Comparison of Different Configurations for a Shoreline Pond Electrode Station in the Case of an HVDC Transmission System—Part II: Electric Field Study for Frames of Non-Linear Novel Electrode Arrangement Based on a Simplified Analytical Model

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The paper presents new electrode frame designs (as compared to the typical straight ones, described in CIGRE Β4.61/2017) as well as novel spatial frame arrangement solutions, thus moving beyond their parallel placement to the longitudinal axis of a breakwater (or variations of it, as described in the previous paper, present by the authors). All the aforementioned designs and arrangements are thoroughly studied with respect to the electric field analysis (emphasizing the near-field).

Abstract

According to CIGRE, the usual arrangement of electrodes in a shoreline electrode station for HVDC interconnections is straight with the following form: forming straight frames with the electrodes at equal distances and placing the frames parallel to the longitudinal axis of the breakwater, successively at fixed distances between them. In a previous paper by the authors, 10 alternative configurations of placement of such straight frames were examined to determine which placements mainly affect the near-field results. In particular, radial or circumferential arrangements of the straight frames on a central base in the open sea improve the overall field results, such as the absolute potential and electrode station resistance to remote earth, satisfying the requirements of the maximum electric field strength. In this paper, the nonlinear configuration of the frames will be studied from an electric field perspective at the level of a preliminary study forming innovative configurations in order to check their suitability with respect to the relevant requirements of the CIGRE guidelines B4.61/2017. These arrangements, located in electrode stations, are evaluated and compared with the older configurations for two cases, those of Korakia in Crete and Stachtoroi in Aegina, Attica, for the HVDC Crete-mainland Greece interconnection of 1 GW, ±500 kV.

1. Introduction

When configuring an HVDC interconnection of great length, the non-use of a metallic return leads to a significant cost reduction [1,2,3,4]. In particular, in the case of the bi-directional interconnection between the island of Crete and mainland Greece in the region of Attica (with nominal characteristics of voltage ± 500 kV DC using voltage source converter sources, of power 1 GW, and of length 380 km), a bipolar heteropolar configuration with sea return was chosen, thus curtailing the respective metallic return by 310 km and costs by 15%. However, there are a multitude of technical constraints regarding the location of electrode stations, which are related not only to their proximity to associated works, such as converter stations, and the distribution network for auxiliary services but also to their non-proximity requirements to certain areas (such as archaeological sites, seismic faults, and Natura areas) [5] and to the interaction with other human installations and activities, such as electrochemical corrosion of installations, e.g., pipeline networks [6,7,8,9], railway networks [10], cables metallic sheaths [11], swimming, impact on flora and fauna [1], etc., which are essentially determined by the study of the electric field during the steady and transient state [1,12,13,14,15,16,17]. The electric field study is usually carried out with computational models, such as finite elements [13,16,17,18,19,20,21,22], the inclined layer model [18], finite volume elements [18,23,24], the hemispheroidal model [18,23,24], semi-analytical methods [21,25], cross-platform finite element analysis, solver and multi-physics simulation (COMSOL Multiphysics) [3], current distribution, electromagnetic fields, and grounding and soil structure analysis (CDEGS) [9,10,26]). In particular, specialized computational packages are expensive and require time-consuming training. In addition, computational models require accurate topographic spatial models and knowledge of the specific resistivity of seawater/soil in the area under study, which requires costly and time-consuming measurements with petrophysical-geophysical-electric methods [13,15,16,17], such as electric potential measurements in sea, drillings, electric soundings, transient electromagnetic soundings [27], electrical resistivity and magnetotelluric tomography [28,29]. Therefore, during the preliminary study phase, for reasons of cost and time, the use of analytical methods is chosen, such as the use of point sources of electric current, as formulated in CIGRE B4.61 675:2017 [1] or through its variation in IEC TS 62344:2013 [12], which is based on the Rusk methodology [30,31,32,33,34] and in its simplified form on the Uhlmann methodology [34,35], in the hybrid method of the present authors [4,36], which unifies and extends the point source into a linear source and subsequently into multiple sources through superposition. The image method is used more rarely [37].

The paradox is that while there are several methods for calculating the electric field strength and its effects, the electrode configurations are limited. The guidelines of CIGRE B4.61 675: 2017 [1], as well as [14,34,38], suggest the configuration of the shoreline electrode station with the electrodes placed in straight frames with fixed distances between them and with the frames placed sequentially, parallel to the longitudinal axis of the breakwater. This was initially proposed by the Hellenic Independent Power Transmission Operator (IPTO) for the Attica–Crete interconnection [2] as well as in the Muskrat Falls Project [15,16,17]. Especially in the latter, the Nalcor Energy—Hatch study [17] found that the electrodes at the edges of a straight arrangement, with equal distances between them, are charged more than the intermediate ones. This led Nalcor Energy—Hatch to change the distances between the electrodes from uniform to semi-uniform placement along the entire length of the electrodes of the electrode station, to uniform placement per frame, to non-uniform placement per frame, and to uniform non-linear placement (placement not in a straight line), concluding that, in their case, the most suitable form in terms of equal distribution of currents in the electrodes is semi-uniform placement along the entire length of the electrodes of the electrode station (with denser placement of the edge electrodes and sparser for the intermediate ones). In this way, the non-uniformity of the current density between electrodes was limited but without resolving it and without providing the mathematical basis for solving the problem. In addition, the non-standard form of frames creates additional difficulties in both construction and maintenance. In ref. [4,36], in order to address the expected non-uniformity, the use of a corrective incremental factor in the electric current density of all electrodes was proposed without essentially solving the problem of non-uniformity but by addressing the dimensioning more conservatively. However in ref. [36], for the first time, the geometry was analysed for the following electrode station configurations against the basic configuration of straight frames in a row, parallel on the longitudinal axis of the protective dam (first configuration) [1,2,14,15,16,17,34,38]:

• Straight frames in a row, each placed vertically on the longitudinal axis of the dam (2nd configuration);
• Straight frames in two overlapping rows, each parallel to the longitudinal axis of the dam and aligned with each other (third configuration);
• Straight frames in two overlapping rows, each parallel to the longitudinal axis but non-overlapping on the vertical axis of the dam (fourth configuration);
• Straight frames in two successive rows, each vertical to the longitudinal axis of the dam and aligned with each other (fifth configuration);
• Straight frames in two successive rows, each vertical to the longitudinal axis of the dam and non-overlapping on the vertical one (sixth configuration);
• Straight frames in perimetrical placement to the protective dam, adapting to the outline of the pond under construction (seventh configuration);
• Straight frames adapted to a T-shaped protective dam (to increase the inner outline of the dam) (eighth configuration);
• Straight frames adapted radially to a central base with guides in open sea (ninth configuration);
• Straight frames adapted perimetrically to a central base in open sea (10th configuration).

Configurations 2 through 8 can be viewed as modifications of the fundamental arrangement, whereas configurations 9 and 10 represent entirely new designs. The latter differ primarily because they are positioned around a central base in open waters. For every configuration examined, both the distinct characteristics of the suspension and lifting mechanisms and the specific engineering demands—such as dam construction, pond dredging, embankment shaping, and quay wall construction—were documented. Following this stage, the analytical framework previously described for determining the electric field gradient, ground potential rise, and resistance to remote earth at electrode stations [4,36] was applied to the newly proposed layouts concerning the electrode stations at Stachtoroi and Korakia regarding the HVDC interconnection between Crete and Attica. Field measurements indicate, first and foremost, that evaluating a single frame with a linear electrode arrangement does not adequately represent an electrode station composed of multiple frames. Reassessment of the relevant field parameters is therefore essential. The ninth and 10th configurations notably enhance the performance outcomes, particularly with regard to the absolute potential values and the station’s resistance to remote earth—improvements largely attributable to the absence of a dam. Moreover, these configurations yield acceptable maximum electric field strengths within the inactive frame area, remaining close to or below the 2.5 V/m mark. Although they do not minimize the zone of influence compared with standard layouts, this limitation is of lesser consequence given their open-water positioning.

Conventional offshore electrode arrangements [21]—for instance, copper conductors laid above the seabed—were dismissed because existing station electrodes also operate as anodes. Other concepts, such as embedding electrode parts within concrete boxes or using fiberglass-based flat electrode meshes parallel to the seabed [39,40], have also been proposed. Nevertheless, these alternatives were not adopted by IPTO [2] since they do not permit direct visual monitoring of electrode condition and make maintenance or replacement difficult.

In this paper, the preliminary study of shoreline pond electrode stations with different configurations of electrode frames (other than the classical linear) is conducted, which was already examined in refs. [4,15,16,17,36] covering the limits of the potential gradient for the protection of marine life and divers in marine electrodes in steady and transient states and of the absolute potential in relationship with remote earth for the steady state. To calculate the electric field gradient, ground potential rise and resistance to remote earth of electrode stations, the method proposed in ref. [2] and utilized in ref. [36] is used, which means that for the near-field, a linear current source is used for each electrode that extends cylindrically in a sea/soil or dam zone of constant circumference. The respective mathematical background of a point current source in the form of an appropriate wedge of sea on the ground is used for the far-field (unifying and expanding the theoretical background in refs. [1,12]). In the case of more than one electrode, the application of this method is extended through superposition (the methodology is recorded in Appendix A). Specifically, in the present case, instead of straight frames, bow frames, suitably adapted circumferentially to a central base in open sea (a variant of the 10th configuration of ref. [36]), and frames of a special shape with the placement of vertical electrodes in one or more circular arrangements resembling a modified “birthday cake” will be examined. Subsequently, these frame bases will be placed appropriately on the seabed, using as an example the electrode stations at Stachtoroi and Korakia for the HVDC interconnection of Crete—Attica. Interesting conclusions emerge from this study since using the same number of electrodes, with a different placement of the frames of non-linear arrangement of equidistant electrodes, achieves better field results, reducing the absolute potential and remote earth resistance of the electrode station. In the future, the above data will be supplemented by the third part, which concerns the construction of civil engineering projects for the respective placement of the electrodes.

The structure of the paper is as follows: in Section 2, the pre-existing structure of electrode stations in ref. [36] is initially given, and the mathematical background for the configuration of the new frame forms and electrode stations is fully developed. This is followed by both the electric field analysis and the structural configuration of the new frame forms in Section 3 and of the electrode stations in Section 4. In Section 5, the proposed solutions are compared with the previous configurations presented in ref. [36], and the final conclusions follow in Section 6. For the sake of completeness and ease of reading, the theoretical background of the analytical field model of ref. [4,36] is presented in Appendix A.

2. Configuration of Non-Linear Frames and of Respective Electrode Stations

2.1. Pre-Existing Structure

Due to the previous experience in near-shore electrodes [1,4,12,14,15,16,17,34,36,38], the electrodes are placed behind a breakwater in a linear arrangement. In particular, the electrodes form rectilinear frames, usually at equal distances [1,4,14,15,36] with or without an inclination to the vertical axis. Indicatively, in Figure 1, each frame is made up of a number of electrodes N_{el_frame} (13 in this case), with a length equal to L_el and a diameter equal to d_el (twice the radius r_el), which are placed with an inclination λ (base: height) in series with each other at a distance ℓ_p (measuring from each bar centre). The upper edge of the frame/electrodes is at depth h_u, the middle of the frame/electrodes is at depth h_m, the lower edge of the frame/electrodes is at depth h_d, while the seabed depth is h_sb. Therefore, the water zone of the near-field intensity calculation method L, the total frame length ℓ_f and the respective vertical suspension depths are calculated as follows:

$$L=L_{el}\times cos\left(atan\lambda \right)$$

(1)

$${\mathcal{l}}_f=\left(N_{el\_frame}-1\right)\times {\mathcal{l}}_p$$

(2)

$$h_m=h_u+0.5\times L_{el}\times cos\left(atan\lambda \right)$$

(3)

$$h_d=h_u+L_{el}\times cos\left(atan\lambda \right)$$

(4)

The height above the upper edge of the electrode h_u is such that the entire electrode is always submerged in the water, regardless of ripples outside the breakwater (for the classic design case). If a breakwater is not used, it must take an appropriate value so that the electrode is not found outside the water or, even better, outside the breaking wave water (which presents lower density/conductivity). In a different geometric design, the respective sizes have to be recalculated.

Next, the respective frames are placed in series forming an imaginary axis parallel to the longitudinal axis of the breakwater from the inner/protected side [1,12,14,15,16,17,34,36,38]. Its basic structure is seen in Figure 2, with the following parameters:

• The width of the dam t_d on the xΟx^/ axis;
• The distance of the imaginary axis of the electrode station from the axis of the dam d_r1 on the axis xΟx^/;
• The frame length ℓ_f;
• The estimated width of the critical frame zone d_fv vertical to the dam (along the xΟx^/ axis) resulting from single-frame simulations;
• The distance between consecutive frames on the dam, estimated to ensure the critical zone for diver drop for repairs s_c (on yΟy^/), resulting from single-frame simulations;
• The width d_stv of the electrode station critical zone vertical to the dam (on the xΟx^/ axis) resulting from the electrode station simulations;
• The length of the electrode station critical zone ℓ_cbd on the dam below the centre of the dam (on the yΟy^/ axis), resulting from the electrode station simulations;
• The length of the electrode station critical zone ℓ_cud on the dam above the centre of the dam (on the yΟy^/ axis), resulting from the electrode station simulations.

In ref. [36], nine more frame configurations were examined, which are summarized in Figure 3.

From the comparison of the results of the respective arrangements, as mentioned above, the ninth and 10th configurations have very good results, as they concern the open sea. In the present case, other forms are examined, which differ from the classic rectilinear frames.

2.2. Bow Frames Suitably Adapted Circumferentially to a Central Base in the Open Sea (11th Configuration)

This is a variant of the 10th configuration of ref. [36], where bow frames are used in place of the straight frames. Specifically, the central base is a circle of radius R_Kc, and bow frames are placed with the necessary gap d_r1 from the walls of the base, i.e., the electrodes are located in a circle of gross radius R_K3 (=R_Kc + d_r1). In this way, the requirements of the suspension-lifting-placement mechanism of the frames are limited, as in the classic arrangement in Figure 2. However, this requires that the frames be placed at such a depth that they are always out of reach of breaking waves.

In Figure 4, the respective plan view of the proposed arrangement for six electrode frames is shown (N_frame^/ = 6). The formed angle θ_f between successive frames is calculated by:

$${\theta }_f={\displaystyle \frac{{360}^{°}}{{N_{frame}}^{/}}}$$

(5)

The distance d_sc can be determined as the maximum between the estimated width of the critical zone of frame d_fv-out (vertical to the longitudinal axis of the frame) on the outer side of the frame, d_fv-in on the inner side of the frame, and the estimated length (distance between two consecutive frames placed consecutively along their longitudinal axis) s_c, i.e., three distances estimated to ensure the critical zone in order for a diver to drop for repairs and that are obtained from single-frame simulations. Therefore, it holds that:

$$d_{sc}=max\left\{d_{fv-out},{d_{fv-in},s}_c\right\}$$

(6)

The widths d_Ox and d_Ox^/ of the critical zone of the electrode station along the semi-axes Ox and Ox^/ from the beginning of the axes (which is the centre of the central base) and the lengths d_Oy and d_Oy^/ of the critical zone of the electrode station along the semi-axes Oy and Oy^/ from the beginning of the axes are obtained from the electrode station simulations.

Figure 5 shows the view of the arrangement of vertical bow frames of length ℓ_f in the sea around a central circular base of gross radius R_K3. The upper frame level is at depth h_u, which is greater than the depth reached by the breaking wave water, while the seabed is at h_sb, which is greater than the sum of the upper frame level h_u and the frame/electrode rod height L_el.

Figure 6 shows a bow frame with a number of electrodes per frame N_{el_frame}, with a distance between consecutive electrodes ℓ_p. The analysis of two consecutive electrodes of the frame is carried out next (as shown in Figure 6), essentially enlarging Figure 7. Specifically, these electrodes are placed at a radius R_K3 from the centre O of the central circular base at points A and B, forming the angle θ_el between them. From the analysis of the right triangle OMA (formed by the perpendicular bisector (OM) of the isosceles triangle OAB), in Figure 7, it follows that:

$$sin\left(\angle MOA\right)=sin\left({\displaystyle \frac{{\theta }_{el}}{2}}\right)={\displaystyle \frac{\left(AM\right)}{\left(OA\right)}}={\displaystyle \frac{{\mathcal{l}}_p/2}{R_{K3}}}\Rightarrow {\theta }_{el}=2\times {sin}^{-1}\left({\displaystyle \frac{{\mathcal{l}}_p}{{2\times R}_{K3}}}\right)$$

(7)

Whereas the distance (MN) is equal to:

$$\left(MN\right)=\left(ON\right)-\left(OA\right)=R_{K3}-R_{K3}\times sin\left({\displaystyle \frac{{\theta }_{el}}{2}}\right)$$

(8)

From Figure 6, the angle θ_frame that the frame takes is:

$${\theta }_{frame}=\left(N_{el\_frame}-1\right)\times {\theta }_{el}$$

(9)

Additionally, in Figure 6, the two extreme electrodes of the frame are placed at a radius R_K3 from the centre O of the central circular base at points A’ and B’, forming the angle θ_frame between them. Therefore, similar to Figure 7, in Figure 6, the analysis of the right triangle OM^/A^/ (formed by the perpendicular bisector (OM^/) of the isosceles triangle OA^/B^/) is given by:

$$sin\left(\angle M^{/}O{A}^{/}\right)=sin\left({\displaystyle \frac{{\theta }_{frame}}{2}}\right)={\displaystyle \frac{\left(A^{/}{M}^{/}\right)}{\left(O{A}^{/}\right)}}={\displaystyle \frac{{\mathcal{l}}_f/2}{R_{K3}}}\Rightarrow {\mathcal{l}}_f=\left(A^{/}{B}^{/}\right)=2\times \left(A^{/}{M}^{/}\right)=2\times R_{K3}\times sin\left({\displaystyle \frac{{\theta }_{frame}}{2}}\right)$$

(10)

Whereas the distance (M^/N^/) is equal to:

$${\mathcal{l}}_{pw}=\left(M^{/}{N}^{/}\right)=\left(O{N}^{/}\right)-\left(O{A}^{/}\right)=R_{K3}-R_{K3}\times cos\left({\displaystyle \frac{{\theta }_{frame}}{2}}\right)$$

(11)

Studying Figure 4 again and taking into account the angle θ_f between successive frames and the angle θ_frame that the frame takes, it follows that the angle between the closest edges of two successive frames is equal to θ_f-θ_frame, and the corresponding distance is ℓ_bc. Similar to Figure 6 and Figure 7 and Equations (7) and (10), it results that the distance between the two consecutive frames in Figure 4 is equal to:

$${\mathcal{l}}_{bc}=2\times R_{K3}\times sin\left({\displaystyle \frac{{{\theta }_f-\theta }_{frame}}{2}}\right)$$

(12)

At the same time, for reasons of maintenance of each frame, as already analysed in refs. [2,36], it must at least hold that:

$${\mathcal{l}}_{bc}\ge d_{sc}$$

(13)

Combining Equations (5), (7), (9), (12) and (13), it must hold that:

$${\mathcal{l}}_{bc}=2\times R_{K3}\times sin\left({\displaystyle \frac{{{\theta }_f-\theta }_{frame}}{2}}\right)\ge d_{sc}\Rightarrow {\displaystyle \frac{{{\theta }_f-\theta }_{frame}}{2}}\ge {sin}^{-1}\left({\displaystyle \frac{d_{sc}}{{2\times R}_{K3}}}\right)\Rightarrow$$

$${{\theta }_f-\theta }_{frame}\ge {2\times sin}^{-1}\left({\displaystyle \frac{d_{sc}}{{2\times R}_{K3}}}\right)\Rightarrow {\displaystyle \frac{{360}^{°}}{{N_{frame}}^{/}}}-\left(N_{el\_frame}-1\right)\times {\theta }_{el}\ge {2\times sin}^{-1}\left({\displaystyle \frac{d_{sc}}{{2\times R}_{K3}}}\right)\Rightarrow$$

$${\displaystyle \frac{{180}^{°}}{{N_{frame}}^{/}}}\ge {sin}^{-1}\left({\displaystyle \frac{d_{sc}}{{2\times R}_{K3}}}\right)+\left(N_{el\_frame}-1\right)\times {sin}^{-1}\left({\displaystyle \frac{{\mathcal{l}}_p}{{2\times R}_{K3}}}\right)$$

(14)

where N_frame^/ and N_{el_frame} are predetermined by the electrodes and their charging method, ℓ_p by the distance between consecutive electrodes for access reasons during their maintenance, and d_sc by the electric field analysis of a single frame.

Given the angles θ_f of Equation (5), the angle θ_el between successive electrodes of the same frame and θ_frame of Equation (9), the coordinates of the respective i-th electrode of Figure 6 of the j-th frame in the Cartesian coordinate system of Figure 4 are obtained:

$$x_{el(i,j)}=R_{K3}\times cos\left({\displaystyle \frac{{\theta }_{frame}}{2}}-\left(i-1\right)\times {\theta }_{el}+\left(j-1\right)\times {\theta }_f\right)$$

(15)

$$y_{el(i,j)}=R_{K3}\times sin\left({\displaystyle \frac{{\theta }_{frame}}{2}}-\left(i-1\right)\times {\theta }_{el}+\left(j-1\right)\times {\theta }_f\right)$$

(16)

This configuration essentially represents an adaptation of the central base designs described as cases 9 and 10 in ref. [36]. The main modification lies in the arrangement of bow frames positioned around the perimeter, replacing the radial and circumferential placement of straight frames used in those earlier layouts. From a field result perspective, the potential hazard to divers during maintenance—particularly when the electrode station is operating at full load—can be effectively mitigated by suitably enlarging the R_Kc distance measured from the base centre. Although this design necessitates a distinct civil engineering effort, it eliminates the need for cantilever-type frame guides (a clear advantage over configuration 9). However, it also requires a broader base (a drawback relative to configuration 9 and comparable to configuration 10). The base may be constructed from unreinforced concrete or from solid, modular artificial blocks—each weighing roughly 10 tons or more—at least along the perimeter. The inner sections can be filled with inert materials or sea water, depending on the project’s requirements. Overall, the expected electric field behaviour is considered satisfactory. The suspension, lifting, and positioning systems for the electrodes would function with lever arms typically corresponding to R_K3 (=R_Kc + d_r1) if a single shared system is employed or approximately d_r1 if an independent mechanism is used for each frame. Construction challenges may arise in locations where the seabed has a pronounced slope, particularly in the central base area. When a large R_Kc radius is adopted, it becomes possible to create an internal cavity of radius R₁. This not only reduces the volume of construction material required but also lowers the absolute potential developed by the system. Similar to configurations 9 and 10 in ref. [36], this arrangement requires the underwater connection of the electrode station’s central ground conductor—positioned on the base—to the HVDC interconnection converter. By contrast, this issue is considerably less noted in configurations 1 through 8 of ref. [36].

2.3. Specially Shaped Frames Suitably Adapted to a Circular Arrangement (12th Configuration—“Birthday Cake”)

Alternatively, to achieve better behaviour per frame, the basic idea is for each frame to have the vertical linear electrodes placed in a circular arrangement of radius R_f with equal arcs between them. In this case, per frame, the electrodes will face the same ambient resistance, in contrast to the electrodes of the straight or bow-shaped frames of configurations 1 through 11, thus there is theoretically no asymmetric charging of the electrodes. For local reasons, such as bottom slope and distance from the coast, there will be some variations, but they will be less significant than in the case of the straight frame. Additionally, with the circular arrangement, each electrode cancels out the same electric field from its neighbouring electrodes, in contrast to the straight frames whose outer electrodes are less affected, resulting in higher electric current density values. If each frame consists of N_{el_frame} electrodes, then the arc angle between the electrodes is θ_N:

$${\theta }_N=360°/N_{el\_frame}$$

(17)

If each electrode has a diameter equal to d_el and the electrodes are placed at a distance ℓ_p between them (from rod centre to rod centre), then the necessary radius of the circular arrangement is equal to R_f, which, applying the sine definition to the OAC triangle in Figure 8, results in the following:

$$sin\left(\angle COA\right)=sin\left({\displaystyle \frac{{\theta }_N}{2}}\right)={\displaystyle \frac{\left(OC\right)}{\left(OA\right)}}={\displaystyle \frac{{\mathcal{l}}_p/2}{R_f}}\Rightarrow R_f=0.5\times {\mathcal{l}}_p/sin\left({\displaystyle \frac{{\theta }_N}{2}}\right)$$

(18)

Furthermore, in Figure 8 it is shown that the upper edge of the frame/electrodes is at depth h_u, the middle of the frame/electrodes is at depth h_m and the lower edge of the frame/electrodes is at depth h_d, while the bottom is at depth h_sb, as in Figure 1. Due to the vertical suspension, Equations (3) and (4) apply to the depths. Suspension with an inclination is technically difficult due to the arrangement of the electrodes in the form of a cage and the placement of each electrode on a suitable base.

However, if for construction reasons the electrodes cannot be placed in a circular arrangement, they can be distributed in two or more circular arrangements. In the case of placing the electrodes in two circles of small radius R_f1 and large radius R_f2, the arc angle between two consecutive electrodes of different circular arrangements is θ_N according to Equation (17), forming the plan view of Figure 9. The arc angle between two consecutive electrodes on the same circular arrangement is 2∙θ_N, for which the following apply:

$${\left(OB\right)=\left(OD\right)=\cdots =R}_{f1}$$

(19)

$${\left(OA\right)=\left(OC\right)=\cdots =R}_{f2}$$

(20)

$${\left(BD\right)=\cdots =d}_1$$

(21)

$${\left(AC\right)=\cdots =d}_2$$

(22)

$${\left(AB\right)=\cdots =d}_3$$

(23)

$${\angle AOB=\angle AOF=\angle BOE=\angle FOC=\angle DOE=\cdots =\theta }_N$$

(24)

Taking into account that the electrodes must be placed at a distance ℓ_p between them (from rod centre to rod centre), it is required that the following apply cumulatively:

$$d_1\ge {\mathcal{l}}_p \& d_2\ge {\mathcal{l}}_p \& d_3\ge {\mathcal{l}}_p$$

(25)

Applying the sine definition to the OBD and OAC triangles in Figure 9, the required radii R_f1 and R_f2 respectively result in:

$$sin\left(\angle EOD\right)=sin\left({\displaystyle \frac{{2\times \theta }_N}{2}}\right)=sin\left({\theta }_N\right)={\displaystyle \frac{\left(DE\right)}{\left(OD\right)}}={\displaystyle \frac{\raisebox{1ex}{$d_1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{R_{f1}}}\Rightarrow R_{f1}=0.5\times d_1/sin\left({\theta }_N\right)$$

(26)

$$sin\left(\angle FOC\right)=sin\left({\displaystyle \frac{{2\times \theta }_N}{2}}\right)=sin\left({\theta }_N\right)={\displaystyle \frac{\left(FC\right)}{\left(OC\right)}}={\displaystyle \frac{\raisebox{1ex}{$d_2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{R_{f2}}}\Rightarrow R_{f2}=0.5\times d_2/sin\left({\theta }_N\right)$$

(27)

However, since radius R_f1 is smaller than radius R_f2 according to Figure 9, it necessarily follows that:

$$R_{f1}<R_{f2}\Rightarrow 0.5\times d_1/sin\left({\theta }_N\right)<0.5\times d_2/sin\left({\theta }_N\right)\Rightarrow d_1<d_2$$

(28)

That is, if the distance d₁ is greater than ℓ_p, then due to (28), it is simultaneously satisfied that d₂ is also greater than ℓ_p. From the combination of Equations (25) and (26), the minimum radius R_f1 is equal to:

$$d_1\ge {\mathcal{l}}_p\Rightarrow R_{f1}\ge 0.5\times {\mathcal{l}}_p/sin\left({\theta }_N\right)\Rightarrow {minR}_{f1}=0.5\times {\mathcal{l}}_p/sin\left({\theta }_N\right)<R_{f2}$$

(29)

Applying the law of cosines for the angle θ_N (or, otherwise, for the side AB) in the triangle OAB, the radius R_f2 is calculated through an appropriate quadratic equation:

$${\left(A{\rm B}\right)}^2={\left(OA\right)}^2+{\left(O{\rm B}\right)}^2-2\times \left(OA\right)\times \left(O{\rm B}\right)\times cos\left(\angle AOB\right)\Rightarrow$$

$$d_3^2=R_{f2}^2+R_{f1}^2-2\times R_{f2}\times R_{f1}\times cos\left({\theta }_N\right)\Rightarrow R_{f2}^2-2\times R_{f1}\times cos\left({\theta }_N\right)\times R_{f2}+R_{f1}^2-d_3^2=0\Rightarrow$$

$$R_{f2}={\displaystyle \frac{2\times R_{f1}\times cos\left({\theta }_N\right)\pm \sqrt{{\left(2\times R_{f1}\times cos\left({\theta }_N\right)\right)}^2-4\times \left(R_{f1}^2-d_3^2\right)}}{2}}\Rightarrow$$

$$R_{f2}=R_{f1}\times cos\left({\theta }_N\right)+\sqrt{d_3^2-R_{f1}^2\times {sin}^2\left({\theta }_N\right)}>R_{f1}$$

(30)

where only the solution with the positive root is acceptable, so that the distance R_f2 is greater than R_f1.

Therefore, taking the equality part of Equation (25) (regarding d₃) and Equation (29) and applying them to Equation (30), it follows that:

$${minR}_{f2}=0.5\times {\mathcal{l}}_p/sin\left({\theta }_N\right)\times cos\left({\theta }_N\right)+\sqrt{{{\mathcal{l}}_p}^2-{\left(0.5\times {\mathcal{l}}_p/sin\left({\theta }_N\right)\right)}^2\times {sin}^2\left({\theta }_N\right)}\Rightarrow$$

$${minR}_{f2}=0.5\times {\mathcal{l}}_p\times \left(1/tan\left({\theta }_N\right)+\sqrt{3}\right)\mathrm{for} {N_{frame}}^{/}\ge 4$$

(31)

Therefore, according to Equations (29) and (31), the minimum required radii of the circular arrays are obtained. If a different radius R_f1 is selected, then the minimum necessary radius R_f2 is determined by Equation (30). The use of more circular arrays leads to more difficult mathematical models. In addition, the placement of the electrodes in two or more circular arrays leads to uneven electric current intensity between the electrodes of the inner and outer arrays, which it would be preferable to avoid.

Subsequently, the circular array frames are placed in turn in a circular arrangement of radius R_K4 in order to maintain the advantages found from the circular placement of the frames radially or circumferentially around a central base (ninth and 10th configuration of ref. [36], 11th configuration of the present one, which will be seen in Section 4.2. Additionally, the frames can be placed on underwater structures so as to avoid the construction of artificial islands, significantly limiting the environmental impact. The main disadvantage is that during maintenance, the frames are not directly accessible from land or from a permanent installation on the sea surface, but instead a boat with a diver and a winch is required in order to lift the corresponding frame, or a helicopter with a diver and a “rescue hoist” is needed in order to lift an electrode together with its base.

Figure 10 presents the plan view of the proposed configuration for six frames of circular electrode arrangement. Angle θ_f is formed between the centres of the circular arrangements of successive frames according to Equation (5), and dd_K is the distance at which the centres of the circular arrangements of the frames are apart from each other. The latter is at least equal to the minimum required radius of the critical zone of the frame in order to cover the required electric field limits in steady and transient operation of the electrode station. d_Ox and d_Ox^/ mark the widths of the critical zone of the electrode station along the semi-axes Ox and Ox^/ from the origin of the axes (which coincides with the centre O of the perceived polygon inscribed in a circle of radius R_K4), while d_Oy and d_Oy^/ are the widths of the critical zone of the electrode station parallel along the semi-axes Oy and Oy^/ from the origin of the axes O, as derived from the simulations of the electrode station.

With the help of Figure 10, the necessary radius R_K4 of the perceived polygon is calculated as follows: The centres of the circular arrangements of the N_frame^/ frames are placed at the vertices of the regular polygon inscribed in the circle with centre O and radius R_K4 in such a way that the centrer of successive frames form an arc of angle θ_f. Hence, in the present case, without loss of generality, the following hold:

$${\left(AB\right)=\left(BC\right)=\left(CD\right)=\left(DE\right)=\left(EF\right)=\left(FA\right)=dd}_K$$

(32)

$${\left(OA\right)=\left(OB\right)=\left(OC\right)=\left(OD\right)=\left(OE\right)=\left(OF\right)=R}_{K4}$$

(33)

Point M is the intersection of the perpendicular bisector and the angle bisector in the isosceles triangle OAB. Applying the sine definition to the triangle OAB, the radius R_K4 results in:

$$sin\left(\angle MOB\right)=sin\left({\displaystyle \frac{{\theta }_f}{2}}\right)={\displaystyle \frac{\left(BM\right)}{\left(OB\right)}}={\displaystyle \frac{\raisebox{1ex}{${dd}_K$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{R_{K4}}}\Rightarrow R_{K4}=0.5\times {dd}_K/sin\left({\displaystyle \frac{{\theta }_f}{2}}\right)$$

(34)

Therefore, by successively applying Equations (5) and (34), the calculation of R_K4 is directly carried out. Especially in the case of a regular hexagon, the distance dd_K between the centres of the circular arrangements of the frames is identical to radius R_K4.

It is noted that points A, B, C, D, E, and F of Figure 10 correspond to the centre O of the frame in Figure 8. Therefore, in the case of N_frame^/ frames, with each frame having N_{el_frame} electrode rods, according to Figure 8, the Cartesian coordinates of the i-th rod in the j-th frame of Figure 10 are calculated as follows:

$$x_{el(i,j)}=R_{K4}\times cos\left(\left(j-1\right)\times {\theta }_f\right)+R_f\times cos\left(\left(i-1\right)\times {\theta }_N\right)$$

(35)

$$y_{el(i,j)}=R_{K4}\times sin\left(\left(j-1\right)\times {\theta }_f\right)+R_f\times sin\left(\left(i-1\right)\times {\theta }_N\right)$$

(36)

Similarly, Cartesian coordinates can be calculated for a frame of two (Figure 9) or more circular electrode arrays.

3. Application of the Analytical Methodology for the Calculation of Electric Field Strength in Non-Linear Frames

3.1. General Data

The preliminary design adopts a rubble-mound breakwater configuration similar to that illustrated in Figures 11 and 12 in ref. [4]. For the HVDC link between Crete and Attica, the available lengths are approximately 50 m at the Stachtoroi site in the Argosaronic Gulf and 70 m at Korakia in Crete. The fundamental characteristics of both the interconnection and the electrode stations are provided in Section 3.4 in ref. [4] and Section 3.1 in ref. [36]. In the context of the present study, the parameters most relevant to the analysis of the electrode station’s electric field are summarized below:

• During monopolar steady-state operation, each pole delivers a nominal power of 500 MW at 500 kV, corresponding to a current of 1000 A.
• Under fault-clearing conditions, the maximum transient current reaches 12,800 A, lasting approximately 0.5 s.
• The station employs fully reversible electrodes made of high-silicon iron in a tubular model such as type 4884 SZ (Centertec Z series, ANOTEC), compliant with ASTM A518 G3. Each unit exhibits a connection resistance of 1 mΩ, weighs about 143 kg, and has a diameter of 122 mm (=2∙ r_el) and a length of 2130 mm (=L_el). According to the manufacturer [41], reliable reversible operation is achieved for current densities up to 20 A/m².
• The electrodes are installed vertically. This arrangement provides the greatest water area possible for near-field analysis and also simplifies installation on the vertical walls of concrete breakwaters.
• Stachtoroi area: At this site, located in the Argosaronic Gulf, the plan-view angle θ_g (see Figure 1 of ref. [36] and Appendix A, Figure A1) is set at 210°, as determined from the station layout in ref. [4] (Figure 11). The corresponding water angle θ_w (Figure 2 of ref. [36]; Appendix A, Figure A2) is 0.272°, consistent with the worst-case conditions for the Aegina region listed in Table 1 of ref. [4].
• Korakia area: For the Korakia site in Crete, the plan-view angle θ_g (see Figure 1 of ref. [36] and Appendix A, Figure A1) is 248° according to the electrode station’s position shown in ref. [4] (Figure 12), while the respective water angle θ_w (Figure 2 of ref. [36]; Appendix A, Figure A2) equals 2.29° (far-region/worst-case scenario in Table 1 of ref. [4]).
• Although measured electrical resistivity ranges between 0.167 and 0.212 Ω∙m in both regions, a conservative value of 0.25 Ω∙m is used. This choice compensates for potential reductions in salinity due to freshwater inflows—particularly relevant near Korakia, where two small rivers (dry for most of the year) are present. Electrodes will also be located below the maximum depth of breaking water, as this exhibits much higher resistivity (approximately 2.00 Ω∙m).
• An indicative electrical resistivity of 100 Ω∙m is measured at the dam, accounting for water exchange through internal voids [15,16].
• For reliability purposes, the station incorporates five frames (N_frame = 5) plus one as spare.
• An infinite value for soil electrical resistivity is assumed to simplify computations, providing conservative (worst-case) results.
• The water sector defined by the angle θ_g (Figure 1 of ref. [36]; Appendix A, Figure A1) is neglected to streamline calculations—again leading to conservative outcomes.
• The influence of the breakwater/dam shown in Figure 1 of ref. [36] (Appendix A, Equation (A1)) is also ignored. This simplification affects the estimation of both the electric field strength and the absolute potential within or up to the dam boundary. The absolute potential on the dam surface can be approximated by multiplying the seawater electric field strength by the ratio of the two resistivities (ρ_d/ρ_w).

The last three assumptions are consistent with those employed in the earlier analyses of the electrode station base configurations (see Section 5.1 in ref. [4] and Sections 3 and 4 in ref. [36]). In any case, the analytical method proposed in ref. [4] and utilized in ref. [36] concerns the preliminary design of a shoreline electrode station, in which there are no analytical data on various parameters in space and time, such as ground resistivity, etc. Because of this, simplifications are made to the arrangements and parameter values are taken such that the calculated values of electric field strength, the absolute electric potential with respect to remote earth (infinity) and the equivalent ohmic resistance of the electrode station with respect to remote earth take the most unfavourable values. The ultimate objective is to prevent the development of significant potential differences between any two points—which could lead to electrochemical corrosion of metallic structures—and to avoid hazardous electric field strengths in the vicinity of the electrode station, thereby ensuring the safety of humans and other living organisms under both steady-state and transient conditions. At the same time, the application of this method enables the preliminary study to be done quickly and much more economically, being accepted in principle in CIGRE B4.61 675:2017 [1] or through its variation in IEC TS 62344:2013 [12].

3.2. Electric Field Study of a Bow Frame

As in the case of the classic frame, the approach is the same. Therefore, the total number of electrodes required N_{min_el} is 67 as results from the application of Equation (1) in ref. [36] (see Appendix A, Equation (A1)) assuming a total electric current Ι_{tot_st} of 1100 A at steady state and under overload conditions. Hence, for each frame, the necessary electrodes, N_{el_frame}, are 13, amounting to a total of 78 (including those in the reserve). Considering the increment factor β (=6.1%) [4], and implementing Equations (2) and (3) in ref. [36] (see Appendix A, Equations (A2) and (A3)), the final current density values J_{fu_lo_st} (under full load conditions) and J_{mt_st} (under periodic maintenance) are equal to 18.33 A/m² and 22.00 A/m², respectively. That is, under conditions of full load and normal operation, the manufacturer’s requirements of 20 A/mm² current density are met [41], while during maintenance there is a slight excess, which, however, does not create any technical problems in terms of electrode corrosion due to the short time period that it lasts (of the order of a few hours). In the transient state, for a current I_tot-tr of 12.8 kA, the final current density values J_{fu_lo_tr} (under full load conditions) and J_{mt_tr} (under periodic maintenance) are equal to 213.28 A/m² and 255.93 A/m², respectively. During both the steady-state and transient periods, the worst case is during maintenance. Furthermore, due to the proportional relationship between current and electric field strength, according to Equations (4)–(7) of ref. [36] (i.e., Equations (A4)–(A7) of Appendix A), the steady-state is mainly considered, since the ratio of transient and steady-state currents is 11.63 (=12.8/1.1), compared to the ratio of transient and steady-state electric field strengths, which is 12 (=15/1.25) (according to what is described in Appendix A, Section 2.5 of ref. [4], CIGRE B4.61 675:2017 [1] (p.65), and IEC TS 62544:2013 [12] (p.32), whereas the electric field strength for continuous operating conditions in water should be smaller than or equal to 1.25 V/m (for marine mammals) and for transient operating conditions in water should be smaller than or equal to 15 V/m).

This frame is placed on a canvas of 40 m (Ox semi-axis) by 60 m (yOy’ axis) with a step of 0.05 m and with the seventh electrode placed on the semi-axis Ox at the position of the first frame of Figure 4 or alternatively at point N’ of Figure 6 in the area of Korakia (with a computational mesh of 801 × 1201 = 962,001 points). The electrodes are vertical, so the effective length L is equal to 2.13 m. For the case of a bow frame configuration of a radius R_K3 = 17.0 m and distance between consecutive electrodes ℓ_p = 0.5 m, the angle between consecutive electrodes θ_el is equal to 1.69° using Equation (7), the angle of the bow frame θ_frame is equal to 20.22° using Equation (9), the length of the bow frame chord ℓ_f is equal to 5.969 m using Equation (10), the width of the bow frame ℓ_pw is equal to 0.264 m using Equation (11), and the distance between consecutive frames ℓ_bc is equal to 11.567 m using Equation (12). These values are independent of the field results.

From this simulation, the electric field strength on the Oxy plane (Figure 11) and the area of electric field strength, with values greater than 1.25 V/m (Figure 12), are obtained under steady-state operation conditions during maintenance. The maximum electric field strength E_max is equal to 22.624 V/m (against the 31.87 V/m that was the case with one electrode, 229.82 V/m for when the 13 electrodes are placed at the same spot or 24.053 V/m for the case of the rectilinear frame under the same conditions according to [4]) due to the superposition of the electric fields of the 13 electrodes. Which means there is a slight decrease in field values, i.e., a small improvement, compared to the rectilinear frame. The distance d_fv-out where it stays below the limit of 1.25 V/m along the Ox axis on the outer side of the frame is determined at 10.808 m, the distance d_fv-in on the inner side of the frame at 10.736 m, and the distance s_c along the Oy axis at 8.538 m. Therefore, the critical zone distance ds_c is determined through Equation (6) to be equal to 10.808 m, which is smaller than ℓ_bc (=11.567 m) and satisfies the basic condition of Equation (13) and slightly smaller than the 10.901 m of the corresponding straight frame, as per [4]. This is easily evident from Figure 6, where the length of the bow frame chord is slightly reduced compared to the straight frame. This results in the electric field strength increasing slightly along the bow frame chord, because the placement of the electrodes is denser along the yOy’ axis. In contrast, perpendicular to the chord axis, the electric field strength decreases because along the xOx’ axis, the electrodes are spatially spread (see Figure 6). The changes in the electric field strength values are small but ultimately decreasing.

From the corresponding simulation in transient operating conditions during maintenance, the maximum electric field strength E_max is calculated as 263.184 V/m against the 22.624 V/m of the previous steady-state charge, and it has a corresponding ratio equal to 11.82 against the theoretical ratio of 11.63, which is due to the analysis of the simulation grid. The distance d_fv-out, where it stays below the limit of 15 V/m along the Ox axis on the outer side of the frame, is determined at 10.436 m, the distance d_fv-in on the inner side of the frame at 10.383 m, and the distance s_c along the Oy axis at 8.205 m (reduced from the corresponding values of the previous state by 0.373, 0.353, and 0.333 m, respectively). Therefore, the critical zone distance ds_c is determined according to Equation (6) to equal 10.436 m, which is smaller than ℓ_bc (=11.567 m) and satisfies the basic condition of Equation (13) more comfortably. Similarly, the area of critical electric field strength areas, with values greater than 15 V/m, is slightly reduced to 371 m² compared to 395 m² in the previous state. The reductions are proportional to the quotient between the ratio of transient and steady-state currents and the corresponding ratio of electric field strengths, being (the quotient) equal to 0.969 (=11.63/12). In this way, this study is covered by the field analysis in the steady state in the maintenance phase.

Table 1. Simulation results with method “C” of ref. [4] (see Appendix A) of a linear current source for a bow frame for a different radius R_K3 of 13 electrodes, with ℓ_p = 0.5 m for the area of Korakia (ρ_S = ∞, without dam, L = 2.13 m, ρ_w = 0.25 Ω∙m, θ_g = 248°, J_{mt_st} = 22 A/m², E_{limit_S} = 1.25 V/m).

RK3 [m]	θel (°)	θframe (°)	ℓf [m]	ℓpw [m]	ℓbc [m]	Emax [V/m]	dfv-in ref. [m]	dfv-out [m]	sc [m]	ℓκ [m]	Sc [m2]	Equation (13)
Point	-	-	-	-	-	229.82	11.215		22.430	22.430	503.12	-
15	1.91	22.92	5.960	0.299	9.539	24.224	10.715	10.796	8.542	23.044	394.808	No
16	1.79	21.49	5.965	0.280	10.554	23.080	10.726	10.803	8.539	23.044	394.875	No
17	1.69	20.22	5.969	0.264	11.567	22.624	10.736	10.808	8.538	23.045	394.880	Yes
18	1.59	19.10	5.972	0.249	12.578	24.268	10.745	10.813	8.537	23.046	394.893	Yes
19	1.51	18.09	5.975	0.236	13.589	23.167	10.753	10.818	8.535	23.046	394.853	Yes
20	1.43	17.19	5.978	0.225	14.599	22.404	10.761	10.822	8.534	23.047	394.853	Yes
21	1.36	16.37	5.980	0.214	15.607	23.591	10.767	10.825	8.534	23.047	394.828	Yes
2000	0.01	0.17	6.000	0.002	1994.802	24.053	10.900	10.900	8.525	23.051	394.855	Yes

presents the results of the electric field simulation for a fixed distance between consecutive electrodes ℓ_p = 0.5 m in the Korakia area for a bow frame of variable radius R_K3. It records the calculated values of the angle between consecutive electrodes θ_el, the angle of the bow frame θ_frame, the length of the bow frame chord ℓ_f, the width of the bow frame ℓ_pw and the distance between consecutive frames ℓ_bc by Equations (7)–(12), respectively. Furthermore, recorded are the values determined through the field analysis of the distance d_fv-out (where the electric field strength stays below the limit of 1.25 V/m) along the Ox axis on the outer side of the frame as well as the values of the respective distance d_fv-in on the inner side, of the distance s_c (along the Oy axis), and the total length of the critical frame zone ℓ_k on the dam, consisting of 13 electrodes (on the yOy’ axis). Finally, Table 1 shows the values of the corresponding area S_c of the critical zone with an electric field that is above the limit of 1.25 V/m (i.e., the yellow area of Figure 12) and (if satisfied) meets the basic condition of Equation (13). The radius R_K3 increases by a 1-metre step, starting from 15 m, until a radius of a suitable value is found, which satisfies the condition of Equation (13). The radius of 2000 m corresponds practically to the “infinite” radius, i.e., to the rectilinear frame (according to Table 10 in ref. [4]). It is found that:

• As the radius R_K3 increases, the sizes of the angles θ_el and θ_frame decrease, which is expected based on Equations (7) and (9), respectively.
• As the radius R_K3 increases, the chord of the bow frame ℓ_f increases slightly from 5.96 m to 6.00 m, which is the length of the straight frame (for R_K3 = ∞ with (N_{el_frame} − 1) × ℓ_p = (13−1) × 0.5 = 6.00 m).
• As the radius R_K3 increases, the width of the bow frame ℓ_pw decreases from 0.30 m to 0 m for the straight frame (for R_K3 = ∞).
• As the radius R_K3 increases, the distance between consecutive frames ℓ_bc increases significantly, tending, for large values, to the corresponding radius R_K3.
• The maximum electric field strength value is limited between the values 22.4 and 24.2 V/m, i.e., the same or slightly lower value than that of the straight frame.
• As the radius R_K3 increases, the distances d_fv-out and d_fv-in where the electric field strength stays below the limit of 1.25 V/m (along the Ox axis on the outer and inner sides of the frame, respectively) increase, tending to 10.900 m of the straight frame from smaller values.

• The distance d_fv-out is always greater than d_fv-in, but the difference decreases as the radius R_K3 increases.
• As the radius R_K3 increases, the distance s_c, where the electric field strength stays below the limit of 1.25 V/m along the Oy axis, decreases slightly, tending to 8.525 m, with a small difference of only a few cm.
• As the radius R_K3 increases, the total length of the critical frame zone ℓ_k on the dam (on the yOy’ axis) increases slightly, tending to 23.05 m with a small difference of less than 1 cm.
• The area of the critical electric field strength areas, with values greater than 1.25 V/m, remains practically constant at 395 m² regardless of the radius R_K3.

Table 2. Simulation results with method “C” of ref. [4] (see Appendix A) of a linear current source for a bow frame for a different ℓ_p of 13 electrodes for the area of Korakia (ρ_S = ∞, without dam, L = 2.13 m, ρ_w = 0.25 Ω∙m, θ_g = 248°, J_{mt_st} = 22 A/m², E_{limit_S} = 1.25 V/m) with an R_K3 radius (with a precision of 1 m) that satisfies the condition for Equation (13).

ℓp [m]	RK3 [m]	θel (°)	θframe (°)	ℓf [m]	ℓpw [m]	ℓbc [m]	Emax [V/m]	dfv-in ref. [m]	dfv-out [m]	sc [m]	ℓκ [m]	Sc [m2]
	Point	-	-	-	-	-	229.82	11.215		22.430	22.430	395.138
0.2	14	0.82	9.82	2.397	0.051	11.873	32.573	11.134	11.146	10.066	22.530	395.130
0.3	15	1.15	13.75	3.591	0.108	11.782	27.815	11.037	11.062	9.531	22.654	395.140
0.4	16	1.43	17.19	4.782	0.180	11.679	24.957	10.904	10.949	9.023	22.827	395.000
0.5	17	1.69	20.22	5.969	0.264	11.567	22.624	10.736	10.808	8.538	23.045	394.880
0.6	18	1.91	22.92	7.152	0.359	11.447	22.677	10.535	10.641	8.079	23.310	394.493
0.7	19	2.11	25.33	8.332	0.462	11.322	21.922	10.298	10.448	7.644	23.620	393.853
0.8	20	2.29	27.50	9.509	0.573	11.192	21.523	10.027	10.232	7.233	23.975	393.040
0.9	20	2.58	30.94	10.670	0.725	10.035	20.307	9.698	9.984	6.848	24.366	391.758
1.0	21	2.73	32.74	11.838	0.851	9.896	20.897	9.349	9.722	6.482	24.802	390.008
1.1	22	2.87	34.38	13.004	0.983	9.755	20.625	8.959	9.440	6.136	25.277	387.610
1.2	23	2.99	35.88	14.168	1.118	9.613	19.555	8.524	9.137	5.811	25.789	384.415
1.3	24	3.10	37.25	15.329	1.257	9.468	20.045	8.043	8.814	5.505	26.338	380.318
1.4	25	3.21	38.51	16.488	1.398	9.323	20.060	7.509	8.472	5.217	26.922	375.218
1.5	25	3.44	41.26	17.616	1.603	8.141	19.785	6.866	8.108	4.953	27.521	368.773

lists the corresponding results of the simulation of the electric field at Korakia for different distances between consecutive electrodes ℓ_p for a bow frame, including the radius R_K3 with a precision of 1 m that satisfies the condition for Equation (13) as well as the remaining variables (i.e., θ_el, θ_frame, ℓ_f, ℓ_pw, ℓ_bc, d_fv-out, d_fv-in, s_c, ℓ_k, S_c). It is found that:

• As the distance ℓ_p increases, so does the radius R_K3, for which condition (13) is satisfied, although in some cases it seems to remain constant, but this is due to the simulation step of the radius R_K3, which is taken as 1 m (cases for ℓ_p = 0.9 m and ℓ_p = 1.5 m).
• As the distance ℓ_p increases, the sizes of the angles θ_el and θ_frame and the distances ℓ_f and ℓ_pw increase, while the distance between consecutive frames ℓ_bc decreases.
• As the distance ℓ_p increases, the maximum electric field strength value decreases (in some cases where there is a small increase, this is due to the selected values of the radius R_K3, which is taken as 1 m).
• As the distance ℓ_p increases, the distances d_fv-out, d_fv-in and s_c decrease, while the total length of the critical frame zone ℓ_k on the dam (on the yOy’ axis) increases.
• As the distance ℓ_p increases, the area of the critical electric field strength areas, with values greater than 1.25 V/m, decreases at a small rate.

Table 3. Simulation results with method “C” of ref. [4] (see Appendix A) of a linear current source for a bow frame for a different ℓ_p of 13 electrodes for the area of Stachtoroi (ρ_S = ∞, without dam, L = 2.13 m, ρ_w = 0.25 Ω∙m, θ_g = 210°, J_{mt_st} = 22 A/m², E_{limit_S} = 1.25 V/m) with an R_K3 radius (with a precision of 1 m) that satisfies the condition for Equation (13).

ℓp [m]	RK3 [m]	θel (°)	θframe (°)	ℓf [m]	ℓpw [m]	ℓbc [m]	Emax [V/m]	dfv-in ref. [m]	dfv-out [m]	sc [m]	ℓκ [m]	Sc [m2]
	Point	-	-	-	-	-	171.60	8.374		16.748	16.748	220.301
0.2	11.0	1.04	12.50	2.395	0.065	8.860	24.032	8.267	8.282	7.243	16.881	220.263
0.3	12.0	1.43	17.19	3.587	0.135	8.759	20.759	8.139	8.174	6.730	17.046	220.220
0.4	13.0	1.76	21.16	4.773	0.221	8.646	17.537	7.966	8.028	6.251	17.274	220.028
0.5	14.0	2.05	24.56	5.954	0.320	8.523	17.077	7.747	7.847	5.804	17.563	219.703
0.6	15.0	2.29	27.50	7.132	0.430	8.394	17.039	7.480	7.634	5.390	17.911	219.115
0.7	16.0	2.51	30.08	8.305	0.548	8.260	16.630	7.165	7.390	5.006	18.316	218.038
0.8	16.5	2.78	33.34	9.466	0.693	7.609	16.249	6.784	7.113	4.653	18.771	216.523
0.9	17.0	3.03	36.40	10.620	0.851	6.952	15.861	6.340	6.808	4.323	19.267	214.248
1	18.0	3.18	38.20	11.780	0.991	6.807	15.620	5.845	6.481	4.025	19.831	211.018
1.1	18.5	3.41	40.89	12.924	1.165	6.143	15.197	5.253	6.128	3.751	20.425	206.610
1.2	19.5	3.53	42.32	14.077	1.315	5.994	14.598	4.593	5.753	3.496	21.069	200.728
1.3	20.5	3.63	43.61	15.229	1.467	5.845	14.892	3.825	5.355	3.262	21.753	192.740
1.4	21.0	3.82	45.85	16.358	1.658	5.175	14.654	2.874	4.937	3.052	22.462	182.235
1.5	21.5	4.00	47.98	17.482	1.857	4.503	14.724	1.704	4.503	2.860	23.202	168.098

lists the corresponding results of the simulation of the electric field at Stachtoroi for different distances between consecutive electrodes ℓ_p for a bow frame, including the radius R_K3 with a precision of 0.5 m that satisfies the condition of Equation (13), as well as the remaining variables (i.e., θ_el, θ_frame, ℓ_f, ℓ_pw, ℓ_bc, d_fv-out, d_fv-in, s_c, ℓ_k, S_c). The same general conclusions are found, as in the results of Table 2, with the only difference being that the corresponding values of the parameters related to the field results are smaller due to the smaller angle formed with the ground plane θ_g according to Figure 6 in ref. [4], Figure 1 in ref. [36] and Figure A1 in Appendix A.

In conclusion, the bow frame has slightly better field results compared to the straight one except for the critical electric field strength areas, which remain the same. It is reiterated that, according to [4] and the corresponding study of ℓ_p values from 0.2 to 1.5 m (see the corresponding Tables 9–11), it was found that increasing the length ℓ_p causes a decrease in the maximum value of the electric field strength, an increase in the frame dimensions, a small decrease in the critical length of the frame safety zone and the corresponding surface, but a significant increase in the dimensions of the electrode station. Consequently, the selected value of 0.5 m covers both the access requirements for maintenance and the dimensional limitations of the overall arrangement. This value is also proposed to be used for the corresponding shoreline electrode station configuration.

3.3. Electric Field Study of a Special-Shaped Frame Suitably Adapted to a Circular Arrangement (“Birthday Cake”) and Construction Configuration

Here, an isolated frame of a circular arrangement is studied in order to determine its general behaviour. Although, based on the analysis of Section 3.2, it should consist of 13 electrodes, for reasons of easier standardization, the use of N_{el_frame} = 12 electrodes at a constant radius was chosen so that the arc angle θ_N between consecutive electrodes amounts to 30° based on Equation (17). The most unfavourable field results in terms of boundaries arise from the steady state. The total number of electrodes is 6 × 12 = 72 under full load conditions of six panels, so the current intensity per electrode is I_{el_fu_lo_st} = 1100/72 = 15.28 A and the corresponding electric current density per electrode J_{fu_lo_st} (under full load conditions) are equal to 18.71 A/m² ≈ 19 A/m². In the case where the electrodes are arranged in a symmetrical circular arrangement (Figure 8) under symmetrical operating conditions of all frames, the symmetrical charging electric current density J_{fu_lo_st1} (under full load conditions) is equal to 18.71 A/m² ≈ 19 A/m². While in the case where the electrodes are arranged in two circular arrangements (Figure 9) under symmetrical operating conditions of all frames, the symmetrical charging electric current density J_{fu_lo_st2} (under full load conditions) is equal to 19.86 A/m² considering the increment factor β (=6.1%) [4] and implementing Equations (2) and (3) in ref. [36] (see Appendix A, Equations (A2) and (A3)). That is, the electric current density is less than the limit of the manufacturer’s specifications, which is 20 A/mm² [41]. For the case of the steady-state operation of five frames with 12 electrodes per frame, there is the possibility of non-uniform loading, so the corresponding electric current density (during maintenance J_{mt_st12}) is equal to 23.83 A/m² with a factor β = 6.1%. Here, the limit of the manufacturer’s specifications is violated [41], but because it concerns the maintenance phase, it does not last long (in the order of a few hours). Alternatively, each frame can have N_{el_frame} = 16 electrodes at a fixed radius so that the arc angle θ_N between consecutive electrodes amounts to 22.5° based on Equation (17). The total number of electrodes is 6 × 16 = 96 under the full load conditions of six frames, so the current intensity per electrode is I_{el_fu_lo_st} = 1100/96 = 11.46 A, and the corresponding electric current density per electrode J_{fu_lo_st} (under full load conditions) is equal to 14.04 A/m². Therefore, similar to before, the electric current density in steady-state operation under full load conditions of a six-frame circular array (Figure 8) J_{fu_lo_st1}^/ is equal to 14.1 A/m², in the case of two circular arrays (Figure 9) J_{fu_lo_st2}^/ is equal to 14.89 A/m₂ considering the increment factor β (=6.1%) and in the case of steady-state operation of a non-symmetrical full load of five frames during maintenance J_{mt_st12}^/ is equal to 17.87 A/m² with a factor β = 6.1%. That is, in all cases, the requirements of the manufacturer’s specifications regarding the electric current density are covered [41]. Also, in a similar way to Section 3.2, the most unfavourable situation (with respect to the study of field values) is during maintenance in the steady state.

In the case of placing 12 electrodes in a circular arrangement (with the requirement that the distance between the electrodes be 0.50 m), the corresponding radius R_f is equal to 0.9659 m (=0.5/(2∙sin(30°/2))) according to Equation (18), so, regarding construction, a radius of 1.00 m is selected. The necessary distance of 0.50 m was determined with the following reasoning: First, the distance between the electrodes in both the straight frames of refs. [4,36] and in the bow frames of the present study was chosen to be 0.50 m. In addition, the frame under construction will be placed on a concrete base, around which there will be a wall of unreinforced concrete to protect the electrodes from waves and any possible impact with an anchor from illegally sailing vessels in the area. Because an electrode (placed in the concrete in the sea) would present difficulties for adjusting and supporting/fixing it to the base, a better solution would be to form the electrode on a small concrete base of suitable thickness, i.e., 30 cm in diameter for a Φ122 electrode and 60 cm in depth, with the electrode being placed within a length of 40 cm of the 2.130 m so that when placed in the sea on the base, a solid structure would be created. However, so as to not fracture the unreinforced concrete during the entry/exit of the component, there must be a thickness of at least 20 cm between the electrode bases. Consequently, the corresponding minimum distance is set at 0.50 m (=0.30/2 + 0.20 + 0.30/2). Additionally, the external dimensions of the frame installation base are 2.5 m with slightly sloping sides to keep the total weight of the base less than 100 t in order to easily find barges for the installation. The corresponding indicative arrangement is presented in Figure 13, where certain details are also shown, such as that the base is open from the top side (for diver access for maintenance), with Φ200 pipes for the movement of water in the internal bottom of the base (for the removal of sediments formed by the operation of the electrodes), the hangers for the construction, etc. In addition, the Φ200 pipes must be placed horizontally on two separate levels (as seen in Figure 13) so that the water does not directly hit the electrodes. These holes can also be used for the entry of the cables connecting the electrode rods to the electrode control station. The interior of the circular arrangement of electrodes of equivalent diameter 1.878 m = (2 × 1.000 – 2 × 0.122/2) allows for accessing all electrodes from the interior. The corresponding space, on the external side, ranges from 0.44 m on the low end to 0.84 m on the high end, which makes the corresponding repair more difficult. Of course, on the internal side of the electrodes, the distance between the electrodes of 0.50 m allows for their overall supervision. The corresponding volume of the base amounts to 37.66 m³ (=π × 2.5² × 1.0 + π × (2.5² − 1.75²) × 1.80), that is, with a specific weight of unreinforced concrete of 2500 kg/m³, the weight amounts to 94.15 t. It is noted that the weight of an electrode, together with its elementary base of cement/unreinforced concrete, amounts to 237 kg, which is calculated as 143 kg from the size of the electrode [41] and from a base volume of 0.0375 m³ (=π × 0.15² × 0.6 – π × 0.061² × (2.122 − 1.700)), which a standard helicopter rescue winch of 275 kg can lift. Correspondingly (if the electrodes are connected to each other), for the 12 electrodes the weight amounts to at least 2.844 kg (i.e., practically close to 3 t), while for 16 electrodes it is close to 4 t, with a corresponding requirement for the ship winch.

In the case of placing 16 electrodes in a circular arrangement (with the requirement that the distance between the electrodes be 0.50 m), the corresponding radius R_f is equal to 1.28 m according to Equation (18). If the corresponding circular arrangement in Figure 13 were formed, then the space that would remain towards the outer side for the diver would be very small (from 0.16 m on the low end to 0.56 m on the high end). Because of this, placing the electrodes in a circular arrangement with the specific base is not feasible. Alternatively, increasing the construction radius of the base from 2.50 m to 2.75 m, with a corresponding shift of the average radius of the inner wall from 1.75 m to 2.00 m, in order to maintain accessibility for a diver from the outer side of the electrodes, would result in a corresponding volume of the base of 43.90 m³ (=π × 2.75² × 1.0 + π × (2.75² − 2.0²) ×1.80), i.e., to a weight of 109.75 t, which makes it difficult to find a suitable barge, etc. Besides, it is sufficient to increase the central radius from 2.50 m to 2.60 m (with a corresponding shift of the average radius of the inner wall from 1.75 m to 1.85 m) to surpass a weight of 100 t (corresponding volume 40.11 m³ and weight 100.28 t). Because of this, 16 electrodes cannot be placed in a circular arrangement with an electrode distance of 0.50 m.

However, if the required distance between the electrodes becomes 0.39 m, then the corresponding radius R_f is equal to 0.9995 m, as per Equation (18). The necessary distance of 0.39 m can be achieved if, instead of a small concrete base for each Φ122 electrode with a diameter thickness of 30 cm and a depth of 60 cm, a suitable synthetic material of high electrical conductivity and high mechanical and thermal resistance is used in order to bear all the stresses that it will receive during its technical life, both from the electrode and from the concrete base of the frame. Such synthetic materials can form a smaller base around the electrode with a thickness of about 3 mm so that the entire diameter of the electrode base does not exceed 0.19 m (>0.122 + 2 × 0.003) and the corresponding depth is 0.50 m (>2.123 − 1.700 + 2 × 0.003). Based on this case, Figure 14 is formed. Of course, the corresponding synthetic materials that can be found in the market for civil engineering work have not been tested in such projects, and the use of flame to adapt the synthetic material to the electrode should be avoided due to possible destruction of surface paint/electrode treatments. However, if the above obstacle is overcome, then a thickness of unreinforced concrete of at least 20 cm is ensured between the electrode bases, and consequently the corresponding minimum distance is set at 0.39 m (=0.19/2 + 0.20 + 0.19/2). The remaining technical characteristics of the structure are the same as in Figure 13, including the volume and weight of the electrode frame base, except for the water renewal pipes, whose diameter is reduced to Φ160 so that the water does not directly impact the electrodes while ensuring the renewal of the water at the inner bottom of the base for the removal of sediments formed by the operation of the electrodes thanks to the increase in their number from 24 to 32, keeping the total available lateral water renewal surface practically constant (from 0.7540 m² to 0.6433 m²).

Alternatively, the placement of 16 electrodes can be done in two circular arrays, provided the distance between the electrodes is 0.50 m, in order to avoid the use of synthetic materials. In this case, the radius of the inner circular arrangement R_f1 is equal to 0.653 m (=0.5/(2∙sin(22.5°))) according to Equation (26). If the radius of the inner circular arrangement R_f1 is set at 0.75 m, then the accessibility of the diver from the inner side of the electrodes is ensured, with a space of 1.38 m (=2 × 0.75 − 2 × 0.122/2) in diameter. Therefore, the radius of the outer circular arrangement R_f2 is equal to 1.102 m according to Equation (28), with θ_N = 22.5°, R_f1 = 0.75 m, and d₃ = 0.5 m, and consequently is set at 1.10 m. If the inner radius of the electrode protection wall is set at 1.60 m (instead of the original 1.50 m in Figure 13), then the corresponding space on the outer side of the electrodes ranges from 0.44 m on the low end to 0.84 m on the high end, which allows for the corresponding electrode repair while leaving the same access space as the basic arrangement in Figure 13. The outer dimensions of the frame mounting base amount to a radius of 2.5 m with slightly sloping sides and a total height of 2.80 m, as shown in Figure 15, while the 32 water renewal pipes have a diameter of Φ160 so that the water does not directly hit the electrodes but ensures the renewal of the water at the inner bottom of the base. The respective volume of the base amounts to 36.656 m³ (=π × 2.5² × 1.0 + π × (2.5² − 1.80²) × 1.80), meaning the weight amounts to 91.64 t, which is less than 100 t.

Further on, an electrode frame with 12 electrodes in a circular arrangement (which are in the steady state charged with an electric current per electrode I_{el_mt_st}= 19.452 A during maintenance) is initially studied since (according to paragraph 2 of ref. [36] and Appendix A) the total electric current intensity per electrode and not the electric current density is taken into account in the analytical model. The entire arrangement is placed in a 60 m by 60 m grid in seawater with a resistivity of 0.25 Ωm. The arc of the “left” water dam zone from Figure 1 in ref. [36] (or Figure A1 of Appendix A) for the Korakia area is equal to 160° (compared to the initial 112°, as shown in Figure 12 in ref. [4]), as the electrode base will be placed at a depth of at least 10 m so as not to encounter breaking water (the corresponding zone can reach 8 to 10 m in the worst case scenario, which, together with the active length of the electrode, results in the corresponding requirement of 10 m). The active length of the electrode for vertical suspension is equal to 1.70 m, having subtracted the length of the electrode located inside the base according to Figure 13. From the corresponding solution, we obtain the maximum electric field intensity Ε_max in the frame arrangement, the area of the electric field intensity (with values greater than 1.25 V/m), the corresponding radius of the above area r_κ per required length ℓ_p between the electrodes, and the necessary radius of the circular arrangement R_f. Alternatively, the corresponding radius r_κ^/ can be obtained for an electric field intensity limit of 2.5 V/m. In

Table 4. Simulation results with method “C” of ref. [4] (see Appendix A) of a linear current source for a “birthday cake” frame for different ℓ_p for the area of Korakia/Stachtoroi (ρ_S = ∞, without dam, L = 1.70 m, ρ_w = 0.25 Ω∙m, θ_g = 200°, E_{limit_S} = 1.25 V/m for r_κ, E_{limit_S} = 2.5 V/m for r_κ^/), whereas “*” is the proposed construction of Figure 13, Figure 14 and Figure 15.

	12 Electrodes in a Circular Arrangement, Ιel_mt_st = 19.452A				16 Electrodes in a Circular Arrangement, Iel_mt_st = 14.589 A				16 Electrodes in Two Concentric Circular Arrangements, Iel_mt_st = 14.589 A
ℓp [m]	Rf [m]	rK [m]	rK/ [m]	ℓp [m]	Rf [m]	rK [m]	rK/ [m]	ℓp [m]	Rf [m]	rK [m]	rK/ [m]	ℓp [m]	Rf [m]
0.2	0.3864	9.8351	4.9177	32.543	0.5126	9.8338	4.9170	25.513	0.2613	0.4146	9.8338	4.9170	30.716
0.3	0.5796	9.8351	4.9177	26.943	0.7689	9.8338	4.9170	20.601	0.392	0.6219	9.8338	4.9170	26.290
0.4	0.7727	9.8351	4.9177	24.392	1.0252	9.8338	4.9170	18.548	0.5226	0.8293	9.8338	4.9170	22.462
0.5	0.9659	9.8351	4.9177	22.852	1.2815	9.8338	4.9170	17.240	0.6533	1.0366	9.8338	4.9170	20.602
0.6	1.1591	9.8351	4.9177	21.857	1.5377	9.8338	4.9170	16.476	0.7839	1.2439	9.8338	4.9169	19.441
0.7	1.3523	9.8351	4.9177	21.121	1.794	9.8338	4.9170	15.979	0.9146	1.4512	9.8338	4.9168	18.041
0.8	1.5455	9.8351	4.9177	20.580	2.0503	9.8338	4.9170	15.524	1.0453	1.6585	9.8338	4.9166	17.618
0.9	1.7387	9.8351	4.9177	20.135	2.3066	9.8338	4.9170	15.176	1.1759	1.8658	9.8338	4.9159	16.718
1.0	1.9319	9.8351	4.9177	19.743	2.5629	9.8338	4.9171	14.925	1.3066	2.0731	9.8338	4.9146	16.692
1.1	2.125	9.8351	4.9179	19.482	2.8192	9.8338	4.9196	14.701	1.4372	2.2804	9.8337	4.9118	16.327
1.2	2.3182	9.8351	4.9183	19.263	3.0755	9.8338	4.9197	14.507	1.5679	2.4878	9.8337	4.9065	15.812
1.3	2.5114	9.8351	4.9192	19.078	3.3318	9.8338	4.9265	14.341	1.6985	2.6951	9.8336	4.8969	15.395
1.4	2.7046	9.8351	4.9214	18.918	3.5881	9.8338	4.9460	14.215	1.8292	2.9024	9.8335	4.8800	14.942
1.5	2.8978	9.8351	4.9261	18.779	3.8444	9.8338	4.9932	14.132	1.9598	3.1097	9.8333	4.8504	15.375
*	1.000	9.8351	4.9177	22.725	1.000	9.8338	4.9170	18.920	0.750	1.100	9.8338	4.9170	20.049

, the corresponding results are listed, changing the distance between the electrodes and rods ℓ_p from 0.20 m to 1.50 m. Similarly, the same procedure is repeated for the Korakia area for the frame with 16 electrodes in a circular arrangement of necessary radius R_f, which are in the steady state and are charged with an electric current intensity per electrode I_{el_mt_st} = 14.589 A during maintenance and for the electrode frame in two circular arrangements of the two necessary radii R_f1 and R_f2, based on the minimum distances.

If the corresponding study is carried out for the Stachtoroi area, the electrode frame base will be placed at isobaths of at least 5 m, as the breaking water zone can reach 3 m under the most unfavourable conditions, which together with the active length of the electrode results in the requirement of 5 m. However, due to the steep bottom slope, as shown in 11 in ref. [4], the electrode frame bases will extend at least 15 m into the isobath, with the consequence that the arc of the “left” water zone barrier of Figure 1 in ref. [36] (or Figure A1 of Appendix A) will range from 155° to 180°⁺, with a typical value equal to 160° (compared to the initial 150°). Therefore, the numerical results of one electrode frame base of the Stachtoroi area are identical to those of Korakia, as recorded in Table 4. Additionally, the numerical results for the construction structures of Figure 13, Figure 14 and Figure 15 are recorded in Table 4, as are the graphical representations of the electric field intensity on the Oxy plane (for the frame structures of Figure 13, Figure 14 and Figure 15 in Figure 16, Figure 17 and Figure 18) and the area of the electric field intensity with values greater than 1.25 V/m for the structure in Figure 13, namely the 12-electrode frame base in a circular arrangement of radius R_f = 1 m (the rest are visually identical) in Figure 19.

The following conclusions emerge from the corresponding study of the results:

• In a circular arrangement of electrodes, when the distance between them changes, the necessary radius r_κ required to reduce the electric field intensity below 1.25 V/m practically does not change. Additionally, it is found that this radius is practically the same, equal to 9.84 m, in all three construction structures studied (frame of 12 electrodes in a circular arrangement, frame of 16 electrodes in a circular arrangement, and frame of 16 electrodes in two concentric circular arrangements).
• Regarding the electric field intensity limit, equal to 2.5 V/m, it is observed that with the increase in the distance between the electrodes, there is a minimal increase in the necessary radius r_κ^/ (of the order of a few mm), which is required to comply with this limit for the case of a frame with a circular arrangement (as shown in Table 4). In contrast, in the case of a frame with two circular arrays, a reduction in the corresponding radius (of the order of a few mm) is observed, though this does not practically change the radius of around 4.92 m.
• From the comparison with the corresponding results of the straight frames (Tables 9–11 in ref. [4]) and the bow frames (Table 1, Table 2 and Table 3), it is found that the straight and bow frames change their behaviour more with the change in the distance of successive electrodes.
• In terms of electric field intensity, when the distance of the electrodes ℓ_p increases, the maximum intensity of the electric field calculated with the analytical method “C” in ref. [4] (see Appendix A) decreases. This generally expresses the distribution of the electric field in the nearby space. It is found that increasing the number of electrodes reduces the levels of electric field intensity (comparison of Figure 16 and Figure 17, i.e., a circular arrangement structure of 12 electrodes with a corresponding one of 16 electrodes), while placing the same number of electrodes in two circular arrangements leads to more intense fluctuations in the electric field (comparison of Figure 17 and Figure 18, i.e., a structure of 16 electrodes in one circular arrangement and two circular arrangements). In the calculations based on Equations (4)–(7) in ref. [4] (Equations (A4)–(A7) of Appendix A), the angle θ_g is taken into account, which significantly limits the active surface of the zone of action of each electrode, which in combination with the small thickness of the electrode water zone, which is equal to the active length of the electrode, leads to much higher electric field intensity values. Due to the superposition of the 12 or 16 different electrodes, higher electric field intensity values are obtained around the electrodes than those that would result if one were to study the interface of a single electrode.
• Essentially, a distance of at least 10 m is required between the circular array frames so that the developing electric field intensity from one frame falls below 1.25 V/m. Of course, when one frame is not operating, the other five are fully operational, so the corresponding distance should be significantly greater, as has been established from studies of straight frames in various arrangements (see [4,36]).

It is also noted that:

• As with linear and bow frames, increasing the length ℓ_p causes a decrease in the maximum value of the electric field strength, an increase in the frame dimensions (expressed through the corresponding outer radius R_f for the case of one circular arrangement and R_f2 for the case of two circular arrangements) and a very small change in the critical length of the safety zone of the birthday cake frame and the corresponding surface. Thus, again, a value of 0.5 m is chosen, which covers both the access requirements for maintenance and the possibility of easy implementation of the corresponding concrete base.
• If an isolated frame of a circular arrangement consists of N_{el_frame} = 13 electrodes, then the electric current per electrode is equal to I_{el_mt_st} = 17.955 A instead of 19.452 A for 12 electrodes. That is, the increase in the number of electrodes by 8.33% yields a decrease in the electric current per electrode by 7.70% (similarly to the density current). In the case of a length ℓ_p equal to 0.5 m, the maximum electric field intensity Ε_max becomes 21.2780 V/m, i.e., smaller by 6.89% than the corresponding value of 22.8516 V/m for the 12 electrodes. The necessary radius of the circular arrangement R_f increases to 1.0446 m, i.e., by 8.15% compared to the case of 12 electrodes. The radius r_κ for the area where the electric field intensity values are greater than 1.25 V/m remains unchanged and equal to 9.8351 m. The same occurs with radius r_κ^/ for the area where the electric field intensity values are greater than 2.5 V/m, taking a value equal to 4.9177 m. Thus, apart from the small reduction in density current and maximum electric field intensity, the use of 13 electrodes does not offer any other benefit other than an increase in material both in terms of electrodes by 8.33% and concrete by 5.17%.

• If an additional electrode is added at the centre of the circular arrangement (of either 12 or 16 electrodes), then the necessary radius r_κ does not change, but the calculated electric field intensity is reduced. However, the available access space for the diver is reduced by half, so this addition is not considered necessary. Furthermore, due to its location (in the centre of the circle), this electrode encounters a different resistance with respect to remote earth, so, in fact, a current of different intensity will pass through.
• Among the three construction structures in Figure 13, Figure 14 and Figure 15, the structure with the 12 electrodes in a circular arrangement of radius 1.0 m is the simplest, without substantial differences from the other two in terms of field effects and the necessary radius to achieve an electric field strength below 1.25 V/m. Taking into account that close to the frame, locally higher electric field strength values are presented compared to the other two structures, for the rest of the study, the structure of Figure 13 is taken as the least favourable of the three. However, it is also the lightest during the construction phase (taking into account the 100 t limit of the installation barge).

4. Application of the Analytical Methodology for the Calculation of Electric Field Strength in the Electrode Station with Various Configurations

4.1. General Remarks

To determine the electric field strength in the area close to the electrode station, the superposition methodology is employed during periodic maintenance at a steady state, since it is then that the least favourable results are observed concerning safety distance and as regards to the point electric field criterion of 1.25 V/m. The whole array is adapted on an appropriate simulation canvas which, in the basic configuration, is 160 m (on the xOx’) by 160 m (on the yOy’), as in Figure 4 and Figure 10. The step is 0.05 m by 0.05 m. The simulation involves the uniform loading of the station (be it six frame or five frame) by setting, in turn, the respective frames out of operation, thus determining the electric field strength all over the Oxy as well as the area where it reaches values higher than E_{limit_S} = 1.25 V/m. The results include the following, depending on the mode of operation:

• The electric current density J_st with respect to the peripheral surface or the current per electrode I_el;
• The widths d_Ox and d_Ox^/ of the critical zone of the electrode station along the semi-axes Ox and Ox^/ from the beginning of the axes (which is the centre of the central base in Figure 4 or the centre of the imaginary polygon in Figure 10);
• The lengths d_Oy and d_Oy^/ of the critical zone of the electrode station along the semi-axes Oy and Oy^/ from the beginning of the axes;
• The maximum electric field strength in the area of the non-operational frame during maintenance E_off;
• The maximum electric field strength of the arrangement E_max.

To calculate the electrode station to remote earth resistance R_el, the maximum value of the absolute potential V_{rel_max} and the absolute potential on the beginning of the axes O V_O, the corresponding simulation is extended to a canvas of 150 km (Ox axis) by 160 m (yOy’ axis) and with a step of 0.10 m from 60 m up to 100 m, 1.0 m from 100 to 200 m, 5.0 m from 200 to 1000 m, 10 m from 1000 to 10,000 m, and 100 m from 10 km to 150 km on the Ox axis and by 0.10 m on the yOy’, as in Figure 4 and Figure 10. It is pointed out that the maximum absolute potential V_{rel_max} is calculated for y, in which the maximum value of all V_max(y) occurs, where each one has resulted from its respective integration (along a semi-axis parallel to Ox).

4.2. Eleventh Electrode Station Configuration—Bow Frames Adapted Perimetrically to a Central Base

This configuration employs the standard bow frame arrangement illustrated in Figure 4, Figure 5 and Figure 6 in §3.2. Each frame accommodates 13 electrodes (N_{el_frame} = 13) spaced 0.50 m apart (ℓ_p). The bow’s chord and the overall “gross” radius of the central base (R_K3) are determined according to site-specific conditions and required electric field results. For Korakia, the minimum spacing between consecutive frames (d_sc) has been computed from Equation (6) as 10.808 m. This requirement is satisfied by adopting a central base “gross” radius of R_K3 = 17.0 m (see Table 1 and Table 2), which also includes the d_r1 distance (=1 m, indicatively)—separating each suspended frame from the base itself. In the case of Stachtoroi, Equation (6) yields a smaller minimum frame spacing of d_sc = 7.847 m. The selected base radius of R_K3 = 14.0 m (from Table 3) similarly encompasses the suspension distance d_r1. Simulation results derived from these configurations are presented in

Table 5. Simulation results for the calculation of electric field strength using a linear current source (ρ_S = ∞, without dam, ρ_w = 0.25 Ω∙m) and absolute potential using a linear current source (for the near field) and point current source (for the far field) (ρ_S = ∞, ρ_w = 0.25, ρ_d = 100 Ω∙m) with different supply methods of the 11th electrode station configuration (circular base with bow frames, in the area of Korakia, Crete (ℓ_p = 0.50 m, R_K3 = 17.0 to 28.5 m, L = 2.13 m, θ_g = 248°, θ_w = 2.29°) and in the area of Stachtoroi, Attica (ℓ_p = 0.50 m, R_K3 = 14.0 to 21.5 m, L = 2.13 m, θ_g = 210°, θ_w = 0.272°) with E_{limit_S} = 1.25 V/m.

		Supply Method	Jst [A/m2]	dOx [m]	dOx/ [m]	dOy [m]	dOy/ [m]	Εoff [V/m]	Εmax [V/m]	Vrel_max [V]	Rel [Ω]	VO [V]
Korakia, Crete	RK3 = 17.0 m, R1 = 0 m	6 frames operating	18.33	56.01	−56.01	56.03	−56.03	-	20.03	3761	3.223	−2747
		5 frames operating (except for no. 6)	22.00	55.53	−58.04	58.42	−52.86	2.11	24.74	12,591	10.788	−3141
		5 frames operating (except for no. 5)	22.00	58.04	−55.53	58.42	−52.86	2.11	24.74	−12,372	10.601	−6230
		5 frames operating (except for no. 4)	22.00	58.62	−50.58	57.07	−57.07	2.11	24.57	−16,274	13.944	−7037
		5 frames operating (except for no. 3)	22.00	58.04	−55.53	52.86	−58.42	2.11	24.74	−12,372	10.601	−6230
		5 frames operating (except for no. 2)	22.00	55.53	−58.04	52.86	−58.42	2.11	24.74	12,591	10.788	−3141
		5 frames operating (except for no. 1)	22.00	50.58	−58.62	57.07	−57.07	2.11	24.57	16,505	14.141	9295
	RK3 = 28.5 m, R1 = 0 m	5 frames operating (except for no. 6)	22.00	57.45	−59.93	58.84	−52.86	1.24	24.47	15,449	13.237	−5745
		5 frames operating (except for no. 1)	22.00	42.48	−60.37	57.68	−57.68	1.24	24.04	19,379	16.604	9620
	RK3 = 28.5 m, R1 = 14 m	5 frames operating (except for no. 6)	22.00	57.45	−59.93	58.84	−52.86	1.24	24.47	15,449	13.237	−6482
		5 frames operating (except for no. 1)	22.00	42.48	−60.37	57.68	−57.68	1.24	24.04	13,415	11.494	5954
Stachtoroi, Attica	RK3 = 14.0 m, R1 = 0 m	6 frames operating	18.33	41.9	−41.9	41.82	−41.82	-	15.1	−2041	1.748	−1385
		5 frames operating (except for no. 6)	22.00	41.55	−43.49	43.75	−39.29	1.94	18.68	8733	7.483	−1538
		5 frames operating (except for no. 5)	22.00	43.49	−41.55	43.75	−39.29	1.94	18.68	−8309	7.119	−3850
		5 frames operating (except for no. 4)	22.00	43.92	−37.04	42.73	−42.73	1.94	18.71	−11,203	9.599	−4453
		5 frames operating (except for no. 3)	22.00	43.49	−41.55	39.29	−43.75	1.94	18.68	−8309	7.119	−3850
		5 frames operating (except for no. 2)	22.00	41.55	−43.49	39.29	−43.75	1.94	18.68	8733	7.483	−1538
		5 frames operating (except for no. 1)	22.00	37.04	−43.92	42.73	−42.73	1.94	18.71	11,636	9.970	6919
	RK3 = 21.5 m, R1 = 0 m	5 frames operating (except for no. 6)	22.00	42.92	−44.76	43.95	−39.48	1.24	18.37	10,505	9.001	−3079
		5 frames operating (except for no. 1)	22.00	31.69	−45.09	43.09	−43.09	1.24	17.9	13,432	11.509	7203
	RK3 = 21.5 m, R1 = 11.5 m	5 frames operating (except for no. 6)	22.00	42.92	−44.76	43.95	−39.48	1.24	18.37	8514	7.295	4148
		5 frames operating (except for no. 1)	22.00	31.69	−45.09	43.09	−43.09	1.24	17.9	10,505	9.001	−3600

. Structurally, the central base is designed as a hollow concrete shell with an outer radius R_Kc = R_K3−d_r1 and an inner radius R₁, the internal cavity being filled with seawater. Figure 20 illustrates the distribution of the electric field strength and delineates the region where the field exceeds 1.25 V/m, representing the most unfavourable operating scenario at Korakia assuming five active frames and one under maintenance (sixth frame out of operation).

Findings for the Korakia area:

• The critical zone extends up to 58.62 m from the centre in all directions depending on how unevenly distributed the station load is (strongly approximating the results in the case of frames positioned perimetrically to a central base). This practically mirrors the behaviour of configuration 10 in ref. [36], which employs straight frames (zone breadth =58.54 m). The slightly less favourable results of a few centimetres are due to the fact that in the bow configuration, the electrodes are essentially arranged in a circular way (of radius 15 m), while in the 10th configuration in ref. [36], the straight frame is a chord of the corresponding circle of radius 15 m, so (except for the outer electrodes of each frame) the rest are located at distances less than 15 m in relation to the centre of the circle. When the structure is positioned at the 12.0 m isobath, the base centre remains more than 60 m offshore, and minimal dredging ensures that the frames lie at depths exceeding 10 m—well below the zone affected by breaking waves (as illustrated in Figure 21).
• The maximum electric field strength of the electrode station deviates by 9.72% from that of a single bow frame—24.74 V/m versus 22.62 V/m (Table 1 and Table 2)—which is the smallest variation observed among all configurations summarized in Table 10 of ref. [36].
• Figure 20b illustrates areas between frames that present notable decreases in the electric field strength below 1.25 V/m. In the zone where one frame is inactive, the field strength does not exceed 2.11 V/m, which is below the 2.5 V/m safety limit. Consequently, diver safety during maintenance is not compromised. If a stricter limit of 1.25 V/m is required even around inactive frames, increasing the base radius to 28.5 m achieves this outcome. This result was established through a series of iterative simulations using base radii from 16.5 m upward in 0.5 m increments. It proves that the inner diameter of the base has negligible influence on electric field values.
• The maximum absolute potentials and the station resistance to remote earth were found to be 16.51 kV and 14.14 Ω, respectively. These are slightly lower relative to configuration 10 in ref. [36], which recorded 16.83 kV and 14.42 Ω for straight frame arrangements around a central base in open sea. Compared with all other designs in Table 10 in ref. [36], the present configuration performs better, except for the ninth configuration, which exhibits lower values (11.10 kV, 9.51 Ω) for radially positioned straight frames in open sea. This layout also demonstrates good performance under symmetrical operating conditions.
• At the centre point O of the central base, the absolute potentials that are observed are of lower absolute values than the corresponding maximum absolute potentials. This reduction results from field interaction effects, since the highest potential levels occur in the vicinity of the active frames.
• Regarding field behaviour, when a single frame is inactive, the results show patterns that closely resemble one another due to the symmetrical features of the array. Any minor discrepancies in electric field strength are attributable to the Cartesian coordinate system employed and the step size used in the simulation. Moreover, when determining the absolute potential, the specific integration path adopted in the calculation is also a decisive factor.

• Increasing the gross radius of the base R_K3 from 17.0 m to 28.5 m slightly reduces the results of the electric field (from 24.74 to 24.47 V/m), ensuring that the electric field strength in the area of the frame under maintenance is below the limit of 1.25 V/m. In the case of a solid base, however, larger maximum absolute potentials and the corresponding values of the electrode station resistance, with respect to remote earth, develop, being equal to 19.38 kV and 16.61 Ω, respectively. If the base becomes hollow (e.g., with an internal radius of 14 m), the respective values are significantly reduced (15.45 kV, 13.24 Ω), even to smaller values than those of the original base (R_K3 = 17.0 m).

Findings for the Stachtoroi area:

• The critical zone, measured from the centre of the arrangement, extends up to 43.92 m in all directions. This depends on how unevenly loaded the station is and practically matches the results for radially arranged frames around a central base. The behaviour is nearly identical to that of configuration 10 of Table 11 in ref. [36], which used straight frames and produced a similar value (43.90 m). The differentiations emerge from the same reasons analysed in the case of Korakia. The installation may be positioned at the 5.5 m isobath (the sum total of the 3.0 m breaking wave depth and the 2.13 m electrode height). Under these conditions, the critical electric field zone extends 28.35 m inland, while the centre of the configuration lies at the 8.8 m isobath (see Figure 22), coinciding exactly with its position in Figure 33 in ref. [36]. Alternatively, if the base is placed 44 m offshore (distance from the centre of the arrangement to the nearest shoreline), the corresponding isobath would be approximately 14 m, ensuring that the entire critical electric field zone remains fully within water.

• The difference between the maximum electric field strength for the entire electrode station and that of a single bow frame is 9.56% (18.71 V/m versus 17.08 V/m, see Table 3). This variation is smaller than that of all other configurations listed in Table 11 in ref. [36] and is also slightly below the value obtained for configuration 10, which features straight frames arranged around a central base in open sea (18.77 V/m).
• In the area of an inactive frame, local electric field strength values exceed 1.25 V/m, yet they are never measured over 1.94 V/m, remaining comfortably within the 2.5 V/m threshold. Thus, no safety hazards arise for divers performing maintenance operations. To maintain electric field strengths below 1.25 V/m (if so desired), a central base radius of approximately 21.5 m is required, as verified through successive numerical trials with radii ranging from 14.0 m up at 0.5 m steps.
• The developing maximum absolute potentials and the corresponding values of the electrode station resistance, with respect to remote earth, are ίσα με 11.64 kV and 9.98 Ω, respectively, slightly smaller than those in configuration 10 with straight frames placed around a central base in open sea in Table 11 in ref. [36] (12.07 kV, 10.34 Ω). Additionally, this value is better compared to all other configurations in ref. [36] except for the ninth configuration with straight frames placed radially to a central base in open sea (7.21 kV, 6.18 Ω of Table 11 in ref. [36]). It also behaves very well during symmetrical operation.
• At the centre point O of the central base, the absolute potentials that are observed are of lower absolute values than the corresponding maximum absolute potentials. This reduction results from field interaction effects, since the highest potential levels occur in the vicinity of the active frames.
• When a single frame is not operating, the electric field results remain practically unchanged owing to the symmetrical geometry of the array in regard to its centre O. The respective values along the d_Ox, d_Ox/, d_Oy and d_Oy/ directions would be identical if the Cartesian coordinate system were rotated accordingly. The minor deviations (limited to the fourth decimal) regarding the electric field strength magnitudes are attributed to the elementary canvas square (0.05 m × 0.05 m square elements).
• The numerical discrepancies observed in the calculation of the absolute potential are more significant than those associated with the electric field strength. This occurs because both the integration path of the electric field strength and the step size used in the numerical process significantly affect the results. The step length is not constant, varying with distance due to the considerable total completion length—approximately 150 km. Consequently, even when identical loading conditions are applied to five different frames, the calculated values are not the same. To ensure safety, the analysis consistently adopts the least favourable value.
• Increasing the gross radius of the base R_K3 from 14.0 m to 21.5 m slightly reduces the results of the electric field (from 18.71 to 18.37 V/m), ensuring that the electric field strength in the area of the frame under maintenance is below the limit of 1.25 V/m. In the case of a solid base, however, larger maximum absolute potentials and the corresponding values of the electrode station resistance with respect to remote earth develop, being equal to 13.43 kV and 11.51 Ω, respectively. If the base becomes hollow (e.g., with an internal radius of 11.5 m), the corresponding values are significantly reduced (10.51 kV, 9.00 Ω) to even smaller values than those of the original base (R_K3 = 14.0 m).

To sum up, configuration 11 demonstrates slightly superior performance compared with configuration 10 in ref. [36], which employs straight frames arranged around a central base in open sea. Overall, it achieves notable improvements in both absolute potential and electrode station resistance to remote earth, with the only exception being configuration 9 in ref. [36], which utilized radially positioned straight frames on a central base. Among all configurations examined, configuration 11 delivers the most favourable field characteristics, thus eliminating diver safety concerns during single-frame maintenance when the station is operating under full-load conditions. An increase in the central base radius generally yields lower electric field strength results and, in most cases, reduces absolute potentials and electrode station resistance to remote earth. For configurations with larger outer radii, the introduction of an internal cavity within the base is important in regard to the aforementioned resistance—a relationship also confirmed for the ninth and 10th configurations in ref. [36].

4.3. Twelfth Electrode Station Configuration—“Βirthday Cake” Frames Placed at the Vertices of an Imaginary Regular Hexagon

In the present case, the array is formed by N_frame^/= 6 frames with N_{el_frame} = 12 or 16 electrode rods in a circular arrangement of radius 1.00 m placed at the vertices of a perceived regular hexagon inscribed in a circle of radius R_K4, which has a typical value of 17 m and can vary from 10 to 25 m by 1.0 m. In the case of the study of the Korakia area, the arc of the “left” water-dam zone of Figure 1 of ref. [36] (or Figure A1 of Appendix A) is equal to 160° (instead of 112°), as the base of the electrode frame will be placed at isobaths of at least 10 m so as to not encounter breaking water, taking into account the isobaths in Figure 21. Correspondingly, in the case of the study of the Stachtoroi area, the arc of the “left” water-dam zone of Figure 1 of ref. [36] (or Figure A1 of Appendix A) is equal to 160° (with a range of values of 150–180°⁺) instead of 150°, as the base of the electrode frame will be placed at isobaths of at least 5 m so as to not encounter breaking water and will extend to 15 m, taking into account the isobaths and the large inclination at the bottom of Figure 22. Based on the loading data of §3.3, the electric field for the electrode station is studied, and the distances d_Ox, d_Ox/, d_Oy, and d_Oy/ (from the origin of the axes) are determined both for the electric field intensity limit above 2.5 V/m and above 1.25 V/m (except for the parameters E_off, E_max, V_{rel_max}, R_el and V_O, as described in §4.1).

Initially, the case is studied where the six frames of 12 electrode rods in a circular arrangement with a radius of 1.00 m are placed at the vertices of a regular hexagon of radius R_K4 = 17.0 m. This distance was determined in order to reconcile the minimum necessary distance of 10 m (of Table 4 of §3.3) and the double distance, which is estimated to be needed, in order (in the non-operating frame) to achieve an electric field intensity within the limits as well as from the need to not overlap the bottom levelling excavations (for the placement of the adjacent frame bases), which will facilitate the civil engineering work, especially for the Stachtoroi area with the large bottom slope. The relevant results with uniform loading of the electrode station of six or five frames (with the sixth, fifth, fourth, third, second or first frame out of operation) are recorded in

Table 6. Simulation results for the calculation of electric field strength using a linear current source (ρ_S = ∞, without dam, ρ_w = 0.25 Ω∙m) and absolute potential using a linear current source (for the near field) and point current source (for the far field) (ρ_S = ∞, ρ_w = 0.25, ρ_d = 100 Ω∙m) with different supply methods for the 12th electrode station configuration (six “birthday cake” frame of a circular arrangement of radius R_f = 1.00 m of 12 electrodes (L = 1.70 m) placed circumferentially around a perceived regular hexagon inscribed in a circle of radius R_K4 = 17.0 m according to Figure 10) in the area of Korakia, Crete (θ_g = 200°, θ_w = 2.29°) and in the area of Stachtoroi, Attica (θ_g = 200°, θ_w = 0.272°) with E_{limit_S} = 2.5/1.25 V/m.

	Supply Method	6 Frames Operating	5 Frames Operating (Except for No. 6)	5 Frames Operating (Except for No. 5)	5 Frames Operating (Except for no. 4)	5 Frames Operating (Except for No. 3)	5 Frames Operating (Except for No. 2)	5 Frames Operating (Except for No. 1)
	Iel_st [A]	15.278	19.452	19.452	19.452	19.452	19.452	19.452
2.50 V/m	dOx [m]	25.685	27.191	27.892	28.001	27.892	27.191	18.069
	dOx/ [m]	−25.685	−27.892	−27.191	−18.069	−27.191	−27.892	−28.001
	dOy [m]	23.262	25.706	25.706	25.609	24.728	24.728	25.609
	dOy/ [m]	−23.262	−24.728	−24.728	−25.609	−25.706	−25.706	−25.609
1.25 V/m	dOx [m]	47.154	48.908	51.187	51.685	51.187	48.908	42.963
	dOx/ [m]	−47.154	−51.187	−48.908	−42.963	−48.908	−51.187	−51.685
	dOy [m]	46.944	51.423	51.423	50.238	46.089	46.089	50.238
	dOy/ [m]	−46.944	−46.089	−46.089	−50.238	−51.423	−51.423	−50.238
	Εoff [V/m]	-	1.824	1.825	1.826	1.824	1.825	1.826
	Εmax [V/m]	19.572	24.194	24.194	24.196	24.194	24.194	24.196
Korakia	Vrel_max [V]	3041.00	3536.02	3477.64	3463.69	3477.64	3536.02	4025.55
	Rel [Ω]	2.765	3.030	2.980	2.968	2.980	3.030	3.449
	VO [V]	1157.58	1172.47	1164.50	1163.49	1164.50	1172.47	1421.81
Stachtoroi	Vrel_max [V]	3166.92	3666.87	3608.48	3594.53	3608.48	3666.87	4156.40
	Rel [Ω]	2.879	3.142	3.092	3.080	3.092	3.142	3.561
	VO [V]	1282.77	1303.32	1295.35	1294.33	1295.35	1303.32	1552.66

, as are the graphs of the electric field strength in the Oxy plane in Figure 23a,b; the area of the electric field strength with values greater than E_{limit_S} = 2.5 V/m in Figure 24a,b; and the area of the electric field strength with values greater than E_{limit_S} = 1.25 V/m in Figure 25a,b with uniform loading of the electrode station of six or five frames with the first one out of operation (the rest due to the symmetry of the configuration and the simplifying methodology will give the same results). From the study of the results, the following emerges:

• The radius of the critical zone, from the centre of the arrangement, reaches up to 51.42 m in all directions depending on the uneven loading of the station. This is better for the Korakia area, which is of the order of 56 m, and worse for the Stachtoroi area, which is of the order of 44 m according to [36].
• In Figure 24, areas are observed between the electrode frames, where the electric field strength is significantly reduced below 2.5 V/m and in Figure 25 below 1.25 V/m. In the area of the frame that is set out of operation (case of Figure 24b and Figure 25b), it is below 2.5 V/m, i.e., there is no safety issue for the diver during their maintenance, but is above 1.25 V/m, reaching the value of 1.82 V/m.
• From Figure 24 and Figure 25, it is observed that within the electrode station arrangement, there are areas with considerably reduced electric field strength values (below the limits of 2.5 or 1.25 V/m). This is due to the mutually cancelling electric fields of the diametrically opposite electrodes, both at the level of the arrangement of the six frames of the electrode station and at the level of the arrangement of electrodes within each frame, as is also clearly confirmed by Figure 19 for a single frame.
• The deviation in the maximum developing electric field strength of the electrode station compared to a single “birthday cake” frame (24.20 V/m against 22.73 V/m, according to Table 4) is of the order of 9.56%, which is moderate.
• The deviation in the maximum developing electric field strength of the electrode station compared to a single straight frame (24.20 V/m έναντι 24.06 V/m) is of the order of 0.57%, which is practically negligible and is better than all other cases for the Korakia area (Table 10 in ref. [36] and Table 5). The deviation in the maximum developing electric field strength of the electrode station compared to a single straight frame for the Stachtoroi area (24.20 V/m versus 17.96 V/m) is quite significant, of the order of 37.72%, which is greater than all other configurations (Table 11 of ref. [36] and Table 5).
• The developing maximum absolute potentials and the corresponding values of the electrode station resistance with respect to remote earth are considerably better (4.03 kV, 3.45 Ω) compared to the original structure (22.77 kV, 19.5 Ω according to Table 15 in ref. [4]) and all other modifications for the Korakia area according to Table 10 in ref. [36] and Table 5.

• The developing maximum absolute potentials and the corresponding values of the electrode station resistance with respect to remote earth are considerably better (4.16 kV, 3.56 Ω) compared to the original structure (17.30 kV, 14.82 Ω, according to Table 15 in ref. [4]) and all other modifications for the Stachtoroi area according to Table 11 in ref. [36] and Table 5.
• The reduction in the maximum value of the absolute potentials is due (beyond the symmetrical structure of the arrangement) to the use of a much smaller concrete “dam” thickness.
• The absolute potentials, which develop at the centre O of the perceived regular hexagon, take smaller absolute values than the corresponding maximum absolute potentials due to field interactions, where the larger values appear near the frames in operation.
• In Stachtoroi, the maximum value of the electric field strength increased compared to the classical designs, but due to symmetry, there are no pronounced changes in the absolute potential between the different loadings.
• The worst values of the electric field results in Stachtoroi, are due to the fact that there is no significant improvement in the angle, in terms of arc, of the “left” water dam zone in Figure 1 of ref. [36] (or Figure A1 of Appendix A) (in contrast to Korakia), while there is a significant reduction in the active length of the water layer zone (which is typically equal to the active length of the electrode). However, in the present case, it is extremely unfavourable because the respective frames will be found at a greater depth (than with all other modifications), with the consequence that the actual water zone is underdimensioned, which in the Stachtoroi area is at least three times larger in the shallows and ten times larger at the most remote base.
• For the Korakia area, by appropriately positioning the frames at an isobath of 10.0 m and above, as shown in Figure 26, it is ensured that the centre O of the electrode station is more than 70 m away from land, and with minimal dredging, the upper ends of the frames are at a depth of more than 7 m outside of breaking water.
• For the Stachtoroi area, with the appropriate placement of the frames at isobaths of 5.0 m and above (as shown in Figure 27), it is ensured that the centre O of the electrode station is more than 40 m away from land and has minimal dredging, while the upper edges of the frames are at least at a depth of 2.2 m so that they never encounter breaking water.
• Spatially, it is found that the installation in Korakia is located much further from the coast, with the closest electrode bases being placed, on average, at isobaths of 10.4 m, while in Stachtoroi, the closest bases are located near the initial position of the dam placed on average at isobaths of 6.5 m. Correspondingly, the most remote bases in Korakia are placed at isobaths of 13.5 m, on average, compared to 16 m in Stachtoroi, which is due to the steep inclination of the seabed near the coast at Stachtoroi.
• The results of the field behaviour, in the case of non-operation of a frame, are completely similar due to the symmetry of the arrangement around the centre O. The respective results in d_Ox, d_Ox/, d_Oy and d_Oy/, would be the same with the appropriate rotation of the Cartesian coordinate system. The small numerical difference in the fourth significant digit in the magnitudes of the electric field strength is due to the equivalent quadratic study element of the phenomenon 0.05 m × 0.05 m.

The case is then investigated in which the six frames of 16 electrode rods in a circular arrangement of radius 1.000 m are placed at the vertices of a regular hexagon of radius R_K4 = 17.0 m, with the consequence that the distance between the electrodes is limited to 0.39 m instead of 0.50⁺ m. The relevant results with uniform charging of the electrode station of six or five frames (with the sixth, fifth, fourth, third, second or first frame out of operation) are listed in

Table 7. Simulation results for the calculation of electric field strength using a linear current source (ρ_S = ∞, without dam, ρ_w = 0.25 Ω∙m) and absolute potential using a linear current source (for the near field) and point current source (for the far field) (ρ_S = ∞, ρ_w = 0.25, ρ_d = 100 Ω∙m) with different supply methods of the 12th electrode station configuration (six “birthday cake” frames of a circular arrangement of radius R_f = 1.00 m of 16 electrodes (L = 1.70 m) placed perimetrically around a perceived regular hexagon inscribed in a circle of radius R_K4 = 17.0 m according to Figure 10) in the area of Korakia, Crete (θ_g = 200°, θ_w = 2.29°) and in the area of Stachtoroi, Attica (θ_g = 200°, θ_w = 0.272°) with E_{limit_S} = 2.5/1.25 V/m.

	Supply Method	6 Frames Operating	5 Frames Operating (Except for No. 6)	5 Frames Operating (Except for No. 5)	5 Frames Operating (Except for No. 4)	5 Frames Operating (Except for No. 3)	5 Frames Operating (Except for No. 2)	5 Frames Operating (Except for No. 1)
	Iel_st [A]	11.458	14.589	14.589	14.589	14.589	14.589	14.589
2.50 V/m	dOx [m]	25.511	27.189	27.889	27.998	27.889	27.189	18.066
	dOx/ [m]	−25.511	−27.889	−27.189	−18.066	−27.189	−27.889	−27.998
	dOy [m]	23.082	25.703	25.703	25.606	24.725	24.725	25.606
	dOy/ [m]	−23.082	−24.725	−24.725	−25.606	−25.703	−25.703	−25.606
1.25 V/m	dOx [m]	46.664	48.601	51.151	51.678	51.181	48.901	42.955
	dOx/ [m]	−46.664	−51.181	−48.901	−42.955	−48.901	−51.181	−51.678
	dOy [m]	46.443	51.416	51.416	50.231	46.052	46.082	50.231
	dOy/ [m]	−46.443	−46.082	−46.082	−50.231	−51.416	−51.416	50.231
	Εoff [V/m]	-	1.824	1.824	1.826	1.824	1.824	1.826
	Εmax [V/m]	16.304	20.390	20.390	20.346	20.390	20.390	20.346
Korakia	Vrel_max [V]	2257.28	2651.65	2607.86	2597.40	2607.86	2651.65	3018.74
	Rel [Ω]	2.052	2.272	2.234	2.226	2.234	2.272	2.587
	VO [V]	859.05	879.23	873.25	872.49	873.25	879.23	1066.21
Stachtoroi	Vrel_max [V]	2350.19	2749.77	2705.98	2695.52	2705.98	2749.77	3116.86
	Rel [Ω]	2.137	2.356	2.319	2.310	2.319	2.356	2.671
	VO [V]	951.951	977.353	971.376	970.613	971.376	977.353	1164.329

. From the study of the results in Table 7 and the comparison with those in Table 6, the following emerges:

• The radius of the critical zone from the centre of the electrode station reaches up to 51.68 m in all directions depending on how uneven the loading of the station is, which is practically the same as the results of the station with frames of 12 electrodes (a reduction of just under 1 mm). That is, there is no change in the boundaries of the radii of the zones where the electric field strength is significantly reduced below 2.5 V/m or below 1.25 V/m. This is due to the circular symmetry of the arrangement of the electrodes both at the frame level and, mainly, overall.
• The deviation of the maximum developing electric field strength of the electrode station compared to a single “birthday cake” frame (20.39 V/m against 18.92 V/m, according to Table 4) is of the order of 7.77%, which is moderate. With this arrangement, without increasing the dimensions of the structure, the field behaviour of the array was improved (for both Korakia and Stachtoroi) by increasing the number of electrode rods by 33%.

• The developing maximum absolute potentials and the corresponding values of the electrode station resistance, with respect to remote earth, are considerably better (3.02 kV, 2.59 Ω) compared to the 12-electrode structure (4.03 kV, 3.45 Ω according to Table 6) as well as to the original structure (22.77 kV, 19.5 Ω according to Table 15 in ref. [4]) and all other modifications for the Korakia area according to Table 10 in ref. [36] and Table 5.
• The developing maximum absolute potentials and the corresponding values of the electrode station resistance, with respect to remote earth, are considerably better (3.12 kV, 2.67 Ω) compared to the 12-electrode structure (4.16 kV, 3.56 Ω according to Table 6) and compared to the original structure (17.30 kV, 14.82 Ω according to Table 15 in ref. [4]) and all other modifications for the Stachtoroi area according to Table 11 in ref. [36] and Table 5.
• The worst-case scenario in terms of absolute potential and electrode station resistance to remote earth is calculated when the first frame is unloaded.

Also, the case of loading five frames of 12 electrodes per frame, with the first frame out of operation, is investigated for radii R_K4 of the circle—in which the regular hexagon is inscribed—from 10 to 25 m with a step of 1.0 m. The relevant results are listed in

Table 8. Simulation results for the calculation of electric field strength using a linear current source (ρ_S = ∞, without dam, ρ_w = 0.25 Ω∙m) and absolute potential using a linear current source (for the near field) and point current source (for the far field) (ρ_S = ∞, ρ_w = 0.25, ρ_d = 100 Ω∙m) for the 12th electrode station configuration (six “birthday cake” frames of a circular arrangement of radius R_f = 1.00 m of 12 electrodes (L = 1.70 m) placed perimetrically around a perceived regular hexagon inscribed in a circle of radii R_K4 varying from 10 to 25 m according to Figure 10) in the area of Korakia, Crete (θ_g = 200°, θ_w = 2.29°) and in the area of Stachtoroi, Attica (θ_g = 200°, θ_w = 0.272°) with E_{limit_S} = 2.5/1.25 V/m for steady state operation of five frames (excepting no. 1) with I_{el_mt_st} = 19.452 A (worst-case scenario).

	2.5 V/m			1.25 V/m					Stachtoroi			Korakia
RK4 [m]	dOx [m]	|dOx/| [m]	dOy [m] = |dOy/|	dOx [m]	|dOx/| [m]	dOy [m] = |dOy/|	Eoff [V/m]	Emax [V/m]	Vrel_max [V]	Rel [Ω]	VO [V]	Vrel_max [V]	Rel [Ω]	VO [V]
10	20.215	26.065	25.228	46.486	50.797	49.621	3.124	25.245	5635	4.829	2437	5505	4.716	2306
11	19.388	26.204	25.272	46.105	50.928	49.704	2.836	25.013	5295	4.537	2248	5164	4.425	2117
12	18.805	26.388	25.279	45.692	51.057	49.791	2.597	24.819	5017	4.299	2088	4486	3.844	1957
13	18.413	26.609	25.247	45.243	51.183	49.880	2.395	24.656	4788	4.102	1950	4657	3.990	1820
14	18.170	26.874	25.215	45.754	51.308	49.971	2.222	24.516	4592	3.935	1950	4461	3.823	1700
15	18.046	27.192	25.251	44.216	51.433	50.062	2.072	24.395	4427	3.793	1727	4296	3.681	1597
16	18.018	27.567	25.383	43.623	51.558	50.151	1.941	24.289	4283	3.670	1635	4152	3.558	1504
17	18.069	28.009	25.609	42.963	51.685	50.238	1.826	24.196	4156	3.561	1553	4026	3.449	1422
18	18.183	28.494	25.920	42.224	51.815	50.319	1.724	24.113	4046	3.467	1479	3915	3.355	1349
19	18.349	29.043	26.307	41.386	51.951	50.392	1.632	24.039	3948	3.383	1413	3817	3.271	1283
20	18.559	29.642	26.757	40.429	52.094	50.456	1.550	23.973	3860	3.307	1354	3729	3.195	1223
21	18.805	30.289	27.262	39.533	52.245	50.508	1.476	23.913	3780	3.239	1299	3650	3.127	1168
22	19.082	30.975	27.814	38.775	52.408	50.544	1.408	23.858	3708	3.177	1250	3577	3.065	1119
23	19.384	31.698	28.405	38.139	52.584	50.562	1.346	23.808	3642	3.120	1204	3511	3.008	1073
24	19.707	32.451	29.029	37.611	52.776	50.559	1.290	23.763	3582	3.069	1162	3451	2.957	1032
25	20.049	33.232	29.682	37.176	52.987	50.534	1.238	23.721	3528	3.023	1124	3397	2.911	992.8

. From the study of the results in Table 8, the following emerge (when the radius R_K4 increases):

• A small decrease in the maximum value of the electric field strength;
• A significant decrease in the electric field strength in the area of the non-operating frame E_off (falling below 1.25 V/m for R_K4 = 25 m);
• A small increase in the radius of the zone (in the form of the distances |d_Ox/|, d_Oy, |d_Oy/|), within which the electric field strength is less than 1.25 V/m or 2.5 V/m;
• A significant decrease in the maximum value of the absolute potential, in the absolute potential at the centre O of the regular hexagon (at the vertices of which the electrode bases are placed) as well as in the electrode station resistance with respect to remote earth.

Additionally, the case of loading five frames of 12 electrodes per frame (with the first frame out of operation) is investigated for a radius R_K4 = 17 m and with a different active length (based on model “C” as described in refs. [4,36], Appendix A). The reason for considering a different length is that—in addition to its decisive role in the results—its reduction from 2.13 m to 1.70 m was based on the fact that only the part exposed to the water participates and not the part located within the inert material (an extremely unfavourable consideration) and furthermore that the water layer through which the electric current flows from each electrode is limited to a height equal to the rod, which is again very unfavourable, as the electrode bases are located at great depth on a sloping bottom, ensuring a water zone greater than the length of the rod. Moreover, the presence of this frame base structure made of inert material, such as unreinforced concrete, around the 12 (or 16) electrodes leads to a radial redistribution of the electric field strength in the outer area of the base practically in a uniform manner over the entire height of the base, with the consequence that the water zone height of 1.70 m is actually quite limited. The relevant results are recorded in

Table 9. Simulation results for the calculation of electric field strength using a linear current source (ρ_S = ∞, without dam, ρ_w = 0.25 Ω∙m) and absolute potential using a linear current source (for the near field) and point current source (for the far field) (ρ_S = ∞, ρ_w = 0.25, ρ_d = 100 Ω∙m) for the 12th electrode station configuration (six “birthday cake” frames of a circular arrangement of radius R_f = 1.00 m of 12 electrodes of varying lengths (L = 1.70 to 15.0 m) placed perimetrically around a perceived regular hexagon inscribed in a circle of radius R_K4 = 17 m according to Figure 10) in the area of Korakia, Crete (θ_g = 200°, θ_w = 2.29°) and in the area of Stachtoroi, Attica (θ_g = 200°, θ_w = 0.272°) with E_{limit_S} = 2.5/1.25 V/m for steady state operation of five frames (excepting no. 1) with I_{el_mt_st} = 19.452 A (worst case scenario).

	2.5 V/m			1.25 V/m					Stachtoroi			Korakia
L [m]	dOx [m]	|dOx/| [m]	dOy [m] = |dOy/|	dOx [m]	|dOx/| [m]	dOy [m] = |dOy/|	Eoff [V/m]	Emax [V/m]	Vrel_max [V]	Rel [Ω]	VO [V]	Vrel_max [V]	Rel [Ω]	VO [V]
1.70	18.069	28.009	25.609	42.963	51.685	50.238	1.826	24.196	4156	3.561	1553	4026	3.449	1422
2.13	15.073	24.335	21.887	31.359	41.736	40.321	1.458	19.331	3328	2.852	1250	3224	2.762	1146
2.50	13.633	22.598	20.184	25.280	36.031	34.363	1.242	16.453	2842	2.435	1072	2753	2.359	982.9
5.00	10.532	19.096	16.789	13.633	22.598	20.184	0.621	8.227	1436	1.230	550.3	1391	1.192	505.5
10.0	9.643	18.149	15.872	10.532	19.096	16.789	0.310	4.113	724.9	0.621	282.3	702.8	0.602	260.2
15.0	9.562	18.067	15.779	9.812	18.332	16.044	0.207	2.742	486.0	0.416	191.0	471.4	0.404	176.3

.

From the study of the results in Table 9, the following results emerge (when increasing the length of the water zone L from 1.70 m to 15 m):

• A significant reduction in the maximum electric field strength value where, for Korakia, it is already lower compared to all modifications, while for Stachtoroi, for a length equal to 2.13 m, it falls lower compared to all configurations while it is slightly higher than the corresponding values of the perimetrical placement of bow frames around a circular central base (11th configuration in this paper), the perimetrical placement of straight frames around a central base (10th configuration in ref. [36]), and the placement of straight frames in a row perpendicular to the dam (second configuration in ref. [36]). For a larger L, the electric field decreases even further.
• A very significant reduction in the electric field strength in the area of the frame that does not operate (falling below 1.25 V/m for a zone length of 2.5 m).
• A reduction of the radius of the zone within which the electric field strength is less than 1.25 V/m or 2.5 V/m (with a greater rate of reduction for 1.25 V/m), achieving a radius of 41.74 m (for a zone length of 2.13 m), which is the respective water zone of all previous modifications, with the result that a smaller zone of influence is achieved for both Korakia (with a radius of the order of 56 m) and for Stachtoroi (with a radius of the order of 44 m).
• A significant reduction in the maximum value of the absolute potential, the absolute potential at the centre O of the regular hexagon (at the vertices of which the electrode bases are placed) and the electrode station resistance with respect to remote earth.
• The results of Table 9 are accurate for large values of water zone length L near the base but not inside the base, where the zone is actually limited to the actual length of the electrode. In addition, inside the base, there is no effect of its wall, leading to the uniformity of the electric field surrounding the base.

In conclusion, for the Korakia area, the positions of the electrode bases are quite far from the initial position of the electrode control station, in the order of 120 m, and are placed further offshore than in any other solution. In terms of field values, it shows the smallest effects on land (compared to all other proposed solutions with a dam) because it is moved away from the coast by another 80 m approximately.

For the Stachtoroi area, the positions of the electrode bases are quite close to the initial position of the dam, increasing the distances of the cables from the initial position of the electrode control station by 20 m. In terms of electric field values, with the excessively strict active electrode length of 1.70 m, it shows a larger radius of influence than the other solutions. However, even with the strict application of an active electrode length of 2.13 m, the radius of influence is limited, in relation to the other solutions, to 41 m from the centre of the O arrangement. The fact that the entire arrangement ultimately extends to depths of 4 to 16 m will lead to an even smaller zone of influence (as shown by the results in Table 9), reaching expected electric field influence radii below 1.25 V/ m, of the order of 20 m. Furthermore, as a solution in terms of absolute potentials, it presents the smallest possible values so far.

From a construction point of view, the configuration of the frame bases is done with unreinforced concrete, obviously in much smaller quantities than in all other cases. The main disadvantage of this solution is the maintenance cost, as it requires a surface vessel, which with the appropriate winches and divers will raise the electrode (or the frame to be repaired) onto the deck of the ship, and then its re-installation will follow. In case of an emergency, a helicopter with a rescue hoist can also be used to lift/install an electrode.

5. Discussion—Comparison of Different Frames and of Different Electrode Station Configurations

For the comparison of the proposed frames,

Table 10. Summary of the simulation of field results for a singular frame.

					Korakia				Stachtoroi
Frame	Figure	Technical Characteristics	Nel_frame [-]	Iel_mt [A]	dsc [m]	Emax [V/m]	Sc [m2]	No. of table	dsc [m]	Emax [V/m]	Sc [m2]	No. of Table
Straight	1	ℓp = 0.5 m, ℓf = 6.0 m	13	17.955	10.901	24.06	394.88	10 in ref. [4]	7.951	17.96	219.69	11 in ref. [4]
Bow	5, 6	ℓp = 0.5 m; RK3 = 17 m, ℓf = 5.969 m (Korakia); RK3 = 14 m, ℓf = 5.954 m (Stachtoroi)	13	17.955	10.808	22.62	394.88	1, 2	7.847	17.08	219.70	3
Basic “Birthday cake”	8	Rf = 1.000 m, ℓp = 0.518 m	12	19.452	9.835	22.73	303.88	4	9.835	22.73	303.88	4
“Birthday cake”	8	Rf = 1.000 m, ℓp = 0.390 m	16	14.589	9.834	18.92	303.80	4	9.834	18.92	303.80	4
“Birthday cake”	9	Rf1 = 0.750 m, Rf2 = 1.100 m, d1 = 0.574 m, d2 = 0.842 m, d3 = 0.498 m	16	14.589	9.834	20.05	303.80	4	9.834	20.05	303.80	4

is formed, which lists the type of frame, its technical characteristics, the number of electrodes per frame (N_{el_frame}), and the loading current of each electrode at steady state under maintenance conditions I_{el_mt}. Also included are its basic dimension—frame length ℓ_f or frame radius R_f and other dimensioning features, the maximum developing electric field intensity E_max, the estimated distance ds_c that ensures a diver can drop down for repairs outside the critical zone and the corresponding area S_c of the critical zone with an electric field above 1.25 V/m, obtained from single-frame simulations.

From the study of the results of Table 10, the following results emerge:

• The smallest value of distance ds_c, which secures the diver from the critical zone, concerns the basic “birthday cake” frame, which presents a reduced value compared to the bow frame by 9.0% and compared to the straight frame by 9.75% for the case of Korakia. In contrast, for the case of Stachtoroi, the distance ds_c of the bow frame presents a reduced value compared to the straight frame by 1.3% and compared to the “birthday cake” frame by 20.2%. This differentiation is due to the combination of the effects of the changes in the arc of the “left” water barrier zone of Figure 1 of ref. [36] (or Figure A1 of Appendix A) and the electric current of the frame. In particular, in the case of Korakia, the corresponding arc for the straight and bow frames is taken as 112° and for the “birthday cake” frame it is taken as 160°, i.e., it increases by 42.8%. In the case of Stachtoroi, the corresponding arc for the straight and bow frames is taken as 150° and for the “birthday cake” frame it is taken as 160°, i.e., the relative increase is considerably smaller by 6.7%. The electric current of the “birthday cake” has an 8.3% higher value compared to the other two frames. Based on Equations (4)–(9) in ref. [36] (or Equations (A4)–(A9) of Appendix A), the increase in the arc leads to a decrease in the electric field strength, in an inversely proportional manner, while the increase in the current leads to an increase in the electric field strength accordingly. This has the consequence that, in the case of Korakia, the large increase in the arc dominates over the increase in current, resulting in the “birthday cake” frame presenting better results, in contrast to the case of Stachtoroi.
• The smallest value of the area of the critical zone S_C, with an electric field strength above 1.25 V/m, concerns the basic “birthday cake” frame, which, due to the symmetry of the arrangement with respect to two axes, presents a reduced value compared to the other two frames by 23% for the case of Korakia. In contrast, for the case of Stachtoroi, the area of the critical zone S_C of the straight frame presents a reduced value compared to the “birthday cake” frame by 27.7%. The relative values between the straight and bow frames are identical. This behaviour is again due to the aforementioned interaction of the arc of the “left” water dam zone of Figure 1 in ref. [36] (or Figure A1 of Appendix A) and the electric current of the frame.
• The lowest value of the maximum electric field strength E_max concerns the bow frame, which presents a reduced value compared to the basic “birthday cake” frame by 0.5% and compared to the straight frame by 6.0% (for the case of Korakia) and by 24.86% and 4.9%, respectively, for the case of Stachtoroi. Here, in addition to the aforementioned interaction for the maximum value, the discretization grid of the study area near the electrodes also plays a part, which in the present simulation has been set at 0.05 m × 0.05 m.
• From the comparison between the “birthday cake” frames, there is a substantial improvement only in terms of the maximum electric field strength E_max, especially with the use of 16 electrodes in a circular arrangement, which however lags behind the bow frame and the straight frame of Stachtoroi. Nonetheless, the basic structure of Figure 13 is preferable for construction reasons because the frame of 16 electrodes in a circular arrangement requires electrodes with a base of synthetic material (untested solution) from Figure 14, while the frame of 16 electrodes in two circular arrangements from Figure 15 has much more limited room for repair with regard to the diver.
• No frame seems to clearly dominate over another except for the fact that the bow frame always gives slightly better results than the straight frame in terms of distance ds_c and maximum electric field strength E_max.
• To sum up, the field behaviour of an individual frame is influenced by the number of electrodes per frame, N_{el_frame}, its basic dimension (frame length ℓ_f or frame radius R_f and other dimensioning features), and the location of the frame relative to the shore assuming that the electrode type and the loading electric current of each frame at a steady state under maintenance conditions are the same. Because the parameters interact, e.g., placing more electrodes reduces the electric current per electrode but increases their density, there is no clear mechanism for controlling the effect on the electric field strength.

Table 11. Summary of the simulation of field results for an electrode station with six frames for the area of Korakia, where “*” denotes the calculation of the corresponding circular disk area (based on Table 10 in ref. [36]).

No.	Configuration	dOx [m]	|dOx/| [m]	dOy [m]	|dOy/| [m]	SC [m2]	Εoff [V/m]	Εmax [V/m]	Vrel_max [V]	Rel [Ω]	No. of Table
1	Straight frames in a row, placed parallel to a 70 m dam	50.29	50.29	67.81	67.81	13,641	3.13	26.09	22,772	19.51	10 & 15 in ref. [4]
2	Straight frames in a row, placed vertical to a 70 m dam	48.70	54.70	65.75	65.75	13,597	2.86	24.98	22,475	19.26	1 & 10 in ref. [36]
3	Straight frames in two overlapping rows, placed parallel to the axis of a 70 m dam, aligned with each other	49.26	60.25	61.10	61.10	13,382	2.98	26.12	26,978	23.12	2 & 10 in ref. [36]
4	Straight frames in two overlapping rows, placed parallel to the axis of a 70m dam, non-overlapping on the vertical axis of the dam	49.01	60.01	62.41	62.41	13,608	3.09	26.13	26,142	22.40	3 & 10 in ref. [36]
5	Straight frames in two successive rows, placed vertical to the axis of a 70 m dam, aligned with each other	46.64	67.64	58.35	58.35	13,336	3.31	26.29	25,705	22.02	4 & 10 in ref. [36]
6	Straight frames in two successive rows, placed vertical to the axis of a 70 m dam, non-overlapping on the vertical axis of the dam	46.74	67.74	58.89	58.89	13,483	3.42	26.25	25,942	22.23	5 & 10 in ref. [36]
7	Straight frames in perimetrical placement to a dam, adapted to a pond outline of 192.5 m	44.60	74.04	55.80	62.92	14,085	1.55	24.78	21,873	18.74	6 & 10 in ref. [36]
8	Straight frames adapted to a T-shaped dam	-	-	-	-	-	-	-	-	-	-
9	Straight frames adapted radially to a central base RK1 = 11 m	58.26	58.26	58.09	58.09	13,537 10,663 *	2.59	26.13	11,096	9.507	8 & 10 in ref. [36]
10	Straight frames adapted perimetrically to a central base RK2 = 16.5 m	58.54	58.54	58.36	58.36	13,666 10,766 *	2.21	24.94	16,831	14.42	9 & 10 in ref. [36]
11	Bow frames adapted perimetrically to a central circular base RK3 = 17.0 m	58.62	58.62	58.42	58.42	13,698 10,795 *	2.11	24.74	16,505	14.14	5
12a	“Birthday cake” frames with Rf = 1.00 m placed on the vertices of a regular hexagon, inscribed in circle of RK4 = 17.0 m – L = 1.70 m	51.69	51.69	50.23	50.23	10,385 8394  *	1.83	24.20	4026	3.45	9
12b	“Birthday cake” frames with Rf = 1.00 m placed on the vertices of a regular hexagon, inscribed in circle of RK4 = 17.0 m – L = 2.13 m	41.74	41.74	40.32	40.32	6732 5473 *	1.46	19.33	3224	2.76	9

and

Table 12. Summary of the simulation of field results for an electrode station with six frames for the area of Stachtoroi, where “*” denotes the calculation of the corresponding circular disk area (based on Table 11 in ref. [36]).

No.	Configuration	dOx [m]	|dOx/| [m]	dOy [m]	|dOy/| [m]	SC [m2]	Εoff [V/m]	Εmax [V/m]	Vrel_max [V]	Rel [Ω]	No. of Table
1	Straight frames in a row, placed parallel to a 50 m dam	36.34	36.34	52.30	52.30	7602	2.95	19.73	17,300	14.82	11 & 15 in ref. [2]
2	Straight frames in a row, placed vertical to a 50 m dam	35.72	41.72	48.92	48.92	7577	2.87	19.08	18,397	15.76	1 & 11 in ref. [36]
3	Straight frames in two overlapping rows, placed parallel to the axis of a 50 m dam, aligned with each other	36.78	44.78	45.70	45.70	7455	3.05	20.03	21,476	18.40	2 & 11 in ref. [36]
4	Straight frames in two overlapping rows, placed parallel to the axis of a 50 m dam, non-overlapping on the vertical axis of the dam	36.95	44.95	46.27	46.27	7579	3.35	20.14	21,581	18.49	3 & 11 in ref. [36]
5	Straight frames in two successive rows, placed vertical to the axis of a 50 m dam, aligned with each other	33.91	51.91	43.35	43.35	7441	3.25	20.16	20,235	17.34	4 & 11 in ref. [36]
6	Straight frames in two successive rows, placed vertical to the axis of a 50 m dam, non-overlapping on the vertical axis of the dam	33.98	51.98	43.74	43.74	7520	3.37	20.12	20,506	17.57	5 & 11 in ref. [36]
7	Straight frames in perimetrical placement to a dam, adapted to a pond outline of 123.0 m	32.55	48.12	44.94	51.61	7789	2.26	19.49	17,842	15.29	6 & 11 in ref. [36]
8	Straight frames adapted to a T-shaped dam	34.07	46.48	43.75	55.85	7990	2.28	19.69	18,312	15.69	7 & 11 in ref. [36]
9	Straight frames adapted radially to a central base RK1 = 8 m	43.57	43.57	43.43	43.43	7569 5964 *	2.48	19.92	7214	6.181	8 & 11 in ref. [36]
10	Straight frames adapted perimetrically to a central base RK2 = 14 m	43.90	43.90	43.73	43.73	7679 6055 *	1.97	18.77	12,069	10.34	9 & 11 in ref. [36]
11	Bow frames adapted perimetrically to a central circular base RK3 = 14.0 m	43.92	43.92	43.75	43.75	7686 6060 *	1.94	18.71	11,636	9.97	5
12a	“Birthday cake” frames with Rf = 1.00 m placed on the vertices of a regular hexagon, inscribed in circle of RK4 = 17.0 m – L = 1.70 m	51.69	51.69	50.23	50.23	10,385 8394 *	1.83	24.20	4156	3.56	9
12b	“Birthday cake” frames with Rf = 1.00 m placed on the vertices of a regular hexagon, inscribed in circle of RK4 = 17.0 m – L = 2.13 m	41.74	41.74	40.32	40.32	6732 5473 *	1.46	19.33	3328	2.85	9

(for the areas of Korakia and Stachtoroi, respectively) list the most unfavourable absolute values according to the method of electrode positioning of the electrode station for the following parameters:

• The width d_Ox of the critical zone of the electrode station (along the Ox semi-axis from the beginning of the axes), the width d_Ox/ (along the Ox^/ semi-axis), the length d_Oy (along the Oy semi-axis), and the length d_Oy/ (along the Oy^/ semi-axis) for an electric field strength limit above 1.25 V/m;
• The indicative equivalent area of the zone of influence S_C (calculated through Equation (40) in ref. [36]);
• The maximum electric field strength E_off (within the area of the frame that is out of operation);
• The maximum electric field strength of the arrangement E_max;
• The maximum absolute potential V_{rel_max};
• The electrode-station-to-remote-earth resistance R_el.

It is pointed out that especially for configurations 9 to 12, the indicative equivalent area of the zone of influence S_C can be estimated as the corresponding area of a circle with a diameter of the largest value of d_Ox, d_Ox/, d_Oy, or d_Oy/ (which are practically identical). The best results, per parameter, in Table 11 and Table 12 are marked in bold.

From the study of the relevant results of Table 11, for the area of Korakia, the additional results that are drawn by the addition of the new configurations are:

• Based on all quantitative criteria, the 12th configuration using a 12-electrode “birthday cake” frames of a circular arrangement of radius R_f = 1.000 m placed at the vertices of a regular hexagon inscribed in a circle of radius R_K4 = 17.0 m, with an electrode length taken at L = 2.13 m, is advantageous over all other configurations.
• If the electrode length is limited to L = 1.70 m (which is excessively strict), then the 12th configuration remains superior in all parameters except for widths d_Ox and d_Ox/, where it is proven inferior to the seventh and first configurations by a few meters but with no significant consequence. For example, in the 12th configuration, there is no dam, and the placement of the bases is done in the sea, as shown in Figure 26.

From the study of the relevant results of Table 11 for the area of Stachtoroi, the additional results that are drawn by the addition of the new configurations are:

• The 12th configuration using 12 electrode “birthday cake” frames in a circular arrangement of radius R_f4 = 1.000 m placed at the vertices of a regular hexagon inscribed in a circle of radius R_K4 = 17.0 m with an electrode length of L = 2.13 m is generally advantageous over all other configurations except for widths d_Ox and d_Ox/. There, it is proven inferior to the seventh and first configurations but with no significant consequence, since, in the 12th configuration, there is no dam and the placement of the bases is done in the sea, as shown in Figure 27. Furthermore, it is disadvantageous compared to the second configuration in ref. [36] (placement of straight frames in a row perpendicular to the dam), the 10th configuration in ref. [36] (perimetric placement of straight frames around a central base), and the 11th configuration (bow frames on a circular central base) in terms of the maximum electric field strength of the arrangement E_max by 0.25, 0.56 and 0.62 V/m, respectively (i.e., there is a small variation). In contrast, regarding other criteria, such as the maximum absolute potential V_{rel_max} and the electrode-station-to-remote-earth resistance R_el, it has an overwhelming advantage, achieving values that are half as high as the next best configuration (in this case, configuration 9).
• If the electrode length is limited to L = 1.70 m (which is quite strict), then the 12th configuration is superior only in the maximum electric field strength E_off (within the area of the frame that is out of operation), maximum absolute potential V_{rel_max} and the electrode-station-to-remote-earth resistance R_el. Furthermore, the fifth configuration is advantageous over the lengths d_Oy and d_Oy/ and the indicative equivalent area of the zone of influence S_C.

Essentially, the placement of electrode frames in a circular arrangement (i.e., “birthday cake”) placed at the vertices of an imaginary regular hexagon leads to the best possible field results in terms of safety zones, maximum absolute potential and electrode-station-to-remote-earth resistance except for the maximum electric field strength of the arrangement, provided that the active length of the rod and the water zone, according to the third consideration, is equal to its nominal length. The maximum electric field strength is expected to be even smaller, especially at Stachtoroi, because the water zone is considerably larger than 2.13 m (based on Section 4.3), and the breaking water is limited to a maximum of 1.5 to 2 m.

In any case, the configuration of an electrode station is influenced by many parameters, and it cannot be said from the outset that one electrode arrangement is overwhelmingly superior to the others. However, two main conclusions emerge:

• The arrangement of straight electrode frames placed behind a dam, as suggested by the technical guidelines CIGRE B4.61 675:2017 [1] and IEC TS 62344:2013 [12], is not the only option.
• A corresponding preliminary study and determination of the most suitable arrangement per electrode station location area (in this paper, based on electric field criteria) is required.

6. Conclusions

This paper presents the electric field analysis of new frame forms and shoreline electrode stations for HVDC interconnections with the use of analytical method for calculating the electric field gradient, ground potential rise and resistance to remote earth of electrode stations in ref. [4] and in ref. [36], where, for the near electric field, a linear current source is used for each electrode, which extends cylindrically over a zone of constant sea/land or dam thickness, while for the far field, the corresponding mathematical background of a point current source is used in the form of a suitable wedge of sea on land. In the case of more electrodes than one, the application of the method is extended through superposition [4,36].

The main contribution in frame configuration (against straight frames) is summarized as follows:

• For the first time, bow frames (Figure 6) are introduced, which are suitable for perimetrical placement around a circular central base in the open sea (Figure 4 and Figure 5).
• For the first time, frames whose electrodes are introduced in one or more circular arrangements (Figure 8 and Figure 9) are presented, which resemble a “birthday cake” and are suitable for underwater placement and with the least possible environmental impact (Figure 10).
• At the same time, the use of this form of electrode frames provides the possibility of direct visual monitoring and maintenance, in contrast to concrete boxes with electrode parts inside or flat electrodes mesh based on fiberglass structure parallel over the seabed [39,40].

From the electric field analysis for the areas of Korakia, Crete, and Stachtoroi, Attica, no one frame seems to clearly dominate over another except for the fact that the bow frame always gives slightly better results than the straight frame in terms of the maximum developing electric field strength E_max and the estimated distance ds_c, which allows a diver to drop down for repairs outside the critical zone, while the corresponding area S_c of the critical zone with an electric field above 1.25 V/m remains the same. In contrast, the innovative design of the “birthday cake” frame is overwhelmingly advantageous in all the aforementioned values in Korakia but not in Stachtoroi due to the interaction of the arc of the “left” water dam zone of Figure 1 in ref. [36] (or Figure A1 of Appendix A) and the electric current of the frame during the calculation process (as explained in §5). Also, from the comparison between the “birthday cake” frames, the basic structure of 12 electrodes in a circular arrangement (see Figure 13) is preferred over 16 electrodes in a circular arrangement (see Figure 14) or in two circular arrangements (see Figure 15) due to the construction simplicity and the larger space for repair by a diver. However, the frames do not operate individually but in electrode station configurations, so it is in this structure that they must be evaluated.

The main contribution to the electrode station configuration is that it introduces the following suggestions:

• Bow frames adapted perimetrically to a central circular base in open sea (11th configuration);
• “Birthday cake” frames (with the electrodes in a circular arrangement) placed at the vertices of a regular polygon inscribed in a circle in the open sea as an underwater structure (12th configuration).

These are completely novel compared with the basic configuration of straight frames in a row and parallel on the longitudinal axis of the protective dam (first configuration) [1,4,14,15,16,17,34,36,38], with the seven classical variants with straight frames and the use of a dam [36], and with two “novel” variants with straight frames placed perimetrically or radially around a central base in the open sea [36].

For each configuration, the distinct design characteristics—including the suspension and lifting mechanisms as well as the repair procedures—were systematically documented together with the construction requirements, such as those involving central base formation or underwater construction works. Further on, the methodology previously employed for the calculation of the electric field gradient, ground potential rise, and resistance to remote earth of electrode stations [4,36] was implemented for the newly proposed installations at the Korakia and Stachtoroi sites regarding the HVDC interconnection between Attica and Crete. Analysis of the field results indicates, firstly, that the field study of a single frame—whether the bow or “birthday cake”—is insufficient to represent the behaviour of an electrode station comprising multiple frames. Instead, all field parameters need be recalculated. Essentially, in terms of electric field results, placing electrode frames in a circular arrangement, i.e., “birthday cake”, at the vertices of a perceived regular polygon (12th configuration) leads to the best possible electric field results (in both areas under study), achieving values that are half of the next best configuration in terms of absolute potential as well as ground resistance of the electrode station and critical zones (expressed through an indicative equivalent area of the zone of influence and the maximum electric field strength within the area of the frame that is out of operation for maintenance reasons). In the Stachtoroi area, it lags behind the widths d_Ox and d_Ox/ of the critical zone of the electrode station and along the semi-axes Ox and Ox^/ from the beginning of the axes compared to the seventh and first configurations, respectively, but it is of no significance because, as in the 12th configuration, there is no dam, and the placement of the bases is done in the sea, as shown in Figure 27. Furthermore, it is inferior to the second, 10th and 11th configurations in terms of the maximum electric field strength of the arrangement E_max with a difference of less than or equal to 0.62 V/m, which occurs inside the electrode base. Also, the 11th configuration presents very satisfactory results, with bow frames around a circular central base having an advantage over the ten previous configurations [4,36] by presenting the lowest value of the maximum electric field strength and the second lowest value of the absolute potential and electrode station resistance to remote earth (after the ninth configuration with straight frames adapted radially to a central polygonal base).

In any case, the configuration of an electrode station is influenced by many parameters, and it cannot be said from the outset that one electrode arrangement is overwhelmingly superior to the other arrangements. It is shown, in a systematic way, that the configuration proposed by the technical guidelines CIGRE B4.61 675:2017 and IEC TS 62344:2013 for straight electrode frames placed behind a dam is not exclusive. Also, in each electrode station placement area, the corresponding preliminary electric field strength study needs to be carried out in order to determine the most suitable arrangement.

Nevertheless, apart from the field results, for the selection of the final form of construction of the electrode station, the method of their construction, the respective cost, as well as the cost of their operation with technical and economic criteria, must be taken into account, which will be addressed in the third part of this series of papers.