Best L 1 Piecewise Monotonic Data Approximation in Overhead and Underground Medium-Voltage and Low-Voltage Broadband over Power Lines Networks : Theoretical and Practical Transfer Function Determination

This paper investigates the efficiency and accuracy of the best L 1 piecewise monotonic data approximation (best L1PMA) in order either to approximate the transfer functions of distribution BPL networks or to reveal the aforementioned transfer functions when various faults occur during their determination.The contribution of this paper is quadruple. First, based on the inherent piecewise monotonicity of distribution BPL transfer functions, a piecewise monotonic data approximation is first applied in BPL networks; best L1PMA is outlined and applied during the determination of distribution BPL transfer functions. Second, suitable performance metrics such as the percent error sum (PES) and fault PES are reported and applied so as to assess the efficiency and accuracy of the best L1PMA during the determination of distribution BPL transfer functions. Third, the factors of distribution BPL networks that influence the performance of best L1PMA are identified. Fourth, the accuracy of the best L1PMA is assessed with respect to its inherent properties, namely, the assumed number of monotonic sections and the nature of faults, that is, faults that follow either continuous uniform distribution (CUD) or normal distribution (ND), during the determination of distribution BPL transfer functions. Finally, best L1PMAmay operate as the necessary intermediate antifault method for the theoretical and practical transfer function determination of distribution BPL networks.


Introduction
Electrical power industry at the beginning of the 21st century is facing a variety of technological and economic challenges.The rising cost of energy, the mass electrification of everyday life, the integration of distributed energy sources, microgrids, demand side management, and climate change are the major drivers that will determine the speed at which the modernization of power distribution grid-that is, either overhead or underground, either MV or LV power grids-will occur [1,2].Taking into consideration the inevitability of this oncoming massive change, SG technologies are fervently embraced by the electrical power industry [3].
Towards that direction, the deployment of BPL networks across the entire distribution grid can help towards the evolution of the vintage power system to an advanced IP-based power network supported by a plethora of SG applications [4][5][6].Actually, distribution BPL networks may provide the required broadband communications framework for different SG services and applications such as better power grid management, home and wide-area networking, advanced metering infrastructure, and more accurate demand response programs [7][8][9][10][11][12].However, distribution BPL networks considerably differ in their topological and circuital characteristics from other wireline communication networks [13,14].Distribution BPL networks suffer from attenuation, multipath, and noise, each significantly affecting their network design [4,[15][16][17].
Apparently, the introduction of appropriate channel models to determine transfer functions of distribution BPL networks at high frequencies remains a challenging issue since the distribution power grid was not originally designed to deliver broadband services and applications.The wellknown hybrid method, which is usually adopted to describe the spectral behavior of different BPL networks, is also employed in this paper [4,[13][14][15][16][17][18][19][20][21].On the basis of its accurate results, a variety of factors affecting transfer functions of distribution BPL networks are identified, namely, the type of distribution power grid, that is, either overhead or underground, either MV or LV, the topology of the distribution BPL network, the MTL configuration, and the applied coupling scheme, that is, how the BPL signal is injected into power lines.However, despite the fact that a plethora of experimental results and theoretical analyses validate the theoretical accuracy of the hybrid method [22][23][24][25][26], a number of practical reasons and "real-life" conditions may create significant differences between experimental measurements and theoretical results during the transfer function determination of distribution BPL networks.On the basis of six fault categories, which are analytically reported in the following analysis, their impact on the transfer functions of distribution BPL networks is evaluated and analyzed.
In order either to approximate the transfer functions of distribution BPL networks or to mitigate the aforementioned faults that occurred, the best L1PMA is first applied in BPL networks.Among the myriad of data approximation methods that has been proposed in the literature, the application of the proposed best L1PMA, which is theoretically presented and experimentally verified in [27][28][29][30][31][32], is adopted in this paper due to its remarkable efficiency to cope with problems that are derived from univariate signal restoration.In fact, best L1PMA avoids the assumption that transfer functions depend on certain critical parameters and only focuses on the smallest change to the transfer function data such that the first differences of the approximated transfer function data are allowed to change their sign only a prescribed number of times [28,29].In accordance with [33], best L1PMA receives as inputs the measured transfer function and the number of monotonic sections that are presented across the transfer function in the examined frequency band while it gives as output the approximated transfer function.Exploiting the inherent piecewise monotonicity of distribution BPL transfer functions, best L1PMA can comfortably accommodate the undulation of the transfer function data due to the attenuation, multipath, noise, and other faults.In addition, a fact worth mentioning is that the Fortran software package, which had been developed in order to implement the best L1PMA, has extensively been verified in various scientific fields [30,33] and is freely available online in [33].
In order to evaluate the approximation accuracy of best L1PMA during the theoretical determination of transfer functions, the performance metric of PES is proposed in this paper.PES calculates the total sum of the relative differences between the approximated transfer function of best L1PMA and the theoretical transfer function in the examined frequency range.Compared against the fault PES, which computes the total sum of the relative differences between the measured transfer function and the theoretical transfer function in the examined frequency range, the impact of several factors on the accuracy of best L1PMA, such as the distribution power grid type, the distribution BPL network topology, the applied coupling scheme, the number of monotonic sections, and the distribution followed by the faults that occurred, is recognized and assessed.
The rest of the paper is structured as follows.In Section 2, a brief presentation of the overhead and underground MV/BPL and LV/BPL networks is shown.Special attention is given to the used MTL configurations and distribution BPL topologies as well as the fundamental knowledge regarding BPL signal propagation and transmission across distribution BPL networks.In Section 3, a presentation of the best L1PMA is given.Implementation details, mathematical analysis, and PES concerning best L1PMA are analytically reported.Section 4 discusses simulations of various distribution BPL networks intending to mark out the efficiency of best L1PMA either to approximate the transfer functions of distribution BPL networks or to mitigate the faults that occurred due to measurements and noise.Section 5 concludes this paper.1(a) and 1(b), respectively.Overhead MV distribution line consists of three parallel noninsulated phase conductors, which are spaced by Δ MV and hang at ℎ MV above ground, whereas overhead LV distribution lines comprise four parallel noninsulated conductors, which are spaced by Δ LV and located at ℎ LV above ground for the lowest conductor.As regards the overhead MV distribution line configuration, it consists of ACSR three-phase conductors [4,13,14].In the case of the overhead LV distribution line, it consists of ASTER three-phase conductors while the upper conductor is the neutral conductor [4].Exact values concerning conductor properties and configuration geometries are reported in [1].

Presentation of
The ground is considered as the reference conductor with conductivity   and relative permittivity   that are assumed equal to 5 mS/m and 13, respectively.The former ground properties define a realistic scenario [4, 13-15, 34, 35].The impact of imperfect ground on broadband signal propagation across power lines was studied in [13,[34][35][36].

The Underground MV and LV Power
Grids.The underground MV and LV lines, which are examined in this paper, are presented in Figures 1(c) and 1(d), respectively.The underground MV and LV lines are the three-phase sector-type PILC distribution-class cable and the threephase four-conductor core-type YJV/XLPE underground LV distribution cable, respectively.Both underground MV and LV distribution lines are considered to be buried 1 m inside the ground with the same ground properties as described in Section 2.1.More analytically, when the underground MV distribution line is concerned, its cable arrangement consists of the three-phase three-sector-type conductors, one shield conductor, and one armor conductor.In the case of underground LV distribution line, the cable arrangement comprises the three-phase three-core-type conductors, one core-type neutral conductor, and one shield conductor.In both underground distribution lines, the shield and the armor are grounded at both ends [4,15,37].Thus, the shield acts as a ground return path and is considered as the reference conductor [4,19,38].Exact values concerning conductor properties and configuration geometries are reported in [1,4,19,[38][39][40][41].Broadband transmission via underground distribution power lines has been detailed in [15,19,39,40,42] where the analytical formulation considered in this paper has been analyzed.
Here, it should be noted that the type of the underground MV cable, which is examined in this paper, cannot be characterized as the typical case.In fact, most modern underground MV cables are usually single-core laid in horizontal or trefoil arrangements.However, the PILC cable is one of the oldest types of power cables being used for almost one hundred years.Shielded PILC cables are still dominant for urban applications in many countries (e.g., most larger North American urban centers, Netherlands, and Malaysia) and are extensively used for power distribution, not only at the MV but also at the LV level [15,37,38].Therefore, the paper focuses on the propagation via PILC cables-rather than via XLPE cables which present lower attenuation-because PILC cables behave closer to the LOS transmission behavior of the majority of existing underground MV grids.Thus, an indicative picture of the real-world situation will be obtained [37,38].

Topologies of Distribution BPL Networks.
In accordance with [1,4,17], average path lengths of 1000 m and 200 m are typically encountered in overhead and underground distribution BPL topologies, respectively.During the following analysis, the distribution BPL network of Figure 2 having  branches is considered.With reference to Figure 2, five indicative overhead distribution BPL topologies of average path length, which are reported in Table I of [1], are examined, namely, (i) overhead urban case A, (ii) overhead urban case B, (iii) overhead suburban case, (iv) overhead rural case, and (v) overhead "LOS" case.Note that this topology corresponds to the Line-of-Sight transmission-that is, no branches are encountered across the BPL signal transmission path.In accordance with [1,4,17], these five indicative topologies are assumed to be common across overhead distribution BPL networks regardless of their type.
Similar to overhead distribution BPL case, five indicative underground distribution BPL topologies of average path length, which are reported in Table II of [1], are examined in this paper, namely, (i) urban case A, (ii) urban case B, (iii) underground suburban case, (iv) underground rural case, and (v) underground "LOS" case.Similar to overhead distribution BPL networks, these five indicative topologies are assumed to be common across underground distribution BPL networks regardless of their type [4,16,17].
With reference to Figure 2, the distribution and branching TLs are assumed to be identical while their interconnections are considered to be fully activated.In addition, the transmitting and the receiving ends are assumed to be matched to the characteristic impedance of distribution TLs, whereas the branch terminations are assumed to be open circuits [4,13,14,[16][17][18][43][44][45].

The Basics of the BPL Signal Propagation in Overhead
and Underground Distribution BPL Networks.Analytically described in [1, 4, 13-21, 39, 40, 46-49], the standard TL analysis, which involves two conductors, can be extended to the MTL analysis, which involves more than two conductors, through a straightforward matrix approach.Actually, compared to the two-conductor line of standard TL analysis, which supports one forward-and one backward-traveling wave, a MTL structure with  + 1 conductors supports  pairs of forward-and backward-traveling waves.Each pair of forward-and backward-traveling waves is referred to as a mode and is characterized by its propagation constant.As the distribution MTL configurations of Figures 1(a)-1(d) are concerned, overhead and underground MV MTL structures may support three ( = 3) modes, whereas overhead and underground LV MTL structures may support four ( = 4) modes.

The Basics of the BPL Signal Transmission in Distribution
BPL Networks.In accordance with [1,14,20,21,34,35], the BPL signal can be injected into overhead and underground distribution lines through the following two coupling scheme types, namely, (i) WtG and StP coupling schemes (in the case of WtG coupling schemes, the signal is injected into one conductor of the examined overhead MTL configuration and returns via the ground; the WtG coupling scheme between conductor  and the ground will be denoted by WtG  ; in the case of StP coupling schemes, the signal is injected into one conductor of the examined underground MTL configuration and returns via the shield; the StP coupling scheme between conductor  and the shield will be denoted by StP  ) and (ii) WtW or PtP coupling schemes (the signal is injected between two conductors of the examined MTL configuration; in the case of overhead MTL configurations, WtW coupling schemes between conductors  and  will be denoted by WtW - , whereas, in the case of underground MTL configurations, PtP coupling schemes between conductors  and  will be denoted by PtP - ).

Differences between Theory and Practice during the Determination of Transfer Functions of Distribution BPL
Networks.Although a plethora of experimental results and derived theoretical analyses validate the theoretical accuracy of the aforementioned modeling approach [22][23][24][25][26], a set of practical reasons and "real-life" conditions may create significant differences between experimental measurements and theoretical results during the transfer function determination of distribution BPL networks.More specifically, these fault factors can be classified into the following six fault categories, namely [48,52,57], (i) difficulties of isolating specific MTL parameters from time-and frequency-domain reflectometry measurements, (ii) the sensitivity and accuracy of the used measurement equipment, (iii) resonant effects and cross talk phenomena due to parasitic capacitances and inductances, (iv) the weakness of including specific wiring and grounding practices, (v) practical impedance deviations of branch terminations and transmitting/receiving ends, and (vi) the lack of absolutely isolating the effect of noise during the transfer function computations.Taking into consideration these fault categories and ( 2), the measured coupling transfer function   {⋅} is determined by where  denotes the measurement frequency and () synopsizes the total fault that occurred due to the aforementioned six fault categories.
To mitigate the faults that occurred, which can seriously impair the accuracy of transfer function modeling, the best L1PMA is adopted so that the theoretical transfer functions can be restored.

Best L1PMA
3.1.Brief Presentation of Best L1PMA.During the last 70 years, the monotonic problem has attracted significant interest from many academic fields, such as engineering, health, economics, and statistics, as well as from various applications, including signal restoration, spectroscopy, image processing, and art [29,[58][59][60][61][62][63][64][65].Among the various proposed monotonic data approximation methods, the application of the best L1PMA, which is theoretically presented and experimentally verified in [27][28][29][30][31][32], successfully copes with problems that are derived from univariate signal restoration.This is due to the fact that this approximation method focuses on the smallest change to the data such that the first differences of the smoothed data change sign a prescribed number of times.
When the application of best L1PMA is concerned with the intention of restoring theoretical coupling transfer functions of distribution BPL networks behind the total faults that occurred due to the six fault categories of Section 2.6, some advantages that are gained by implementing this approximation method are given; namely, (i) observing the behavior of transfer functions of distribution BPL networks [1,4,[13][14][15][16][17][66][67][68][69][70][71], piecewise monotonicity is a property that always occurs due to the intrinsic BPL propagation and transmission drawbacks, such as "LOS" attenuation and multipath environment; exploiting this inherent BPL network property, best L1PMA is decomposed into optimal monotonic computations in the sections between adjacent turning points (local extrema) of transfer functions [29,32]; (ii) contrary to spline and wavelet approximation methods presented in [72,73], best L1PMA avoids the assumption that transfer functions depend on certain critical parameters; actually, the smoothing process is a straightforward data projection when best L1PMA is adopted [28,29]; (iii) already demonstrated in [28,29,32], best L1PMA is particularly suitable for mitigating large and uncorrelated data errors; since best L1PMA is based on the minimization of the moduli sum of the data errors, the existence of few gross errors among the data makes no difference to the best fit [74]; therefore, the undulation of the coupling transfer function data due to the existence of deep spectral notches and other faults can be comfortably accommodated through the application of the best L1PMA; and (iv) the Fortran software package that is applied in order to implement the best L1PMA has extensively been verified in various scientific fields [30,32,[63][64][65] and is freely available online in [33].
Based on the specifications of Fortran software package of [33], best L1PMA receives as inputs the measured coupling transfer function column vector H  , the measured frequency column vector f, the number of monotonic sections  sect that is provided either by the user or automatically by the computer, and the type of the first monotonic section (i.e., either increasing or decreasing).Note that, in the case of the coupling transfer functions of distribution BPL networks, the first monotonic section is always decreasing.Then, by executing the Fortran software package, which is a straightforward procedure, best L1PMA automatically derives the optimal local extrema as well as the best fit.Actually, best L1PMA gives as output the  × 1 approximated coupling transfer function column vector H  ≡ H  (f) = [  ( 1 ) ⋅ ⋅ ⋅   (  )]  whose elements   (  ),  = 1, . . ., , are the approximated coupling transfer functions for the given measurement frequencies   ,  = 1, . . ., , respectively.In fact, best L1PMA calculates the approximated coupling transfer function column vector H  by minimizing the sum of the moduli of the errors that are given by subject to the piecewise monotonicity constraints where the integer numbers   ,  = 1, . . .,  sect , define the positions of the local extrema of the best L1PMA satisfying the conditions On the basis of ( 8), PES helps towards (i) the evaluation of the approximation accuracy of the best L1PMA to the theoretical coupling transfer function when different numbers of monotonic sections are considered and (ii) the evaluation of the restoration degree of the theoretical coupling transfer function offered by best L1PMA when faults occur due to the six fault categories.With respect to (3), to assess the best L1PMA mitigation efficiency towards the faults, PES of ( 8) is compared against the fault PES that is given by

Numerical Results and Discussion
Various types of distribution BPL networks are simulated with the purpose of investigating the efficacy of the best L1PMA during the transfer function restoration when various faults occur.More specifically, the efficiency of the best L1PMA is assessed based on the changes incurred by a number of factors, such as the type of distribution power grid, the distribution BPL topology, the applied coupling scheme, and the nature of faults (i.e., fault distributions).
As regards the simulation specifications, the BPL frequency range and flat-fading subchannel frequency spacing are assumed to be equal to 1-30 MHz and 1 MHz, respectively.Therefore, the number of subchannels  in the examined frequency range is equal to 30.This implies that the theoretical maximum number of monotonic sections that can occur during the application of the best L1PMA is equal to 29 [29,30].
To simplify the analysis without, however, harming its generality, in the case of overhead and underground MV/ BPL networks, only WtG 1 /StP 1 and WtW 1-2 /PtP 1-2 coupling schemes will be examined, hereafter.As it is usually done [1,4,17,19,20,46], the selection of representative coupling schemes is a typical procedure for the sake of reducing manuscript size.
4.1.Distribution BPL Topologies and Best L1PMA.Prior to understanding the significant fault restoration delivered by adopting best L1PMA, there is a need for recognizing best L1PMA accuracy in approximating theoretical coupling transfer functions of distribution BPL topologies.
In Figures 3(a)-3(e), the theoretical coupling transfer function is plotted versus frequency for the five indicative overhead MV/BPL topologies, respectively, when WtG 1 coupling scheme is applied.In each figure, the best L1PMA is also plotted for a number of representative monotonic sections (i.e.,  sect = 2,  sect = 5, and  sect = 20).In Figures 3(f)-3(j), similar curves to Figures 3(a)-3(e) are shown when WtW 1-2 coupling scheme is adopted.In Figures 4(a)-4(j), similar plots to Figures 3(a)-3(j) are given but for overhead LV/BPL networks.
In Figures 5(a)-5(e), the theoretical coupling transfer function is plotted versus frequency for the five indicative underground MV/BPL topologies, respectively, when StP 1 coupling scheme is adopted.In each figure, the best L1PMA is also plotted for the same number of representative monotonic sections as in Figures 3(a Observing Figures 3(a)-3(j), 4(a)-4(j), 5(a)-5(j), and 6(a)-6(j), several interesting remarks can be pointed out: (i) Already mentioned in [1, 4, 9, 13-17, 19, 71, 75], the spectral behavior of the transfer functions depends drastically on the frequency, the type of the examined distribution power grid, the applied coupling scheme, the physical properties of the MTL configuration, and the path length as well as the number and the length of the branches.
(ii) The presence of branches along the end-to-end transmission path causes signal reflections and a multipath environment that further creates spectral notches in the coupling transfer functions.The number, the extent, and the depth of the spectral notches that occurred depend on the number and the length of the branches.It should be noted that the spectral notches are superimposed on the "LOS" attenuation.
(iii) Based on the behavior of spectral notches in the coupling transfer functions, distribution BPL topologies can be categorized into three channel classes as follows [1,4,15,16,71]: (a) Bad Channel Class.BPL transmission in many urban zones (i.e., urban cases A and B) belongs to this channel class.This channel class is characterized by topologies with a high number of branches of small electrical length.Due (iv) Regardless of the applied coupling scheme and the examined distribution power grid type, best L1PMA remarkably approximates the examined coupling transfer functions of all the channel cases as follows: (a) Best L1PMA curves practically coincide with the coupling transfer functions of "LOS" channel class.The absence of spectral notches and the monotonous form of "LOS" attenuation allowed the best L1PMA to perfectly fit the coupling transfer functions of this channel class by simply using only few monotonic sections (i.e., lower than 5).
(b) Best L1PMA accurately describes the behavior of the coupling transfer functions of good channel class as well as their spectral notches.Actually, the shallow spectral notches of coupling transfer functions are treated by the best L1PMA as monotonous sections of decreases and increases.Hence, best L1PMA succeeds in including in its best fit the vast majority of the local extrema.Therefore, the number of monotonic sections, which is required by the best L1PMA to accurately describe the coupling transfer functions of good channel class, is higher than the respective one of "LOS" channel class.
(c) Best L1PMA satisfactorily describes the spectral behavior of bad channel class.Although the coupling transfer functions of this channel class are characterized by deep and frequent spectral notches, best L1PMA succeeds in including the majority of these local extrema in its fits.In  order to adequately describe these intense spectral fluctuations, it is obvious that best L1PMA exploits all the available monotonic sections; for example, best L1PMA uses more than 20 monotonic sections-see Figures 3(a)-3(j), 4(a)-4(j), 5(a)-5(j), and 6(a)-6(j).
(v) Best L1PMA successfully achieves representing either the local extrema or the tail of coupling transfer functions.In contrast with wavelet or spline approximations [72,73], best L1PMA manages to mitigate perturbations (ripples) into the tail of the approximation when a great number of extrema occur [28,59,60,76].
From the previous analysis, it is evident that the number of monotonic sections determines the approximation efficiency of best L1PMA to the coupling transfer functions of distribution BPL networks.In order to highlight this critical role, PES is plotted versus the number of monotonic sections in Figure 7(a) for the five indicative topologies of overhead MV/BPL networks when WtG 1 and WtW 1-2 coupling schemes are considered.In Figures 7(b)-7(d), same plots are given in the case of overhead LV/BPL, underground MV/BPL, and underground LV/BPL networks, respectively.Note that, in the case of underground distribution BPL networks, StP 1 or PtP 1-2 coupling schemes are applied instead of WtG 1 or WtW 1-2 ones, respectively.
From Figures 7(a)-7(d), it is evident that the perfect fit of best L1PMA primarily depends on the number of monotonic sections.Actually, there are an optimal number of monotonic sections above which the improvement of PES remains marginal.This optimal number depends on (i) the distribution power grid type; (ii) the number, the extent, and the depth of spectral notches of the coupling transfer function (i.e., the channel class of the examined distribution BPL topology); and (iii) the applied coupling scheme.As already mentioned, although this optimal number of monotonic sections is low when distribution BPL topologies of "LOS" and good channel class are examined, it presents significant higher values when distribution BPL topologies of bad channel class are investigated.Taking into account the upper limit of the number of monotonic sections, even if best L1PMA does not perform its best fit to the data in order to coincide with the coupling transfer functions of bad channel class, it satisfactorily approximates them.Through the prism of PES and in order to assess the accuracy of best L1PMA, the optimal number of monotonic sections is presented in Table 1 for the cases examined in Figures 7(a)-7(d).Note that the optimal number of monotonic sections is defined for given best L1PMA execution scenario when PES of the examined scenario takes for the first time the value 10 −3 % or less.From Table 1, it is numerically validated that the best L1PMA succeeds in accurately fitting coupling transfer functions of distribution BPL topologies in all the cases examined.PES results indicate that the best L1PMA can successfully approximate either the trend of "LOS" attenuation or the deep spectral notches that occurred due to the multipath environment.As it has already been outlined, the optimal number of monotonic sections increases when distribution BPL topologies of bad channel class are investigated and remains low when distribution BPL topologies of "LOS" and good channel class are examined.
As the optimal number of monotonic sections is concerned, the following remarks are pointed out: (i) In 15% of the cases examined, best L1PMA has achieved fitting coupling transfer functions with optimal number of monotonic sections greater than 20.
(ii) In 25% of the cases examined, best L1PMA has fitted coupling transfer functions with optimal number of monotonic sections lower than 5.
(iii) In general terms, the average PES of best L1PMA is equal to 0.99 ×10 −5 % which is an extremely low value in comparison with the applied PES threshold of 10 −3 %.This fact implies that best L1PMA perfectly fits theoretical coupling transfer functions.
Since the optimal number of monotonic sections uniquely describes the pattern of the coupling transfer function of each distribution BPL topology, this number is going to characterize each distribution BPL topology towards its coupling transfer function restoration when faults occur as presented in the following section.

Best L1PMA against Faults during the Determination of Theoretical Coupling Transfer Function of Distribution BPL Networks.
As it has already been reported in Section 2.6, there are six fault categories that can create significant differences between experimental measurements and theoretical results during the determination of coupling transfer function of distribution BPL networks.As it is shown in this section, based on the approximation efficiency of best L1PMA, the theoretical coupling transfer function of distribution BPL networks can be revealed even though significant faults may occur.On the basis of the optimal number of monotonic sections, which has been determined in Table 1 for each distribution BPL network topology and coupling scheme, best L1PMA achieves mitigating the additive faults  by simply maintaining the monotonicity pattern of each coupling transfer function.In fact, the fault effect can be drastically counterbalanced since best L1PMA relies on the median of the corresponding data within a range.In the following analysis, the total fault that occurred due to the six fault categories {⋅}, which is described in (3), is assumed to follow either CUD with minimum value − CUD and maximum value  CUD or ND with mean  ND and standard deviation  ND .
To examine the impact of faults on the determination of coupling transfer function of distribution BPL networks, in Figures 8(a)-8(e), the theoretical coupling transfer function is plotted versus frequency for the five indicative overhead MV/BPL topologies, respectively, when WtG 1 coupling scheme is applied.In each figure, the measured coupling transfer function is also drawn with faults that are characterized by the following indicative distributions, namely, (i) CUD with  CUD = 6dB and (ii) ND with  ND = 0dB and  ND = 2 dB.Moreover, the best L1PMA is also curved for the two previous measured coupling transfer functions.In Figures 8(f)-8(j), similar curves to Figures 8(a)-8(e) are plotted when WtW 1-2 coupling scheme is adopted.In Figures 9(a)-9(j), 10(a)-10(j), and 11(a)-11(j), similar plots to Figures 8(a)-8(j) are given in the case of overhead LV/BPL networks, underground MV/BPL networks, and underground LV/BPL networks, respectively.Note that, in the case of underground distribution BPL networks, StP 1 or PtP 1-2 coupling schemes are applied instead of WtG 1 or WtW 1-2 ones, respectively.
From Figures 3(a)-3(j), 4(a)-4(j), 5(a)-5(j), and 6(a)-6(j), it has been pointed out that best L1PMA identifies the most important local extrema of theoretical coupling transfer functions, while interpolating the data at these extrema.The increase of the number of monotonic sections allows best L1PMA to make the sum of the squares of the residuals smaller while maintaining the most important local extrema of the theoretical coupling transfer function.In Figures 8(a)-8(j), 9(a)-9(j), 10(a)-10(j), and 11(a)-11(j), best L1PMA relies on the number of monotonic sections provided by the theoretical coupling transfer functions [29].Inversely, given the number of the monotonic sections, best L1PMA first tries to pinpoint the most important local extrema of measured coupling transfer functions and second performs the best fit among the data in every monotonous section.Actually, best L1PMA makes the sum of the squares of the residuals smaller, while maintaining the most important local extrema.
Moreover, in the attempt to provide structure in data assuming that there is no underlying mathematical function, the assumption of the specific number of monotonic sections seems to be quite important and accurate.In fact, this assumption can substitute the trend test algorithm of [77]   that is responsible for initially estimating the turning points and is adopted in this paper.However, the development of a test similar to [77], which will be tailored to measured BPL coupling functions, would be valuable as future research question.
Although the effectiveness of best L1PMA to coupling transfer function restorations is well presented in the case of suburban, rural, and "LOS" topologies, best L1PMA can have many applications as a data smoothing approach.Despite the large number of local extrema of urban case A and B topologies that can occur during the optimization calculation, the set of an upper bound to the number of monotonic sections may allow a smoother global solution in quadratic complexity with respect to the number of data points [28].Based on the online available Fortran software package of [33], the indicative processing time in seconds of both the hybrid method and the best L1PMA for Figures 8(a 2. Despite the high complexity, the computational burden of both the hybrid method and the best L1PMA ranges from 2.230 s to 3.107 s for each distribution BPL topology examined.As the technical characteristics of the system performing the simulations are concerned, it has an Intel Pentium 1.9 GHz CPU and 4 GB RAM.Therefore, the presented data smoothing approach could be a useful intermediate method for real-time processing applications concerning BPL frequency allocation problem [78]. Apart from the visual result of Figures 8(a)-8(j), 9(a)-9(j), 10(a)-10(j), and 11(a)-11(j), the approximation performance of best L1PMA is examined in terms of PES for a number of different fault distributions.More specifically, in Table 3, PES of best L1PMA is compared against fault PES when indicative fault CUDs are applied (i.e.,  CUD = 3 dB,  CUD = 6 dB, and  CUD = 9 dB) for the distribution BPL network scenarios of Figures 8(a)-8(j), 9(a)-9(j), 10(a)-10(j), and 11(a)-11(j).Similarly, in Table 4, PES of best L1PMA is compared against fault PES when indicative fault NDs are applied (i.e.,  ND = 0 dB and  ND = 2 dB,  ND = 0 dB and  ND = 6 dB, and  ND = 0 dB and  ND = 9 dB) for the same distribution BPL network scenarios.
In Tables 3 and 4, PES, fault PES, and their difference are reported for each distribution BPL network scenario examined.For a given distribution BPL network scenario, when the best L1PMA achieves better PES in comparison with the respective fault PES, the difference is presented in bold; otherwise, it is presented in italic.Apart from small and average faults, best L1PMA is also examined when gross fault cases occur (i.e., CUD with  CUD = 9 dB and ND with  ND = 0 dB and  ND = 9 dB).These aggravated cases are denoted by * .From Figures 13(a)-13(d), it is obvious that PES primarily depends on the CUD maximum value and ND standard deviation.Also, for the same value of CUD maximum value and ND standard deviation, PES is more sensitive to the fluctuations of ND standard deviation.
Independently of the fault distribution, PES of overhead distribution BPL networks is more fault vulnerable in comparison with PES of underground distribution BPL networks.This is due to the fact that overhead BPL networks suffer from reflections due to the multipath environment rather than the "LOS" attenuation.Since their average channel attenuation is significantly lower than the respective one of underground BPL networks, the same fault magnitude more crucially harms the coupling transfer functions of overhead BPL networks further deteriorating their PES.Nevertheless, as concerns PES differences between MV and LV BPL networks of the same grid type (either overhead or underground), these differences remain marginal.
Except for the distribution power grid type, significant PES differences are shown among the different topologies.More specifically, in overhead distribution BPL networks, rural and "LOS" topologies are the most fault sensitive whereas, in underground distribution BPL networks, urban case A and B topologies are the most fault resistant.This is due

Figure 2 :
Figure 2: End-to-end distribution BPL topology with  branches.

Figure 3 :
Figure 3: Coupling transfer function of the five indicative topologies of overhead MV/BPL networks when different coupling schemes are applied and various best L1PMAs are adopted.(a)-(e) WtG 1 coupling scheme.(f)-(j) WtW 1-2 coupling scheme.

Figure 5 :
Figure 5: Coupling transfer function of the five indicative topologies of underground MV/BPL networks when different coupling schemes are applied and various best L1PMAs are adopted.(a)-(e) StP 1 coupling scheme.(f)-(j) PtP 1-2 coupling scheme.

Figure 7 :
Figure 7: PES of distribution BPL topologies when various coupling schemes are applied and different numbers of monotonic sections are considered.(a) Overhead MV/BPL networks.(b) Overhead LV/BPL networks.(c) Underground MV/BPL networks.(d) Underground LV/BPL networks.

Figure 10 :
Figure 10: Theoretical, measured, and best L1PMA coupling transfer function of the five indicative topologies of underground MV/BPL networks for two indicative fault distributions (CUD and NU) when different coupling schemes are applied.(a)-(e) StP 1 coupling scheme.(f)-(j) PtP 1-2 coupling scheme.

Overhead and Underground MV/BPL and LV/BPL Networks: MTL Configurations, BPL Topologies, and Propagation/Transmission Analysis
2.1.The Overhead MV and LV Power Grids.The overhead MV and LV distribution lines, which are examined in this paper, are shown in Figures ()]  are the voltages of the  modes supported by the considered distribution MTL configuration (modal voltages), [⋅]  denotes the transpose of a matrix, H  {⋅} is the  ×  modal transfer function matrix whose elements   , {⋅}, ,  = 1, . . ., , are the modal transfer functions, and   , denotes the element of matrix H  in row  of column .

Table 1 :
Optimal number of monotonic sections and PES for distribution BPL networks (OV: overhead; UN: underground).

Table 2 :
Indicative processing time of best L1PMA for distribution BPL networks (OV: overhead; UN: underground).