Review and Progress towards the Common Broadband Management of High-Voltage Transmission Grids: Model Expansion and Comparative Modal Analysis

The need of bridging the digital gap between underdeveloped/developed areas and promoting smart grid (SG) networks urges the deployment of broadband over power lines (BPL) systems and their further integration. The contribution of this paper is fourfold. First, based on the well-established hybrid model of (Lazaropoulos and Cottis 2009, 2010, Lazaropoulos, 2012) and the generic multidimensional network analysis tool presented in (Lazaropoulos 2012, Sartenaer 2004, Sartenaer and Delogne 2006, 2001) an exact multidimensional chain scattering matrix method, which is suitable for overhead high-voltage/broadband over power lines (HV/BPL) networks, is proposed and is evaluated against other theoretical and experimental proven models. Second, the proposed method investigates the overhead HV/BPL transmission grids (overhead 150 kV single-circuit, 275 kV double-circuit, and 400 kV double-circuit multiconductor structures) with regard to their end-to-end signal attenuation. It is found that the above features depend drastically on the overhead power grid type, the frequency, the MTL configuration, the physical properties of the cables used, the end-to-end distance, and the number, the length, and the terminations of the branches encountered along the end-toend BPL signal propagation. Third, the impact of the multiplicity of the branches at the same junction in overhead HV grids is first examined. Based on the inherent long-branch structure and the quasi-static behavior of single/multiple branches with matched terminations of overhead HV grid, a simple approach suitable for overhead HV/BPL channel estimation is presented. Fourth, identifying the similar characteristics among different overhead HV/BPL configurations, an additional step towards the common overhead HV/BPL analysis is demonstrated; the entire overhead HV/BPL grid may be examined under a common PHY framework regardless of the overhead HV/BPL grid type examined. Finally, apart from the presentation of broadband transmission potential of the entire overhead transmission power grid, a consequence of this paper is that it helps towards: (i) the better broadband monitoring and management of overhead HV transmission power grids in an interactive SG network; and (ii) the intraoperability/interoperability of overhead HV/BPL systems under the aegis of a unified transmission/distribution SG power network.


Introduction
Due to ubiquitous nature of the transmission and distribution power grids, the structure of these grids-that is, low-voltage (LV), medium-voltage (MV), and high-voltage (HV) grids-is the key to developing an IP-based power system, offering a plethora of potential smart grid (SG) applications [1][2][3][4][5].In the meanwhile, the deployment of broadband over power lines (BPL) networks through the entire transmission and distribution grid forms a potentially convenient and inexpensive communications medium for delivering broadband last mile access in remote and/or underdeveloped areas [6][7][8].
Since overhead HV power lines are generally the lowest cost method of transmission for large quantities of electric energy, utilities mainly employ overhead HV power grids for new urban, suburban, and rural installations [9][10][11][12][13][14][15][16].Apart from the power delivery, the availability of a reliable communication network on the HV power grid side is important for the support of the significant changes towards upcoming SG transformations in transmission power networks [9,[17][18][19][20].
Due to the need of broadband communications through transmission power grids, the development of accurate channel models at high frequencies along the overhead HV transmission power lines is imperative.As usually done in BPL transmission, a hybrid model is employed to examine the behavior of BPL transmission channels installed on BPL multiconductor transmission line (MTL) structures [1,2,[21][22][23][24][25][36][37][38][39][40][41].Through a bottom-up approach consisting of an appropriate combination of the similarity transformation and MTL theory, the modes that may be supported by an overhead HV/BPL configuration are determined concerning their propagation constants and their characteristic impedances [1-3, 21-25, 35-47].Based on [3,36,42,43], in this paper, a top-down approach is proposed to determine the end-to-end attenuation of overhead HV/BPL channels based on an exact version of multidimensional chain scattering matrix or T-Matrix method.This novel extended chain scattering matrix or T-Matrix (TM2) method, which is already well verified in the case of overhead and underground LV/BPL and MV/BPL distribution power grids [3], combines the accuracy of the generic multidimensional network analysis tool presented in [3,36,42,43] and the simplicity of hybrid model of [1][2][3][21][22][23][24][25].Based on numerical results concerning various simulated overhead HV/BPL topologies, BPL transmission via the overhead transmission power grid is investigated aiming at clarifying the influence of factors, such as the overhead HV power grid type (150 kV, 275 kV, or 400 kV and single or double circuit), the physical properties of the cables used, the MTL configuration, the end-to-end distance, and the number, the length, the terminations, and the multiplicity of the branches encountered along the endto-end BPL signal propagation.
Through the exhaustive comparative analysis of a great number of overhead transmission power grid numerical results, the common nature of overhead HV/BPL systems is disclosed resulting in a common PHY framework as it concerns the BPL signal transmission through their power lines.Combining these special BPL signal transmission characteristics with the quasi-static spectral behavior of single/multiple branches with matched terminations, a simplified channel modeling approach suitable for overhead HV/BPL networks is proposed.
Following the first steps towards overhead HV/BPL system integration made in [1,2], a consequence of the proposed top-down modeling and common PHY framework is that their combination further enhances the intraoperability/interoperability of overhead HV/BPL systems in a SG environment.
The rest of this paper is organized as follows.In Section 2, the overhead HV configurations adopted in this paper are demonstrated.Section 3 resumes the MTL theory and eigenvalue decomposition (EVD) modal analysis concerning overhead HV/BPL transmission.Section 4 deals with signal transmission via overhead HV power lines by the proposed TM2 method which is applied for the evaluation of the end-to-end EVD modal transfer functions.In Section 5, numerical results are provided, aiming at marking out the validity of TM2 method compared to other well-proven models and how the various features of the overhead transmission power grids influence BPL transmission.On the basis of these numerical results, the common PHY framework of overhead HV/BPL systems is further enriched.Section 5 recapitulates the conclusions of this paper.

Overhead HV Transmission Power Networks
The overhead HV power grid differs considerably from transmission via twisted-pair, coaxial, or fiber-optic cables due to the idiosyncrasy of the network structure and the physical properties of the power cables used [1,2,10,11,21,23,26,29,38,48].
According to [2], the electrical power industry classifies overhead HV power transmission systems into three main categories, depending on: (i) the voltage levels (from 150 kV up to 1000 kV), (ii) the number of MTL circuits per each tower (mainly, either single or double circuit); in the case of single-circuit three-phase overhead HV systems, each tower supports three-phase conductors whereas in the case of double-circuit three-phase overhead HV systems, each tower supports six-phase conductors; (iii) the number of neutral conductors per each tower [1,2,15,16,49,50].

MTL Theory and EVD Modal Analysis
Compared to a two-conductor line supporting one forwardand one backward-traveling wave, an MTL structure with n XV + 1 conductors parallel to the z axis as depicted in Figures 1(a)-1(c) may support n XV pairs of forward-and backward-traveling waves with corresponding propagation constants where [•] XV denotes the overhead HV/BPL grid type considered-for example, 150 kV, 275 kV, and 400 kV in the case of 150 kV single-circuit, 275 kV double-circuit, and 400 kV double-circuit overhead HV/BPL configuration, respectively.These waves may be described by a coupled set of 2n XV first-order partial differential equations relating the line voltages V i (z, t), i = 1, . . ., n XV , to the line currents I i (z, t), i = 1, . . ., n XV , while each pair of forward-and backward-traveling waves is referred to as a mode [1-3, 21, 23, 44, 45].
(1) Common mode of overhead BPL transmission (CM XV ) which propagates via the n XV conductors and returns via the ground.γ XV CM ≡ γ XV 1 constitutes the CMXV propagation constant.The spectral behavior of CM 150 kV , CM 275 kV , and CM 400 kV is thoroughly investigated in [1,2].
As it has already been presented in [1][2][3][21][22][23][24][25], the EVD modal voltages V XV,m T and the EVD modal currents T may be related to the respective line quantities T via the similarity transformations [21,23,36,44,45] where [•] T denotes the transpose of a matrix; T XV V and T XV I are n XV × n XV matrices depending on the overhead power grid type, the frequency, the physical properties of the cables, and the geometry of the MTL configuration [3,9,13,14,21,23,36,44,45,51,52,67,68].It is assumed that is the eigenbase of the MTL configuration.Through the aforementioned equations, the line voltages and currents are expressed as appropriate superpositions of the respective EVD modal quantities.From (1) The TM2 method-presented in Section 3-models the spectral relationship between V XV,m i (z), i = 1, . . ., n XV , and where H XV,m {•} is the n XV × n XV EVD modal transfer function matrix whose elements H XV,m i, j {•}, i, j = 1, . . ., n XV , with i = j are the EVD modal cochannel (CC) transfer functions, while those H XV,m i, j {•}, i, j = 1, . . ., n XV with i / = j are the EVD modal cross-channel (XC) transfer functions and H XV,m i, j denotes the element of matrix H XV,m {•} in row i of column j [1-3, 21-23, 25].
Combining (1) and (4), the n XV × n XV transfer function matrix is determined from Based on (6)

Evaluation of the End-to-End EVD Modal Transfer Functions
So far, several methods with converging results have been used to determine the EVD modal transfer functions of BPL transmission channels [1-3, 21-25, 36, 39-43, 47, 66, 69, 70].In this paper, the TM2 method, which is already well verified in the case of overhead and underground MV/BPL and LV/BPL distribution power grids [3], is applied.TM2 method constitutes a refined exact version of scattering matrix (SM) and TM methods-analyzed in [1][2][3][21][22][23][24][25]exploiting the generic multidimensional network analysis tool-presented in [3,36,42,43].This new version of TM method is tailored especially for overhead HV/BPL networks.More specifically, similarly to TM method, in order to apply the TM2 method, an end-to-end overhead HV/BPL connection is separated into segments-network modules, each of them comprising the successive branches encountered-see Figure 2(a)-.

Transmission via the Network Modules.
In the present analysis, signal transmission through the various network modules is taken into account based on the concatenation of respective chain scattering or T-Matrices.A typical overhead HV/BPL end-to-end connection comprises branch-type network modules, as depicted in Figure 2(b), while A and B are assumed matched [1-3, 11, 21-25, 32, 38, 48, 59].These "branch" modules may be considered as a cascade of two submodules.To determine the multidimensional scattering characteristics of these two submodules, the scattering matrix formalism of the generic multidimensional network analysis tool is adopted [3,36,42,43].In detail, these two submodules are as follows [3].
(i) A "transmission" submodule representing a transmission line of length L k with the 2n XV,k × 2n XV,k scattering matrix where matrix with zero elements and S T,k 11 , S T,k 12 , S T,k 21 , S T,k 22 are the matrix elements of the S T,k as evaluated from (7). γ XV,k , U XV,k , and n XV,k are the set with elements the propagation constant of the modes, the eigenbase, and the number of conductors, respectively.In the cases examined in this paper and as it has previously been applied in [3], since one type of overhead HV/BPL configuration is used across the overhead HV/BPL end-to-end connection, it is assumed that U XV ≡ U XV,k and n XV = n XV,k are the reference eigenbase and the reference number of conductors, respectively.Receiving end (ii) A "shunt" submodule representing the cascade of [3]: (a) an n XV,bk × n XV,bk reflection matrix K bk (Z bk ) associated with the branch terminations; (b) an 2n XV,bk × 2n XV,bk scattering matrix S T,bk (γ XV,bk , L bk , U XV )-similar expression to (7)representing the branch transmission line of length L bk ; (c) an n XV,bk × n XV,k connection elements C k defining the interconnections between the transmission and branch conductors; with 2n XV,k × 2n XV,k scattering matrix where Z bk is the n XV,bk × n XV,bk branch termination matrix, S S,k 11 , S S,k 12 , S S,k 21 , S S,k 22 are the matrix elements of the S S,k matrix as determined from ( 8), and γ XV,bk and n XV,bk are the set with elements the propagation constant of the modes and the number of conductors, respectively, related to this "shunt" submodule overhead MTL configuration [3,36,42,43].When the number of conductors varies from "transmission" to "shunt" submodules, through the eigenbase change procedure-presented analytically in [3,36,42,43]-and by taking into consideration the reference eigenbase U XV and reference number of conductors n XV , scattering matrices K bk , S T,bk , and C k may be grouped together through their cascading.Hence, the "shunt" scattering matrix of ( 8) may be evaluated.
Based on [3,71,72] and by using ( 7) and ( 8), the 2n XV × 2n XV "transmission" submodule chain scattering matrix T T,k  and the 2n XV × 2n XV "shunt" submodule chain scattering matrix T S,k are determined by [3] where T T,k 11 , T T,k 12 , T T,k 21 , T T,k 22 are the n XV ×n XV matrix elements of the T T,k as determined by (9) and T S,k 11 , T S,k 12 , T S,k 21 , T S,k 22 are the n XV × n XV matrix elements of the T S,k as evaluated from (10).Taking into account ( 9) and ( 10), the 2n XV ×2n XV chain scattering matrix of a "branch" module, T k , is determined from [3,40,41,[71][72][73]] The last module of the overhead HV/BPL connection is the "load" module.Based on (9), the 2n XV × 2n XV chain scattering matrix of the "load" module is given by Through the multiplication rule of (11), the 2n XV ×2n XV overall end-to-end chain scattering matrix is determined from where Combining ( 4) and ( 14), the n XV × n XV EVD modal transfer function matrix is given by the S overall 21 element of the S overall matrix; that is [3], Already proven in distribution BPL networks [3], the proposed TM2 method for overhead HV/BPL networks is extremely powerful since it is able to calculate EVD modal transfer functions associated with very elaborate networks including various types of overhead HV/BPL configurations, any type of interconnection at the branches, and any type of branch termination without assuming specific transmission assumptions as in SM and TM methods.Moreover, in contrast to SM and TM methods, the problem of mode mixture now can be fully investigated through the definition of the EVD modal XC transfer functions-as given in (15)-.

Numerical Results and Discussion
The simulations of various overhead HV/BPL transmission channels aim at investigating: (a) the validity and performance of the proposed TM2 method against other already theoretically and experimentally validated methods; (b) the broadband transmission characteristics of overhead HV/BPL channels and how these are affected by the overhead grid features, such as the type/topology of the overhead power grid and the multiplicity of branches; (c) the common BPL/PHY handling potential between different types of overhead transmission power grid.
For the numerical computations, the 150 kV singlecircuit, the 275 kV double-circuit, and the 400 kV doublecircuit overhead HV transmission line configurations, depicted in Figures 1(a), 1(b), and 1(c), respectively, have been considered.As previously mentioned, the modes supported by the overhead HV/BPL cable configurations may be examined separately.Thus, the following analysis is concentrated only on the EVD modal CC transfer functions.
The following discussion will focus on the transmission characteristics related to the CM XV and to the DM XV s of the overhead BPL systems, as well.Since, as demonstrated in [1,2], the DM XV s of the same overhead power grid type exhibit an almost identical spectral behavior, the transmission characteristics of only one DM XV of each overhead power grid type, say that of DM XV 4 , will be examined, hereafter.The simple overhead topology of Figure 2(c), having N branches, has been considered.In order to validate the accuracy of TM2 method and simplify the following analysis without affecting its generality, the branching cables are assumed identical to the transmission cables and the interconnections between the transmission and branch conductors are fully activated.With reference to Figure 2(c), the transmitting and the receiving ends are assumed matched to the characteristic impedance of the mode considered, whereas the branch terminations Z bk , k = 1, 2, . . ., N, are assumed open circuit-that is, K bk = I n XV,bk , k = 1, 2, . . ., N, where I n XV,bk is the n XV,bk × n XV,bk identity matrix-[1-4, 14, 21-24, 38, 51-53, 67].
In accordance with [2], thousands of km of overhead HV lines are installed in more than 120 countries.These lines stretch from approximately 25 km to 190 km from the generation points before reaching any population centers.Shorter branches in the range of 10 km to 50 km are used in order to connect overhead HV transmission lines either between them or with HV/MV substations [1-4, 10, 14-16, 37, 51-53, 67, 74-77].
To compare the proposed TM2 method with other established methods [1,2,24,25,32,47,77,78] and simultaneously examine the end-to-end channel attenuation behavior for different overhead HV/BPL types and topologies, in Figures 3(a), 3(d), 3(g), and 3(j), the endto-end channel attenuation from A to B is plotted with respect to frequency for the propagation of CM 150 kV for the aforementioned four indicative topologies, respectively, applying the proposed TM2 method, the TM method [1,2,24,25], and the Multipath-Echo Based (MEB) method [32,69,77,78].In Figures 3(b), 3(e), 3(h), and 3(k), similar plots are drawn for the propagation of CM 275 kV and in Figures 3(c), 3(f), 3(i), and 3(l) for CM 400 kV .In Figures 4(a)-4(l), similar curves are given for the propagation of DM XV 4 s.From Figures 3(a)-3(l) and 4(a)-4(l) as it concerns the validity of TM2 method, it is observed that the three methods coincide in all the cases examined yielding excellent results regardless of the overhead power grid type, topology, and mode examined.Moreover, the TM2 method describes the spectral notches accurately either in depth or in spectral extent in any multipath environment.Thus, the TM2 method offers an accurate representation of the attenuation discontinuities caused by branching.It is also easily adapted to various network topologies resulting from various kinds of branches or HV/MV transformers.Due to the unique combination of generality and modular simplicity, the TM2 method will be adopted in the following simulations concerning the behavior of overhead HV/BPL transmission channels.
As for the end-to-end channel attenuation of overhead HV/BPL channels, as it has already been investigated in [1-3, 11, 21-25, 32, 79-81], the spectral behavior of end-to-end channel attenuation depends drastically on the frequency, the mode considered, the physical properties of the cables used, the "LOS" distance, and the number and the length of the branches encountered along the end-to-end transmission path.Furthermore, as it has already been proposed in [1, 2, 22-25, 35, 82] for other BPL channels, overhead HV/BPL channels of transmission power grids may also be classified into three classes depending on the behavior of their spectral notches, namely, "LOS" channels, good channels, and bad channels."LOS" case, rural case, and urban case A will represent "LOS", good, and bad channels, respectively, hereafter.
To demonstrate the effect of the branch length on the channel attenuation, in Figures 5(a Observing Figures 3(a)-3(l), 4(a)-4(l), and 5(a)-5(o), it is evident that, due to reflections and multipath propagation caused by branches, spectral notches are observed in the channel attenuation, which are superimposed on the exponential "LOS" attenuation of each mode.However, due to the large branches of overhead HV/BPL topologies examined, the depth and the extent of the spectral notches are reduced in comparison with corresponding overhead LV/BPL and MV/BPL topologies [1][2][3][21][22][23][24][25].
In addition, from Figures 5(d)-5(f) and 5(j)-5(l) the absolute difference of end-to-end channel attenuation between CM XV and DM XV 4 of the same overhead HV/BPL grid type reveals the significant differences among the modes supported.However, due to the common bus-bar system topologies, the channel attenuation curves between CM XV s and DM XV      relative extent of spectral notches, confirming, thus, the strong correlation between the individual modal channels.In general, DM XV 4 s present higher channel attenuations than CM XV s regardless of the overhead power grid type and topology.Anyway, as it is usually done to reduce the extent of results and simplify the analysis, an indicative picture of the transmission characteristics of the modes can be obtained studying the transmission characteristics of only one mode of each power grid type, say DM 150 kV 4 , DM 275 kV 4 , and DM 400 kV 4 , hereafter, without affecting the generality of the analysis [1][2][3][21][22][23][24][25].
As referred to in [1][2][3][21][22][23][24][25], first, apart from causing spectral notches, the various branches also cause additional stepwise attenuation at each branch encountered along the end-to-end transmission path.Second, as the branch length increases, the stepwise attenuation at each branch tends to its corresponding matched branch termination one.This twofold effect of the branch length on the attenuation discontinuity at each branch is examined in Figures 6(a) and 6(g), where the channel attenuation is plotted versus the distance from the transmitting end-see Figure 2(c), point A-for the propagation of DM 150 kV 4 for good channel class case, Topology 1, Topology 3-same as good channel class case but with matched branch terminations, that is, K bk = 0 n XV,bk , k = 1, 2, . . ., N, where 0 n XV,bk is an n XV,bk × n XV,bk matrix with zero elements, bad channel class case, Topology 2, Topology 4-same as bad channel class case but with matched branch terminations, and the "LOS" transmission case at f = 25 MHz and f = 75 MHz, respectively.In Figures 6(b) and 6(h), similar plots are given  ; respectively, is also drawn with respect to the distance from the transmitting end for the same topologies at f = 25 MHz.In Figures 6(j)-6(l), similar plots are given at f = 75 MHz.
From Figures 6(a)-6(l), it is demonstrated that the attenuation discontinuity at each branch depends primarily on the frequency rather than overhead transmission power grid type and branch length.Indeed, from Figures 6(a)-6(c) and 6(g)-6(i), even though good channel class case, Topology 1, and Topology 3 present significant topological differences, the strong role of the branch length defines their almost identical spectral behavior.Same results occur among bad channel class case, Topology 2, and Topology 4.However, the influence of branch length is an intertopological factor; from Figures 6(d)-6(f) and 6(j)-6(l), it is evident that the channel attenuation differences are lower than 3 dB in the majority of the cases for given overhead power grid topology regardless of the overhead power grid type.
The above results are attributed to the inherent spectral features of overhead HV transmission; as the branch lengths increase, which are already extremely longer in comparison with the respective branch lengths of overhead LV/BPL and MV/BPL grids, the spectral behavior of overhead HV/BPL networks tends to converge to that of overhead ones with matched branch terminations-see Figures 5(a)-5(c), 5(g)-5(i), 6(a)-6(c), and 6(g)-6(i)-.In the case of matched branch terminations, relating ( 7)-( 11) with (8) from [24], T k ( f ) mainly depends on E T,k ( f ) rather than K bk which exhibits static behavior in respect with frequency, say The disclosure of this quasi-static spectral behavior of overhead HV/BPL systems is significant since the main problem of overhead transmission and distribution power grids is their intense time-and frequency-selective channel attenuation variation due to various factors, such as sudden new loads, blackouts, synchronization problems, renewable energy source production, power exchange issues, connection/disconnection of large industrial/sensitive electrical loads, and other cascading failures [1][2][3][21][22][23][24][25][26][27][28][29][30][31].This rule of thumb allows better broadband monitoring and management of overhead HV transmission power grids in an interactive SG network bypassing an elaborate broadband analysis of the overhead transmission network.
Except for the single braches at given junction, similar to overhead LV/BPL and MV/BPL networks, a typical urban overhead HV topology can support 1-5 HV/MV  transformers.These overhead topologies are mainly of radial configuration either with a single branch or with multiple branches at the same junction [3,9,10,15,16,23,28,43,46,47,78].
To demonstrate the effect of multiple branches at given junction on the channel attenuation, in Figure 7  , respectively, is also drawn with respect to frequency for the same topologies.Note that the cases examined correspond to rather special and aggravated transmission cases.However, useful remarks may be pointed out regarding multiple-branch junctions and corresponding matched branch termination networks.
From Figures 7(a)-7(c), it is observed that the multiple branches at each junction cause additional stepwise attenuation deteriorating the existing stepwise attenuation due to single branches.However, the effect of the multiple branches may be cumulatively explained studying the attenuation discontinuity at the each junction.
More analytically, in Figures 8(a) and 8(g), the channel attenuation is plotted versus the distance from the transmitting end-see Figure 2 , respectively, is also drawn with respect to the distance from the transmitting end for the same topologies at f = 25 MHz.In Figures 8(j)-8(l), similar plots are given at f = 75 MHz.
Figures 7(a)-7(c), 8(a)-8(c), and 8(g)-8(i) clearly show the destructive effect of branch presence; the attenuation exhibited by overhead HV/BPL channels when single/multiple branches are added across the BPL signal transmission is dramatically worse than the attenuation of overhead "LOS" ones.Multiple branches critically aggravate the channel performance.Therefore, during the implementation of overhead HV/BPL networks in urban and suburban environments, denser overhead HV/BPL networks are preferable.
Moreover, the quasi-static spectral behavior of overhead HV/BPL systems is also validated in overhead transmission power grids with multiple branches per junction.Observing the above figures, even though Topology 5 and Topology 6 present significant differences regarding their branch terminations, their channel attenuation and attenuation discontinuity are very close in all the cases examined.Same results occur between Topology 7 and Topology 8 permitting to generalize the quasi-static spectral behavior consideration in every overhead HV/BPL grid considered.

Comparative Modal Analysis and Discussion.
Concluding the above comparative modal analysis concerning the behavior of overhead HV/BPL grid and its supported overhead  -may be examined giving results with excellent accuracy for the entire overhead HV/BPL transmission grid.The remarkable convergence of this approximation is maintained also to multibranch junction networks as illustrated in Figures 7(d , for all overhead HV/BPL transmission grid defines the final step of the proposed common handling PHY approach.
Synopsizing the above analysis, several useful conclusions and discussion issues are raised, namely, the following.
(2) The common PHY framework gives a representative image of the overall end-to-end channel attenuation and permits the study of the spectral behavior of overhead HV/BPL networks by examining only one mode supported by the various overhead MTL configurations that constitute the overhead HV transmission power grid.This approach offers (i) a useful tool with low complexity towards BPL/SG systems coexistence and unified network design and (ii) necessary rules of thumb in order to bypass the complicated EVD modal analyses.
(3) Based on the well-established model of [1][2][3][21][22][23][24][25] and the generic multidimensional network analysis tool presented in [3,36,42,43], an extremely powerful multidimensional chain scattering matrix method-TM2 method-is presented.Already proven in distribution BPL networks [3], TM2 method is now able to calculate EVD modal transfer functions associated with very elaborate networks including various types of overhead HV/BPL configurations, any type of interconnection at the branches, and any type of branch termination without imposing special simplifying transmission assumptions.In addition, TM2 method provides significant guidance regarding mode mixture issues via the definition of EVD modal XC transfer functions which describe the crosstalk couplings among the modes supported.Finally, TM2 method is well validated against other theoretically and experimentally proven models, such as MEB and TM methods.
(4) As it has already been proposed in [4-8, 69, 83, 84] for other LV/BPL, MV/BPL, and HV/BPL channels, overhead HV/BPL channels of transmission power grids may be classified into three classes depending on the behavior of their spectral notches: "LOS" channels, good channels, and bad channels.
(5) As the branches become longer, the spectral behavior of the overhead BPL networks tends to converge to the spectral behavior of equivalent circuits which consist of matched branch terminations.This remark is also validated in the multi-branch junction overhead HV/BPL networks.This rule of thumb may help towards better broadband monitoring and management of HV transmission grids in the upcoming SG networks.

Conclusions
This paper has focused on a new exact multidimensional chain scattering matrix method (TM2 method) appropriate for overhead HV/BPL networks and the unified handling potential of overhead HV/BPL configurations through the common PHY framework extension.The applied TM2 method combined with the extended common PHY framework offers a valuable tool towards the integrated BPL transmission network design.Also reported in other LV/BPL and MV/BPL networks, the TM2 method has revealed that the broadband transmission capability of overhead HV/BPL networks depends on the frequency, the physical properties of the overhead MTL configuration used, the "LOS" distance, and the number, the length, the terminations, and the multiplicity of the branches along the end-to-end BPL signal propagation.Overhead HV/BPL channels have been classified into three classes: "LOS" channels, good channels, and bad channels.Managing the common PHY framework and exploiting the inherent long-branch structure of overhead HV grid, a simple quasi-static approach suitable for the channel modeling of overhead HV/BPL transmission grid has been proposed.
Finally, the results validate the low-loss nature of overhead HV/BPL grid over a 25 km repeater span well beyond 100 MHz regardless of the overhead HV power grid type and topology, hence, rendering it as a promising broadband SG partner.
(b).Two sets of three phase conductors-conductors 1, 2, 3 and conductors 4, 5, 6-are spaced by Δ 275 kV p1 .The three phase conductors of each set are suspended one above the other spaced by Δ 275 kV p2 and located at heights h 275 kV p above ground for the lowest conductor.Apart from the six phase conductors of two sets, the upper conductor-conductor 7-is the neutral conductor with radii r 275 kV n , spaced by Δ 275 kV nfrom the highest phase conductor-conductor 3 or 6-, and, consequently, hang at heights h 275 kV n above ground (for more details see[2]).This double-circuit sevenconductor (n 275 kV = 7) overhead HV distribution line configuration is considered in the present work consisting of ACSR conductors [1, 2, 14-16, 51-57].
n XV ×n XV matrix elements of the T overall as evaluated from(13).Easily, the 2n XV ×2n XV overall end-to-end scattering matrix is obtained from[3,71,72] n XV × n XV elements of the S overall matrix as defined from(14).

4 , 4 and
) and 5(g), the endto-end channel attenuation from A to B is plotted versus frequency for good channel class case, Topology 1-same as good channel class case but with three times longer branches (L 1 = 3.75 km, L 2 = 21.25 km, L b1 = 63.3 km), bad channel class case, Topology 2-same as bad channel class case but with three times longer branches (L 1 = 1.15 km, L 2 = 12.125 km, L 3 = 8.425 km, L 4 = 3.3 km, L b1 = 82.8km, L b2 = 51.6 km, L b3 = 99.3 km), and "LOS" transmission case for the propagation of CM 150 kV and DM 150 kV4, respectively.In Figures5(b), 5(c), 5(h), and 5(i), similar plots are drawn for the propagation of CM 275 kV , CM 400 kV , DM 275 respectively.Interesting results concerning the channel attenuation behavior of modes supported by the different overhead HV/BPL configurations may occur via their comparative analysis.Therefore, in Figures5(d)-5(f), 5(j)-5(o), the absolute difference of the end-to-end channel attenuation from A to B is drawn versus frequency for the same topologies between: (i) CM 150 kV and CM 275 kV ; (ii) CM 150 kV and CM 400 kV ; (iii) CM 275 kV and CM 400 kV ; (iv) DM150 kV

Figure 3 :
Figure 3: Application of the coinciding proposed TM2 method (red solid lined square), TM method (blue dashed line X), and MEB method (green dash-dotted triangle) to determine the end-to-end channel attenuation for the propagation of CM XV s versus frequency for different overhead HV/BPL types and topologies (the subchannel frequency spacing is equal to 1 MHz).(a) Urban case A/150 kV single circuit.(b) Urban case A/275 kV double circuit.(c) Urban case A/400 kV double circuit.(d) Suburban case/150 kV single circuit.(e) Suburban case/275 kV double circuit.(f) Suburban case/400 kV double circuit.(g) Rural case/150 kV single circuit.(h) Rural case/275 kV double circuit.(i) Rural case/400 kV double circuit.(j) "LOS" case/150 kV single circuit.(k) "LOS" case/275 kV double circuit.(l) "LOS" case/400 kV double circuit.

Figure 4 :
Figure 4: Same as in Figure 3, but for the propagation of DM XV 4 s.

Figure 6 : 4 . 4 and
Figure 6: Channel attenuation versus the distance from the transmitting end-see Figure 2(c), point A-for good channel class case (yellow solid lined triangle), Topology 1 (green dashed line), Topology 3 (cyan solid lined diamond), bad channel case (red solid lined circle), Topology 2 (blue dashed lined triangle), Topology 4 (magenta dashed lined X), and "LOS" transmission case (black dash-dotted line) (the distance span is equal to 100 m).(a) DM 150 kV

4 and DM 400 kV 4 ,
(a), the endto-end channel attenuation from A to B for the propagation of DM 150 kV 4 is plotted versus frequency for good channel class case, Topology 5-same as good channel class case but with two times more, same branches at each junction (L 1 = 3.75 km, L 2 = 0 km, L 3 = 21.25 km, L b1 = L b2 = 21.1 km), Topology 6-same as Topology 5 but with matched branch terminations, bad channel class case, Topology 7same as bad channel class case but with two times more, same branches at each junction (L 1 = 1.15 km, L 2 = 0 km, L 3 = 12.125 km, L 4 = 0 km, L 5 = 8.425 km, L 6 = 0 km, L 7 = 3.3 km, L b1 = L b2 = 27.6 km, L b3 = L b4 = 17.2 km, L b5 = L b6 = 33.1 km), Topology 8-same as Topology 7 but with matched branch terminations, and "LOS" transmission case.In Figures7(b) and 7(c), similar plots are given for the propagation of DM 275 kV respectively.In order to examine the differences among DM XV 4 s for these overhead HV topologies, in Figures7(d), 7(e), and 7(f), the absolute difference of the channel attenuation between,

4 .
(c), point A-for the propagation of DM 150 kV 4 for good channel class case, Topology 5, Topology 6, bad channel class case, Topology 7, Topology 8, and the "LOS" transmission case at f = 25 MHz and f = 75 MHz, respectively.In Figures 8(b) and 8(h), similar plots are given for the propagation of DM 275 kV4and in Figures6(c) and 6(i) for DM 400 kV Interesting convergence results will be reported to the comparative modal analysis subsection that follows by examining the behavior of DM XV 4 s for different overhead HV topologies; in Figures8(d), 8(e), and 8(f), the absolute difference of the channel attenuation between (i) DM150

Figure 8 : 4 ,
Figure 8: Channel attenuation versus the distance from the transmitting end-see Figure 2(c), point A-for good channel class case (yellow solid lined triangle), Topology 5 (green dashed lined reversed triangle), Topology 6 (cyan dashed lined X), bad channel class case (red solid lined circle), Topology 7 (blue dash-dotted line), Topology 8 (magenta dashed lined square), and "LOS" transmission case (black dashed line) (the distancespan is equal to 100 m).(a) DM 150 kV )-7(f), 8(d)-8(f), and 8(j)-8(l).(iii)The common PHY framework of overhead HV/BPL configurations extends the efforts of[1,2,25] towards overhead transmission power system integration.As demonstrated in Figures 5(j)-5(l), only the transmission characteristics of one of either CM XV or DM XV 4 may be assumed offering common conclusions for all overhead HV/BPL configurations examined with satisfactory accuracy.This consideration of only one mode, say DM 150 kV 4 , the n XV × n XV transfer function matrix H XV {•} of each overhead HV/BPL transmission networks is determined [1-3, 21-25, 39, 66].