, respectively. Continuous variables $P_{i,j}$ represent how much power flows from i to j. Equation (1) defines the objective function for cost minimization, where $L_{i,j}$ approximates the cable length by the Euclidean distance between i, located at ($x_i,y_i$), and j, located at ($x_j,y_j$). Furthermore, $p_t$ represents the purchase price plus installation cost per distance for type t cables.

$${\mathcal{F}}_{\mathit{CTBF}}=\sum _{i\in \mathcal{N}}\sum _{j\in {\mathcal{N}}_0}\sum _{t\in \mathcal{C}}{L}_{i,j}·p_t·u_{i,j,t}$$

(1)

2.1.1. Standard Constraints

Regarding the problem constraints, Equation (2) conveys that every $i\in \mathcal{N}$ must connect to one and only one $j\in {\mathcal{N}}_0$ utilizing one of the available cable types.

$$\sum _{j\in {\mathcal{N}}_0}\sum _{t\in \mathcal{C}}{u}_{i,j,t}=1,\forall i\in \mathcal{N},i\ne j$$

(2)

Crossing cables may cause excessive heating, thus requiring extra cable insulation. In addition, they increase maintenance costs [20]. Given these reasons, crossing prohibition is a common constraint in COP formulations. Consider $i_1,i_2,i_3\in \mathcal{N}$ and $j\in {\mathcal{N}}_0$. If the segments $i_1{i}_2$ and $i_{3}j$ intersect, either Equation (3) or Equation (4) applies. The former is valid if j is not a substation; otherwise, the latter is valid. Observe that the potential intersection of segments must be verified for every combination of $i_1,i_2,i_3,j$ in which the elements differ.

$$\begin{array}{c}{\displaystyle \sum _{t\in \mathcal{C}}\left(u_{i_1,i_2,t}+u_{i_2,i_1,t}+u_{i_3,j,t}+u_{j,i_3,t}\right)\le 1}\end{array}$$

(3)

$$\begin{array}{c}{\displaystyle \sum _{t\in \mathcal{C}}\left(u_{i_1,i_2,t}+u_{i_2,i_1,t}+u_{i_3,j,t}\right)\le 1}\end{array}$$

(4)

Equation (5) ensures power flow conservation, assuming nominal power generation, i.e., 1 p.u. This is crucial to secure operational feasibility during favorable wind scenarios with maximum power production. Equation (6) implies the installed cables can tolerate the maximum transportable power, where $P_t^{\mathit{cap}}$ indicates the power capacity, in p.u., of each cable type t. Note that, if the cable is not installed (i.e., if $u_{i,j,t}=0$), the flowing power is bound to zero. Equation (7) limits the number of feeders (M), i.e., cables connecting directly to a substation. Equations (8) and (9) define the binary nature of the u variables and the lower bounds of the P variables.

$$\begin{array}{c}\sum _{j\in {\mathcal{N}}_0}{P}_{i,j}-\sum _{q\in \mathcal{N}}{P}_{q,i}=1,\forall i\in \mathcal{N}\end{array}$$

(5)

$$\begin{array}{c}\sum _{t\in \mathcal{C}}{P}_t^{\mathit{cap}}·u_{i,j,t}\ge P_{i,j},\forall i\in \mathcal{N},\forall j\in {\mathcal{N}}_0\end{array}$$

(6)

$$\begin{array}{c}\sum _{i\in \mathcal{N}}\sum _{t\in \mathcal{C}}{u}_{i,j,t}\le M,\forall j\in {\mathcal{N}}_0-\mathcal{N}\end{array}$$

(7)

$$\begin{array}{c}{u}_{i,j,t}\in \{0,1\},\forall i\in \mathcal{N},\forall j\in {\mathcal{N}}_0,\forall t\in \mathcal{C}\end{array}$$

(8)

$$\begin{array}{c}{P}_{i,j}\ge 0,\forall i\in \mathcal{N},\forall j\in {\mathcal{N}}_0\end{array}$$

(9)

2.1.2. Proposed Constraints

The constraints in Equations (2)–(7) enable feasible solutions regarding grid power capacity, crossing prohibition, flow conservation, and feeder limitation. However, they do not fully exploit the COP characteristics related to the interactions between cable choices and power transportation. Hence, this work presents additional constraints that leverage COP aspects to tighten the search space and enable faster optimization convergence.

Through Equation (6), the standard formulation defines upper bounds for the transmitted power according to each cable type and capacity. However, although Equation (9) imposes lower bounds of zero on the $P_{i,j}$ variables, one can implement additional lower bounds on the variables according to the cable types, if at least two types are available for installation. Assuming an installed cable is of type $t=c_x,x\ne 1$, it is inevitable that its nominal power transportation is greater than or equal to the capacity of the cable type $c_{x-1}$ plus one, i.e.,

$$\sum _{j\in {\mathcal{N}}_0}\sum _{t\in \mathcal{C},t\ne c_1}\left(P_{t-1}^{\mathit{cap}}+1\right)·u_{i,j,t}\le \sum _{j\in {\mathcal{N}}_0}{P}_{i,j},\forall i\in \mathcal{N}$$

(10)

After all, if the power flowing from i is less than or equal to $P_{t-1}^{\mathit{cap}}$, being of type t is not optimal, as type $t-1$ is less costly and can tolerate the required power flow.

It is noteworthy that, for each $i\in \mathcal{N}$, Equation (6) links each $P_{i,j}$ variable to the related $u_{i,j,t}$ summation across $\mathcal{C}$. In other words, if i connects to j, the power from i can only flow to j and no other component of ${\mathcal{N}}_0$. This binding makes it possible to write the power variables in Equation (10) as summations across ${\mathcal{N}}_0$ without compromising the feasibility of the obtained solutions. This affirmation also applies to the following constraint.

Suppose the WPP has at least two cable types available and an installed cable $ij$ is of type $t=c_x,x\ne 1$. As a direct consequence of Equation (10), if the power flowing from i is greater than or equal to $P_{t-1}^{\mathit{cap}}+1$ and i links to j, the nominal power departing j must be greater than or equal to $P_{t-1}^{\mathit{cap}}+2$ because of the additional power generated by i, i.e.,

$$\sum _{t\in \mathcal{C},t\ne c_1}\left(P_{t-1}^{\mathit{cap}}+2\right)·u_{i,j,t}\le \sum _{q\in \mathcal{N}}{P}_{j,q},\forall i\in \mathcal{N},\forall j\in {\mathcal{N}}_0.$$

(11)

If a cable departing i is of type t, the nominal power transportation reaching i is certainly less than or equal to the capacity of cable type t minus one. Otherwise, component i’s cable would be unable to transport the power equal to the incoming flow added to its own generation of 1 p.u. This statement is mathematically represented by

$$\sum _{j\in {\mathcal{N}}_0}\sum _{t\in \mathcal{C}}\left(P_t^{\mathit{cap}}-1\right)·u_{i,j,t}\ge \sum _{q\in \mathcal{N}}{P}_{q,i},\forall i\in \mathcal{N}.$$

(12)

As seen, Equation (10) imposes lower bounds on the flowing power depending on the cable type; Equation (11) tightens the lower bounds of cables downstream of i, adopting a convention where the substation is downstream of all WTs; Equation (12) tightens the upper bounds of cables upstream of i. The proposed constraints have been developed empirically. Nonetheless, it is emphasized that they do not violate any optimality aspect. Their only limitation concerns Equations (10) and (11), which are only applicable when more than one cable type is available for installation (which is usually the case). Despite increasing the optimization problem size, further sections demonstrate the effectiveness of including the proposed constraints when optimizing medium to large WPPS. To the best of the authors’ knowledge, Equations (10)–(12) have not been reported in the literature.

2.2. Power-Based Formulation

In this model [13], the CG is a directed graph with roots at the substation. Binary variables $y_{j,i,n}$ denote cables linking $i\in \mathcal{N}$ to $j\in {\mathcal{N}}_0$ and transporting the nominal power generation equivalent to n WTs, as illustrated in Figure 2. Note that variables y do not relate directly to the cable types. If power from n WTs flows through a certain cable, its type is implicitly the least costly one capable of supporting n WTs.

Equation (13) accounts for cable purchase and installation costs per distance, in addition to their lengths, to describe the objective function. Nonetheless, PBF accounts for the prices based on n, i.e., the flowing power. The auxiliary vector $\alpha$, yielded by Algorithm 1, represents the price per distance according to each power transportation possibility. In Algorithm 1, $k_{c_x}\in \mathcal{K}$ indicates the maximum number of WTs whose nominal power can be transported by a cable of type $c_x$. For instance, given the number of cable types ($\mathit{nct}$) available for installation, $k_{c_{nct}}$ represents how many WTs can have their nominal generation flowing through a cable whose type has the greatest capacity. Equation (14) yields $\mathcal{K}$, with $P^{\mathit{cap}}$ in p.u. and $\lfloor \rfloor$ denoting the floor operation. In Equation (13), $Q_j$ denotes the highest quantity of WTs that can be upstream of $j\in {\mathcal{N}}_0$, meaning $Q_j=k_{c_{nct}}$ if and only if j is a substation. Otherwise, $Q_j=k_{c_{\mathit{nct}}}-1$.

$$\begin{array}{c}{\mathcal{F}}_{\mathit{PBF}}=\sum _{i\in \mathcal{N}}\sum _{j\in {\mathcal{N}}_0}\sum _{n=1}^{Q_j}{L}_{ij}·{\alpha }_t·y_{j,i,n},\end{array}$$

(13)

$$\begin{array}{c}\mathcal{K}=\lfloor P^{cap}\rfloor \end{array}$$

(14)

Algorithm 1 Steps to associate cable prices with power transportation

1: $\mathit{aux}=1$ 2: for x in 1 to $k_{c_{nct}}$, by $1,$ do 3:       if $x\le k_{c_{aux}}$ then 4:            ${\alpha }_x=p_{c_{\mathit{aux}}}$ 5:       else 6:            $\mathit{aux}=\mathit{aux}+1$ 7:            ${\alpha }_x=p_{c_{\mathit{aux}}}$ 8:       end if 9: end for

Several PBF constraints are analogous to CTBF. As in Equation (2), every $i\in \mathcal{N}$ must connect to one $j\in {\mathcal{N}}_0$, transporting power equivalent to a specific number n of WTs.

$$\sum _{j\in {\mathcal{N}}_0}\sum _{n=1}^{Q_j}{y}_{j,i,n}=1,\forall i\in \mathcal{N},i\ne j$$

(15)

Regarding cable-crossings, Equations (16) and (17) rewrite Equations (3) and (4).

$$\begin{array}{c}\sum _{n=1}^{k_{c_{\mathit{nct}}}-1}\left(y_{i_2,i_1,n}+y_{j,i_3,n}+y_{i_1,i_2,n}+y_{i_3,j,n}\right)\le 1\end{array}$$

(16)

$$\begin{array}{c}\sum _{n=1}^{k_{c_{nct}}}{y}_{j,i_3,n}+\sum _{n=1}^{k_{c_{\mathit{nct}}}-1}\left(y_{i_2,i_1,n}+y_{i_1,i_2,n}\right)\le 1\end{array}$$

(17)

Equation (18) ensures that the number of WTs upstream and downstream of $i\in \mathcal{N}$ is feasible. Since the number of connected WTs is equivalent to the nominal power in p.u., Equation (18) secures power flow conservation, thus relating to Equation (5).

$$\sum _{j\in {\mathcal{N}}_0}\sum _{n=1}^{Q_j}n·y_{j,i,n}-\sum _{q\in \mathcal{N}}\sum _{n=1}^{k_{c_{\mathit{nct}}}-1}n·y_{i,q,n}=1,\forall i\in \mathcal{N}$$

(18)

Equation (19) is analogous to Equation (7).

$$\sum _{i\in \mathcal{N}}\sum _{n=1}^{k_{c_{nct}}}{y}_{j,i,n}\le M,\forall j\in {\mathcal{N}}_0-\mathcal{N}$$

(19)

Equation (20) states that the substations must collect the power from all WTs, with $\left|\right|$ denoting the set cardinality.

$$\sum _{j\in {\mathcal{N}}_0-\mathcal{N}}\sum _{i\in \mathcal{N}}\sum _{n=1}^{k_{c_{nct}}}n·y_{j,i,n}=\left|\mathcal{N}\right|$$

(20)

If $y_{j,i,\tau }=1$, the connections leaving i with n upstream WTs must be less than or equal to $\lfloor (\tau -1)/n\rfloor$. Cerveira et al. [13] utilized this property to propose Equations (21) and (22) to tighten the solution region with no compromise to optimality.

$$\begin{array}{c}{\displaystyle \sum _{j\in {\mathcal{N}}_0}\sum _{\tau =n+1}^{Q_j}⌊\frac{\tau -1}{n}⌋·y_{j,i,\tau }\ge \sum _{q\in \mathcal{N}}\sum _{\tau =n}^{k_{c_{\mathit{nct}}}-1}{y}_{i,q,\tau },}\\ \forall i\in \mathcal{N},\forall n\in 2,\dots ,k_{c_{nct}}-2\end{array}$$

(21)

$$\begin{array}{c}\sum _{j\in {\mathcal{N}}_0-\mathcal{N}}{y}_{j,i,k_{c_{nct}}}\ge \sum _{q\in \mathcal{N}}{y}_{i,q,k_{c_{\mathit{nct}}}-1},\forall i\in \mathcal{N}\end{array}$$

(22)

This section presented CTBF and the proposed equations to enhance its convergence. Additionally, it has described the PBF approach for modeling COP for radial grids. The following section addresses the methods utilized to solve such formulations and compares their computational performance.

3. Methods

The methodology behind some of the parameters from the previous section must be clarified before describing the optimization framework. As mentioned, L is a matrix that contains the distances among all WPP elements approximated by their Euclidean distances. Therefore, $\forall i\in \mathcal{N}$, located at ($x_i,y_i$), and $\forall j\in {\mathcal{N}}_0$, located at ($x_j,y_j$), Equation (23) applies. The WPPs to be analyzed are listed in

Table 1. Wind power plants under study.

WPP	$\left|\mathcal{N}\right|$	${\mathit{P}}_{\mathit{wt}}^{\mathit{nom}}$	M
Ormonde	30	5.0 MW	4
Westermost Rough	35	6.0 MW	6
Horns Rev 3	49	8.3 MW	8
Robin Rigg	58	3.0 MW	8
Thanet	100	3.0 MW	10

, where $\left|\mathcal{N}\right|$ is the cardinality of $\mathcal{N}$, i.e., the number of WTs. The coordinates of the WPP elements (WTs and substations) were extracted from [21] (ORM, WMR, HR3, and RR) and [14,22] (TH). Files with the coordinates are available in [23].

$$L_{i,j}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}$$

(23)

This paper considers prices associated with the purchase and installation of cables. For simplicity, it neglects other expenditure sources, such as power losses. Since the prices are merely constant values to base the quantification of the objective function in Equation (1), the authors argue that the nature of such elements is negligible regarding the assessment of how Equations (10)–(12) can enhance the COP convergence. It is emphasized that purchase and installation are the most significant contributors to cable prices even when applying pre-processing strategies to account for power loss costs, as in [14,18].

In real-world WPP projects, plant developers typically select cables from existing catalogs based on their voltage and power ratings. Different cable sizes imply distinct prices. Due to the difficulty in finding reliable sources providing cable price samples, this paper relies on the procedure presented in [24] (page 196) regarding AC cables to obtain synthetic cable prices. That work assessed several real cables and fitted purchase cost curves as functions of the cable nominal power capacity ($S_n$). Since ref. [24] gathered data from real products offered by cable manufacturers, the authors deem the information representative of real-world cable types.

The price per distance of a cable is given by Equation (24) [24], in which ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$ are equal to $0.411·{10}^6,0.596·{10}^6$, and $4.1$, respectively, for a grid operating at a line-to-line voltage ($V_l$) of 33 kV. Since ref. [24] considers values in MSEK/km, a factor of 0.089 converts the result to M€/km. In addition to the purchase cost calculated by Equation (24), an expense of 0.26 M€/km is incurred for the cable installation process [18]. The synthetic cables come from sixteen current capacity (I) cases, with $S_n$ coming from Equation (25).

Table 2. Synthetic cables for analyses, with $I,S_n$, and cost in [A], [MW], and [M/km], respectively.

Cable	I	${\mathit{S}}_{\mathit{n}}$	Cost	Cable	I	${\mathit{S}}_{\mathit{n}}$	Cost
1	100	5.7158	0.3613	9	900	51.4419	0.7231
2	200	11.4315	0.3786	10	1000	57.1577	0.8359
3	300	17.1473	0.4005	11	1100	62.8734	0.9786
4	400	22.8631	0.4282	12	1200	68.5892	1.1589
5	500	28.5788	0.4631	13	1300	74.3050	1.3868
6	600	34.2946	0.5073	14	1400	80.0207	1.6749
7	700	40.0104	0.5632	15	1500	85.7365	2.0392
8	800	45.7261	0.6338	16	1600	91.4523	2.4996

summarizes the cable data. In a real WPP study, one would replace the synthetic cables with the real (standardized) ones before optimizing the cable connections. It is emphasized that this paper focuses on the computational aspects of optimizing COP subjected to multiple cable types. The synthetic cables are considered sufficient for achieving this goal, as this work investigates multiple combinations of cable choices.

$$\begin{array}{c}\mathrm{Purchase}\cos \mathrm{t}=0.089·\left({\alpha }_1+{\alpha }_2·e^{{\alpha }_3·S_n·{10}^{-8}}\right)\end{array}$$

(24)

$$\begin{array}{c}{S}_n=\sqrt{3}·V_l·I\end{array}$$

(25)

In this work, each WPP in Table 1 considers different optimization cases that utilize subsets of the cables listed in Table 2. This approach aims to evaluate the COP formulations in various scenarios related to the availability of cable types. By taking $P_{\mathit{wt}}^{\mathit{nom}}$ as the power base, one can calculate the cables’ power capacity in p.u. ($P^{\mathit{cap}}$) by computing $S_n/P_{\mathit{wt}}^{\mathit{nom}}$.

Table 3. Cabling cases in each WPP.

Case	Cables	Case	Cables
ORM-1	$\left\{7\right\}$	WMR-1	$\left\{8\right\}$
ORM-2	$\{4, 7\}$	WMR-2	$\{4, 8\}$
ORM-3	$\{2, 4, 7\}$	WMR-3	$\{2, 4, 8\}$
ORM-4	$\{2, 4, 7, 8\}$	WMR-4	$\{2, 4, 8, 10\}$
ORM-5	$\{2, 4, 7, 8, 9\}$	WMR-5	$\{2, 4, 8, 10, 11\}$
ORM-6	$\{2, 4, 7, 8, 9, 11\}$	WMR-6	$\{2, 4, 8, 10, 11, 13\}$
HR3-1	$\left\{11\right\}$	RR-1	$\left\{5\right\}$
HR3-2	$\{8, 11\}$	RR-2	$\{2, 5\}$
HR3-3	$\{6, 8, 11\}$	RR-3	$\{2, 3, 5\}$
HR3-4	$\{6, 8, 11, 14\}$	RR-4	$\{2, 3, 5, 6\}$
HR3-5	$\{6, 8, 11, 14, 15\}$	RR-5	$\{2, 3, 5, 6, 7\}$
HR3-6	$\{6, 8, 11, 14, 15, 16\}$	RR-6	$\{2, 3, 5, 6, 7, 8\}$
TH-1	$\left\{6\right\}$
TH-2	$\{3, 6\}$
TH-3	$\{3, 6, 7\}$
TH-4	$\{2, 3, 6, 7\}$
TH-5	$\{1, 2, 3, 6, 7\}$
TH-6	$\{1, 2, 3, 6, 7, 8\}$

and

Table 4. Cables’ power capacities [p.u.] per WPP.

WPP	Cables	${\mathit{P}}^{\mathit{cap}}$
ORM	$\{2, 4, 7, 8, 9, 11\}$	$\{2.3, 4.6, 8.0, 9.1, 10.3, 12.6\}$
WMR	$\{2, 4, 8, 10, 11, 13\}$	$\{1.9, 3.8, 7.6, 9.5, 10.5, 12.4\}$
HR3	$\{6, 8, 11, 14, 15, 16\}$	$\{4.1, 5.5, 7.6, 9.6, 10.3, 11.0\}$
RR	$\{2, 3, 5, 6, 7, 8\}$	$\{3.8, 5.7, 9.5, 11.4, 13.3, 15.2\}$
TH	$\{1, 2, 3, 6, 7, 8\}$	$\{1.9, 3.8, 5.7, 11.4, 13.3, 15.2\}$

present the cases and their corresponding power capacities. In Table 4, since $P^{\mathit{cap}}$ refers to nominal power, rounding the values down translates to how many WTs each cable can support, as in Equation (14).

Upon completion of the described procedures, COP optimization can take place. The formulations from Section 2 were implemented in the JuMP environment [25], version 1.15.1, in Julia Language, version 1.9.3. Gurobi [26], version 9.5.1, was the utilized solver through the version 1.0.3 Julia wrapper (https://github.com/jump-dev/Gurobi.jl (accessed on 14 July 2025)). Gurobi addresses linear programming problems with barrier or simplex algorithms. Branch-and-bound, combined with presolve, cutting planes, heuristics, and parallelism strategies, tackles mixed-integer parts of the problem.

Branch-and-bound algorithms typically employ randomization when branching from nodes. Therefore, comparing different integer programming formulations can be misleading if different randomization seeds are not taken into account. This paper utilizes five seeds for each case of each WPP in Table 3 to attenuate the effect of randomness on the overall assessment of the solving processes. This was done by setting Gurobi’s “Seed” parameter to $\{1, 2, 3, 4, 5\}$. The five-seed average solving time defines the evaluation metric. Apart from “Seed”, “MIPgap” was the only modified Gurobi parameter, which was set to zero in all simulations to ensure global optimality.

Thanet’s size is highlighted as a final remark on the optimization methodology. As mentioned in Section 1, COP formulations become increasingly complex as the WPP’s size increases. Large WPPs such as Thanet typically require mathematical heuristics to reduce the search region; otherwise, the solving process becomes computationally impracticable [14]. In this context, a heuristic refers to a strategy that exploits expertise on a problem to eliminate solution alternatives that are unlikely to be optimal. Although ORM, WMR, HR3, and RR had their full COP optimized in every case, TH benefited from a simple heuristic that enabled viable convergence times.

A simple COP concept refers to WPP elements tending to connect to neighboring elements. In other words, connections to distant elements are unlikely to be optimal. Hence, all Thanet optimization cases disregard WT-to-substation connections if the distance between such elements exceeds 3 km. Furthermore, WT-to-WT connections are eliminated if their distance exceeds 1.5 km. Algorithm 2 describes the heuristic implementation, which occurs after the decision variables are created and before the solving process begins. Figure 3 illustrates the heuristic. Although all WTs are subjected to the heuristic, the image shows only three WT circles for better visualization.

Algorithm 2 Heuristic for reducing the solution region when optimizing Thanet

1: for $\forall i\in \mathcal{N}$ do 2:      for $\forall j\in \mathcal{N}$ do 3:            if $i==j\vee L_{i,j}>1.5$ then 4:                  Set $u_{i,j,t}$’s upper bound to zero $\forall t\in \mathcal{C}$ 5:                  Set $P_{i,j}$’s upper bound to zero 6:            end if 7:      end for 8:      for $\forall j\in {\mathcal{N}}_0-\mathcal{N}$ do 9:            if $L_{i,j}>3$ then 10:                  Set $u_{i,j,t}$’s upper bound to zero $\forall t\in \mathcal{C}$ 11:                  Set $P_{i,j}$’s upper bound to zero 12:            end if 13:      end for 14: end for

Concerning the Thanet heuristic, note that (i) since part of the candidate solutions are eliminated, there is no global optimality guarantee despite setting “MIPgap” to zero; (ii) in CTBF and PBF, the number of u and y variables ($N_u$ and $N_y$) are given by Equation (26) and Equation (27), respectively. Therefore, Thanet’s CTBF and PBF initially have $10,000\left|\mathcal{C}\right|$ and $10,100k_{c_{\mathit{nct}}}-10,000$ binary variables, respectively. For TH’s case 6, the heuristic brings these numbers from 60,000 to 11,676 in CTBF and from 141,500 to 26,954 in PBF, thus eliminating approximately $80\%$ of the binary variables and significantly weakening the combinatorial explosion; (iii) this heuristic is a simplified version of the methodology in [19]. Readers interested in the complete approach are referred to that work. In this paper, the sole purpose is to enable the solution of the Thanet cases in a reasonable time, so that Equations (10)–(12) can be assessed as valid constraints.

$$N_u=\left|\mathcal{N}\right|·\left(|{\mathcal{N}}_0|-1\right)·\left|\mathcal{C}\right|$$

(26)

$$N_y={\left|\mathcal{N}\right|}^2·\left(k_{c_{\mathit{nct}}}-1\right)+|{\mathcal{N}}_0-\mathcal{N}|·|\mathcal{N}|·k_{c_{\mathit{nct}}}$$

(27)

4. Results and Discussions

This section presents results obtained by solving CTBF and PBF using the methodologies from Section 3. This section refers to Equations (1)–(9) as CTBF1, i.e., the standard CTBF commonly found in the literature. In contrast, CTBF2 considers Equations (1)–(11), hence accounting for the novel constraints in Section 2.1.2. All simulations were conducted in a 64 GB RAM, Intel^® Core^TM i9-11950H workstation. No other performance-critical activity was carried out during the optimization executions, which all ran without initial guesses (warm starts) for the decision variables.

As mentioned in Section 3, this paper utilizes five different branch-and-bound randomization seeds to enable fairer comparisons between the formulations. Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 show the five-seed average time required for Gurobi to solve the CTBF1, CTBF2, and PBF cases of each WPP. The plots also indicate the standard deviation regarding the five-seed solving times.

An overview of Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 demonstrates the impact of a WPP’s size on the time required to optimize COP. From the 30 WTs Ormonde to the 100 WTs Thanet, the order of magnitude of the solving time has grown for each larger WPP. However, within the same WPP, the number of available cables does not necessarily lead to more computational complexity despite implying more decision variables. Robin Rigg’s cases 2 to 3 and Thanet’s 5 to 6 exemplify this affirmation.

ORM and WMR presented cases where CTBF2 led to longer solving times than CTBF1. Small WPPs can be generally solved quickly regardless of the formulation. As shown in Figure 4 and Figure 5, all ORM and WMR cases were globally optimized in under a minute. Therefore, it is possible that the benefit from implementing the proposed additional constraints may not be sufficient to outweigh the increase in the number of problem constraints, thus leading CTBF2 to require more time than CTBF1 to converge.

PBF performed similarly to CTBF1 and CTBF2 in most ORM and WMR cases. However, it took around twice the time in ORM’s case 5 and WMR’s cases 5 and 6. Nonetheless, these WPPs were quickly solved to global optimality regardless of the formulation, indicating that enhancing formulations for small WPPs implies marginal benefits. In contrast, HR3, RR, and especially TH showed more discrepancy in convergence time.

Table 5. Reductions in solving time when switching from CTBF1 to CTBF2.

WPP	Reduction in Average Solving Time [%]
	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
HR3	37.0	14.4	16.7	52.0	44.5	43.2
RR	5.6	46.8	8.7	23.5	3.3	9.0
TH	39.7	73.5	49.9	18.1	85.0	61.5

shows the reduction in mean solving time when changing from CTBF1 to CTBF2 across the six cases for HR3, RR, and TH. This table indicates that a WPP does not necessarily benefit more from CTBF2 as it increases in size. For instance, although RR has more WTs than HR3, the former has been less impacted than the latter in terms of presenting faster convergence. However, TH’s solving times significantly decreased when the proposed constraints were applied. In general, the three WPPs exhibited better convergence upon implementation of Equations (10)–(12).

Figure 9, Figure 10 and Figure 11 show the convergence paths of HR3’s case 4, RR’s case 2, and TH’s case 5 using Gurobi’s “Seed” parameter as equal to one. These cases had the highest relative discrepancy in solving time per WPP between CTBF1 and CTBF2.

Table 6. Time demanded to find (${\mathit{t}}_{\mathit{f}}$) and prove (${\mathit{t}}_{\mathit{p}}$) global optimality.

Case	CTBF1		CTBF2
	${\mathit{t}}_{\mathit{f}}$  [s]	${\mathit{t}}_{\mathit{p}}$  [s]	${\mathit{t}}_{\mathit{f}}$  [s]	${\mathit{t}}_{\mathit{p}}$  [s]
HR3-4	248.8	259.6	70.8	101.8
RR-2	139.3	831.5	239.8	361.2
TH-5	699.6	5804.6	1085.2	716.2

shows the time required for Gurobi to find and prove the global optimum for the three cases. Observe that, by “find”, the authors refer to the solver achieving an incumbent solution that happens to be the global optimum despite still lacking proof of optimality by fully closing the gap.

In HR3’s case 4, CTBF2 was significantly faster than CTBF1 in reaching the global optimum and proving optimality. In RR’s case 2 and TH’s case 5, CTBF1 interestingly took less time to find the global optimum. In RR’s case 2 and TH’s case 5, significantly more time was required for CTBF1 to prove optimality compared to CTBF2. Although these results pertain only to one Branch-and-Bound randomization seed for the specific HR3-4, RR-2, and TH-5 cases, analyses have been conducted for the other four seeds and other cabling cases. Such tests confirmed that, in several situations, CTBF2 required more time than CTBF1 to find the global optimum. Nonetheless, CTBF2 consistently outperformed CTBF1 in proving optimality and converging faster.

In general, the conducted studies indicate that the constraints proposed in Equations (10)–(12) effectively enable faster convergence of COPs modeled through the cable type-based MILP formulation. CTBF2 has been shown to effectively tighten the solution region and enable more rapid convergence, on average, for every case of the medium- and large-size WPPs compared to CTBF1. The proposed approach has been shown not to benefit small WPPs, even worsening some cases. The scalability of the method has been demonstrated by addressing a considerably large WPP comprising 100 WTs.

Despite the analyses in the previous two paragraphs, Figure 6, Figure 7 and Figure 8 clearly show that PBF tends to outperform CTBF1 and CTBF2 for the medium and large WPPs.

Table 7. Reductions in solving time when switching from CTBF2 to PBF.

WPP	Reduction in Average Solving Time [%]
	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
HR3	−168.3	8.9	46.7	37.3	32.5	33.6
RR	−78.2	5.6	6.0	51.1	−45.5	37.2
TH	−1158.9	93.4	78.3	87.3	72.6	53.4

presents the percentage reductions in solving time when changing from CTBF2 to PBF.

One notable aspect in Table 7 is the increase in convergence time for case 1 across the three WPPs, with Thanet presenting a significant difference. These results suggest that CTBF2 outperforms PBF when only one cable type is available for installation, thus classifying a scenario where CTBF2 is preferable over PBF. Apart from such scenarios, Robin Rigg’s case 5 is an outlier in which PBF required more time to reach optimality than the other formulations. For all other cases, PBF performed better than CTBF2 when addressing HR3, RR, and TH, i.e., the WPPs that can actually benefit from optimization enhancements. Especially for TH, PBF achieved global optimality significantly faster than the cable type-based algorithms.

The primary reason for the better performance of PBF over CTBF relates to its inherently tighter search space. Both CTBF formulations address the power flow conservation through the continuous P variables in Equation (5). As a consequence, the binary variables weakly restrict each other concerning the cable types. For example, if WT i links to WT j using a type t cable (i.e., $u_{i,j,t}=1$), the cable departing j can be of any type $\ge t$, depending on other cables that also link to it. Analogously, cables linking to i can be of any type $\le t$ (assuming i is not the most upstream WT in the array). In contrast to CTBF, PBF is a purely binary formulation that addresses power flow through Equation (18) and implicitly yields the cable types. Suppose power flows from i to j carrying n p.u. of power (i.e., $y_{j,i,n}=1$). Then, it is guaranteed that the power reaching i and departing j are equal to $n-1$ and $n+1$, respectively. Although Equation (5) imposes such power relations for CTBF, PBF meets these requirements through the binary variables. This causes the variables to strongly restrict each other, which benefits the branch-and-bound process and enables faster convergence.

Based on the presented analyses, the findings of this paper suggest that PBF is the most computationally efficient algorithm for addressing COP problems targeting medium- to large-scale WPPs employing radial CGs with multiple cable types, as the improvements to CTBF1 were not enough to surpass PBF’s natural advantage.

As additional information not critical to the convergence assessments, Appendix A lists the values of the optimized cost functions concerning all cases in Table 3. The CTBF and PBF results are identical, as the models are equivalent and were all solved globally. The Appendix also shows the CGs obtained for the sixth case of all analyzed WPPs.

5. Conclusions

This work presented novel constraints to enhance a well-known cable type-based COP MILP formulation addressing radial grids. The developed approach leveraged the problem’s characteristics regarding power flow interactions related to cable types. Doing so enabled writing equations that tightened the solution region and yielded global solutions in less time than the standard formulation. The results indicate that small WPPs neither benefit from nor require the proposed constraints, as their global solution can be found in seconds regardless of the formulation, as demonstrated through the Ormonde and Westermost Rough WPPs. In contrast, this paper’s findings suggest that the novel constraints can significantly decrease the solving time for medium and large WPPs. The proposed formulation has improved the average convergence time for all cases in the Horns Rev 3, Robin Rigg, and Thanet investigations. The impact of the proposed formulation varied according to the WPP and cabling case, ranging from 3.3% faster convergence in a Robin Rigg case to 85% in a Thanet case.

Despite best efforts, the improvements to CTBF were not enough to outperform the presented PBF. The conducted investigations indicated a better performance of CTBF over PBF exclusively in cases where the CG utilizes only one cable type. Thus, this work seeks to report the failed experiments to other researchers. This can prevent time from being spent analyzing the same proposed constraints. However, it can also inspire others to gain insights on either reformulating CTBF or proposing different constraints that can boost its performance beyond PBF. Future research possibilities on this topic include enhancing MILP formulations to address ring CGs.