Statistical Channel Modeling of Overhead Low Voltage Broadband over Power Lines (OV LV BPL) Networks – Part 2: The Numerical Results of Class Map Footprints of Real OV LV BPL Topologies, Branch Line Faults and Hook Style Energy Thefts

In [1], the theoretical framework for the interoperability of DHM, iSHM, mSHM, the definition procedure and the class maps has been first presented for OV LV BPL networks. But the main interest of the first paper has focused on the theory of the OV LV BPL topology footprints of TIM, FIIM and HS-DET method on the class maps. In this paper, the numerical results concerning the application of iSHM, mSHM, the definition procedure and the class maps to OV LV BPL networks are first shown. Then, given the iSHM and mSHM class maps, the footprints of TIM, FIIM and HS-DET method databases for the OV LV BPL topologies are highlighted. Finally, a technique for the detection of branch line faults and hook style energy thefts that is based on iSHM and mSHM footprints is proposed.


Introduction
In SG applications, ICT is their key aspect while among the available ICT solutions, BPL technology presents some natural advantages that render the related BPL networks as the most suitable implementation for SG applications [2]. However, BPL networks operate across a hostile medium for communications since transmission and distribution power grid infrastructure and equipment have been designed to deliver power rather than information [1], [3]- [7].
The interaction results of the deterministic framework, which is represented by the DHM [3], [4], [8]- [13], and the recently proposed statistical framework, which is characterized by the SHM operation [14]- [19], for the OV LV BPL networks are presented in this paper with respect to the class maps and the footprint area concept that have been first presented for OV LV BPL networks in [1]. In detail, the numerical results of the application of iSHM and mSHM are presented while the performance of different CASDs among the available ones of iSHM and mSHM is here assessed. Actually, the most successful CASD among the available ones of iSHM is going to be identified with respect to the metrics of percentage change and the average absolute percentage change as the most successful CASD of mSHM is the Empirical one. Then, the definition procedure is going to be applied in OV LV BPL networks enriching the corresponding topology classes with topology subclasses whose respective virtual indicative OV LV BPL topologies are statistically defined in terms of the applied SHM version and its corresponding successful CASD parameters (i.e., MLEs and CDF for iSHM and mSHM CASDs, respectively). On the basis of the most successful CASDs of iSHM and mSHM, the definition procedure gives as output the class maps of OV LV BPL networks that illustrate the borders between the BPL topology classes and also corresponds each CASD parameter pair to its BPL topology subclass average capacity for given power grid type, SHM version, CASD, coupling scheme, IPSD limits and noise levels [1]. By taking into account OV LV BPL topologies, the numerical results concerning the footprints on the class maps, which have been theoretically defined in [1] and may allow the impact investigation of critical events of the operation of OV LV power grids, are here presented, namely: (i) The footprint of the real OV LV BPL topologies is going to first be illustrated on the class maps thus separating the real OV LV BPL topologies from the virtual ones of class maps by exploiting the TIM database [20]; (ii) for given real indicative OV LV BPL topology, the footprint of all the corresponding OV LV BPL topologies with a sole branch line fault is going to be shown on class maps by exploiting the FIIM database [20]; and (iii) for given real indicative OV LV BPL topology, the footprint of all the corresponding OV LV BPL topologies with a single hook for energy theft are going to be demonstrated on class maps by exploiting the HS-DET method [21]. On the basis of the impact investigation of the aforementioned three critical events, the study of iSHM and mSHM footprints can also act as a useful supplementary technique for identifying branch line faults and hook style energy thefts.
The rest of this paper is organized as follows: In Section II, the numerical results concerning the interoperability of DHM, iSHM, mSHM, the definition procedure and the class maps of OV LV BPL networks are demonstrated. The footprints of the aforementioned three critical events during the operation of the OV LV BPL networks are illustrated on the class maps. Apart from the presentation, interesting comparisons with OV MV and OV HV BPL networks are provided while the proposed technique for identifying branch line faults and hook style energy thefts through iSHM and mSHM footprints is provided.

Numerical Results and Discussion
In this Section, numerical results concerning the statistical behavior of OV LV BPL networks are categorized into the following subsections, namely: (i) iSHM: CASD MLEs of iSHM are reported while the performance of iSHM CASDs is assessed in terms of the best percentage change and average percentage change results. The best CASD of iSHM is found with respect to the aforementioned two performance metrics; (ii) mSHM: The performance of Empirical CASD is assessed; (iii) Definition procedure and class maps: The definition procedures for the best CASD of iSHM and the Empirical CASD of mSHM are applied while the corresponding class maps are illustrated; and (iv) Footprints: the footprints of the three OV LV BPL topology databases (i.e., TIM, FIIM and HS-DET method databases) are shown on the iSHM and mSHM class maps.

CASD MLEs of iSHM for the OV LV BPL Networks
In accordance with the iSHM definition [14], [15], the BPMN diagram of iSHM, which describes its operation, is given in Fig. 2(a) of [1]. As the CASD MLEs of iSHM are concerned, these are computed at the MLE computation module of Phase C of the iSHM BPMN diagram thus affecting all the remaining iSHM operation. As already been recognized in [15], [16], the efficiency of the CASDs is based on the respective CASD MLEs that further depend on the capacity estimation of the real indicative BPL topologies that is extended to the case of OV LV BPL networks of this paper.
As the default operation settings have been assumed in [1], MLEs of the Gaussian, Lognormal, Wald, Weibull and Gumbel CASDs of iSHM are reported in Table  1 for the real indicative OV LV BPL topologies of the main subclasses of Table 1 of [1]. Similarly to [15], [16], [18], [19], the graphical analysis concerning the capacity estimation performance of CASDs of iSHM can be securely neglected by simply applying capacity estimation performance metrics such as the percentage change and the average absolute percentage change instead. Indeed, the percentage change and the average absolute percentage change of each iSHM CASD are reported in Table 2 per each indicative OV LV BPL topology main subclass as well as the capacity of each real indicative OV LV BPL topology of main subclasses.
By comparing iSHM CASD MLEs of the real indicative OV LV BPL topologies, which are presented in Table 1, with the respective ones of the real indicative OV MV BPL topologies of Table 1 of [15] and the respective ones of the real indicative OV HV BPL topologies of Table 2 of [19], iSHM CASD MLEs of the real indicative OV LV BPL topologies present more similar behavior to the iSHM CASD MLEs of the real indicative OV MV BPL topologies rather than the ones of the OV HV BPL topologies. This is due to the fact that: (i) the OV LV MTL configuration can be considered as a vertical rearrangement of the OV MV MTL configuration; and (ii) the average transmission path of OV LV BPL topologies has been assumed to be the same to the one of OV MV BPL topologies as well as the topological characteristics. Taking into account the findings of [15], same observations concerning the iSHM CASD MLEs of the indicative OV LV BPL topologies with the ones of OV MV BPL topologies can be expressed, namely: (a) , , , , , ,  and of OV LV BPL topologies increase as the multipath environment of the OV LV BPL topologies becomes more intense; and (b) and of OV LV BPL topologies receive very close values among them that anyway may act as an identity value of the power grid type (i.e., in this case OV LV) rather than OV LV BPL topology identifier.
In accordance with [16], [15], [18], [19], it has been proven that the capacity performance success can be assessed by applying the metrics of capacity, capacity percentage change and average absolute capacity percentage change, as reported in Table 2. More specifically: • iSHM CASD capacity estimation performance for OV LV BPL subclasses presents similarities with the one of OV MV BPL subclasses rather than the one of OV HV BPL subclasses. Since OV distribution BPL topologies are mainly characterized by the significant lower 1km long transmission paths in comparison with the 25km long transmission paths of OV HV BPL topologies, which implies a weaker "LOS" attenuation mechanism, and by the multipath aggravation due to the frequent and relatively short branches of distribution power grids, it is evident that a CASD separation may occur on the basis of the examined power grid level. • Similarly to OV MV BPL topology subclasses, Weibull CASD succeeds in satisfying the 3% absolute threshold of percentage change and average absolute percentage change in all the OV LV BPL topology subclasses examined. Actually, Weibull CASD achieves the best percentage change and average absolute percentage change in all the cases examined (see grey background in the column of Weibull CASD in Table 2). Note that the only CASD that can produce successful capacity estimations but with significantly worse performance in comparison with the one of Weibull CASD, especially in the cases of urban environments, is Lognormal CASD. In contrast with the Gaussian CASD that operates as the best CASD in OV HV BPL networks, Gaussian, Wald and Gumbel cannot successfully operate in OV LV BPL topologies apart from the rural case that anyway resembles to the OV HV BPL signal transmission in terms of the scarce presence of branches. Here, it should be also reminded that the best fit for UN MV BPL networks is Wald CASD as outlined in [15]. In the rest of this paper, only Weibull CASD is going to be considered as the best fit CASD among the available ones of iSHM and to be applied during the following subsections of the definition procedure, class mapping and footprints.

CASD of mSHM for the OV LV BPL Networks
The BPMN diagram of mSHM is illustrated in Fig. 2(b) of [1]. In contrast with iSHM and its supported CASD MLEs, mSHM exploits the Empirical CASD through the Empirical CDF, which acts as the CASD parameter, of the coupling scheme channel attenuation difference for given OV LV BPL topology and coupling scheme. As already been recognized in [16], [18], [19], the efficiency of the mSHM CASD is based on the Empirical CDFs while it depends on the capacity estimation of the real indicative BPL topologies.
As the default operation settings have been assumed in [1] for the computation of the Empirical CDFs of mSHM and similarly to iSHM, the graphical analysis concerning the capacity estimation performance of Empirical CASD of mSHM can be securely neglected while the aforementioned capacity estimation performance metrics are applied instead. Indeed, the percentage change and the average absolute percentage change of mSHM Empirical CASD are reported in Table 3 per each indicative OV LV BPL topology main subclass as well as the capacity of each real indicative OV LV BPL topology of main subclasses, which anyway coincides with the respective one of Table 2. By comparing the results of Table 3 and 2, it is evident that Empirical CASD of mSHM presents a capacity performance success for the main OV LV BPL subclasses (grey background of the last column of Table 3) that can be considered to be comparable with the one of Weibull CASD of iSHM, which anyway demonstrates the best fit results. In all the main OV LV BPL subclasses examined, Empirical CASD succeeds in satisfying the 3% absolute threshold of percentage change and average absolute percentage change. As been mentioned in Sec.2.1, this is an expected result since capacity estimation performance for OV LV BPL subclasses presents more similarities with the one of OV MV BPL subclasses rather than the one of OV HV BPL subclasses. Therefore, apart from the Weibull CASD of iSHM, the Empirical CASD of mSHM is also going to be applied during the following subsections of the definition procedure, class mapping and footprints.

iSHM and mSHM Class Mapping for OV LV BPL Classes for the Default Operation Settings
Already been presented in [1], iSHM and mSHM definition procedures for OV LV BPL topologies are here applied for the class mapping. From Sec.2.2, Weibull CASD of iSHM and Empirical CASD of mSHM are considered to execute the most successful capacity estimations for OV LV BPL topologies and for that reason the corresponding class maps are illustrated in this paper.
As the iSHM class map scenario is concerned, there are 5 pairs of and that are the Weibull CASD MLEs, for the respective real indicative OV LV BPL topologies of the main subclasses of Table 1 of [1]. The iSHM class map of OV LV BPL topologies is plotted in Fig. 1 with respect to and for the default operation settings of [1] when the average capacity of each OV LV BPL topology subclass is considered. Note that as the spacings of the horizontal axis, the spacings of the vertical axis and the capacity borders between the adjacent distribution BPL topology classes are concerned, their computation is in accordance with [18], [19].
As the mSHM class map scenario is concerned, in accordance with [17]- [19], the examined real indicative OV LV BPL topologies of the main subclasses are examined separately during the preparation of mSHM class maps. Among the five real indicative OV LV BPL topologies of the main subclasses of Table 1 of [1], three of them are of interest in this paper and are going to be investigated through mSHM class maps; say, OV LV BPL urban case A, suburban case and rural case. With reference to [1], the mSHM class map of OV LV BPL topologies is plotted in Fig. 2(a) with respect to the indicative OV LV BPL urban case A, horizontal shift and vertical shift for the default operation settings when the average capacity of each OV LV BPL topology subclass is assumed. In the same 2D contour plot, the capacity borders between the adjacent distribution BPL topology classes and the capacity of the reference indicative OV LV BPL urban case A are also shown. In Figs. 2(b) and 2(c), same plots with Fig. 2(a) are shown but for the case of the OV LV BPL suburban case and of the OV LV BPL rural case, respectively.
By observing Fig. 1 and Figs. 2(a)-(c) and by comparing them with the respective ones of [18] and [19], it is evident that iSHM and mSHM class maps of OV LV BPL topologies present more similarities with the respective ones of OV MV BPL topologies rather than of UN MV BPL topologies and of OV HV BPL topologies. Apart from the approximate same capacity border values between the adjacent distribution BPL topology classes, the pattern of iSHM class maps present the same class area notches when the distribution BPL topologies are examined in contrast with the rectangular class areas of UN MV BPL topologies and OV HV BPL topologies. Also, as the mSHM class maps are regarded, the extent and the capacity border continuity of the "LOS" class area -i.e., yellow class area in Figs. 2(a)-(c)-remain very similar between the OV MV BPL topologies and OV LV BPL ones.
Anyway, it should be noted that common issues among all the iSHM and mSHM class maps of transmission and distribution BPL topologies stand that are: (i) the clear distinction between the adjacent BPL topology classes regardless of the type of the examined class map; and (ii) the same general pattern of mSHM class maps where a large "LOS" class area exists at the left of the mSHM class map and two separate aggravated urban case class areas lie at the top and bottom right of the mSHM class map.
On the basis of Fig. 1 and Figs. 2(a)-(c), the footprints of the three OV LV BPL topology databases (i.e., TIM, FIIM and HS-DET method databases), which are going to be studied in Secs. 2.4-2.6, respectively, are graphically superimposed on the aforementioned iSHM and mSHM class maps.

TIM Database Footprint on iSHM and mSHM Class Maps for the Default Operation Settings
In this Section, the footprint of the real OV LV BPL topologies is first illustrated on the iSHM and mSHM class maps of Sec.2.3 thus distinguishing the real OV LV BPL topologies from the virtual ones. With reference to the TIM database [20], real OV LV BPL topologies with their respective theoretical coupling scheme channel transfer functions can be retrieved. However, there is a trade-off during the preparation of the TIM database of interest between the detail degree of the assumed topological characteristics and the execution time of the theoretical coupling scheme channel transfer function and capacity computations through DHM. The TIM BPL topology database specifications that have been reported in [20] for the database preparation and affect the total execution time are the maximum number of branches N, the length spacing Ls for both branch distance and branch length and the maximum branch length Lb while typical lengths of 1000m are assumed for the real OV LV BPL topologies. To maintain realistic total execution times of the footprint representations, two different cases concerning the footprint of real OV LV BPL topologies are here examined; say, the footprints of real OV LV BPL topologies with one and two branches on iSHM and mSHM class maps. Note that the TIM BPL topology database specifications and total execution time that correspond to each scenario and class map type are also reported.
As the iSHM footprint of the real OV LV BPL topologies with one branch are concerned, the maximum number of branches N, the length spacing Ls for both branch distance and branch length and the maximum branch length Lb are assumed to be equal to 1 m, 10 m and 1000 m, respectively. With reference to the iSHM class map of Fig. 1, the iSHM footprint of the real OV LV BPL topologies with one branch is illustrated as superimposed white areas on class maps in Fig. 3(a). Similarly, the iSHM footprint of the real OV LV BPL topologies with two branches is illustrated in Fig. 3(b) when the maximum number of branches N, the length spacing Ls for both branch distance and branch length and the maximum branch length Lb are assumed to be equal to 2, 100m and 1000m, respectively.
As the mSHM footprint of the real OV LV BPL topologies with one branch are regarded, the TIM BPL topology database specifications are assumed to be the same with the ones of the iSHM footprint case. With reference to the mSHM class map of Fig. 2(c), the mSHM footprint of the real OV LV BPL topologies with one branch is illustrated as white areas in Fig. 4(a). Similarly, the mSHM footprint of the real OV LV BPL topologies with two branches is illustrated in Fig. 4(b) when the TIM BPL topology database specifications are assumed to be the same with the ones of the mSHM footprint case.
From Figs. 3(a), 3(b), 4(a) and 4(b), interesting observations concerning the iSHM and mSHM footprint planning of the real OV LV BPL topologies can be pointed out. More specifically: • Since iSHM and mSHM class maps have already been prepared in Sec. 2.3, the total execution time considers the time that is required for the preparation of the TIM database given the TIM database specifications per each case as well as the required statistical and capacity processing. In terms of time, for the preparation of the footprints shown in Figs. 3(a), 3(b), 4(a) and 4(b), the total execution time is equal to 9674s, 7291s, 11646s, and 8567s, respectively. The different TIM database specifications regarding the two cases have been made so that the total execution time can remain relatively low without affecting the generality of the analysis. Anyway, different number of OV LV BPL topologies was expected for the two cases. • With reference to Fig. 3(a), the iSHM footprint of OV LV BPL topologies with one branch is clearly confined in the rural case area. More specifically, as the length of the sole branch of the examined OV LV BPL topology decreases, this forces the presence of deep notches in the corresponding theoretical coupling scheme channel transfer function thus creating a richer multipath environment [12]. Conversely, the consideration of longer branches can impose more frequent but less deep notches in the corresponding theoretical coupling scheme channel transfer functions in comparison with the shorter branches. In accordance with [1], [180], OV LV BPL topologies of rare but less deep notches are characterized by increased values of and and lower capacities. As the relative location of the OV LV BPL topologies with one short branch is concerned in the iSHM footprint, these topologies tend to be located at the lower left regions of the iSHM footprint while the OV LV BPL topologies with longer branches tend to be located at the opposite direction. Same observations can be expressed for the case of OV LV BPL topologies with two branches, where OV LV BPL topologies of two branches remain in the suburban case area with similar branch length behavior regarding the relative location in the class map suburban case area with the case of a single branch.  • With reference to Figs. 4(a) and 4(b), the mSHM footprints of OV LV BPL topologies with one and two branches are clearly confined in the rural and suburban case areas, respectively. Although the number of the examined OV LV BPL topologies remains the same given the number of branches, the main difference between iSHM and mSHM footprints is first their extent; this is due to the fact that the examined real OV LV BPL topologies are characterized by their own CASD MLEs during the iSHM definition procedure and can be straightforward superimposed to iSHM class maps whereas the examined real OV LV BPL topologies are classified among the existing pairs of the mSHM class maps and demonstrated as white spots through the practical approximation of Frobenius distance analyzed in Sec.4.1 of [1]. As the number of spacings of the horizontal and vertical axes of the mSHM class maps of Figs. 2(a)-(c) is both equal to 10, all the white spots of Figs. 3(a) and 3(b) are classified into 8 white spots in Figs. 4(a) and 4(b), respectively. It is evident that as the number of spacings of the horizontal and vertical axes increases so does the number of white spots of the mSHM footprints of the real OV LV BPL topologies since a larger set of available pairs can be examined. • The latter difference between the illustration of iSHM and mSHM footprints also explains the greater total execution times of mSHM footprints, which have been reported, in comparison with the ones of iSHM footprints for given number of branches of the real OV LV BPL topologies. As it is shown, the straightforward CASD MLE computation of iSHM definition procedure remains faster than the comparison among all the available virtual OV LV BPL topologies of each OV LV BPL topology subclass during the preparation of mSHM footprints.

FIIM Database Footprint on iSHM and mSHM Class Maps for the Default Operation Settings
In this Section, the iSHM and mSHM footprints of the real OV LV BPL topologies with a sole branch line fault are first illustrated. For given real OV LV BPL topology, say, OV LV BPL urban case A, one branch line fault is once applied to each of the three branches of the examined OV LV BPL urban case A. With reference to the FIIM database [20], real OV LV BPL topologies with one branch line fault that are based on the OV LV BPL urban case A with their respective theoretical coupling scheme channel transfer functions can be retrieved. Similarly to the TIM database specifications, appropriate FIIM BPL topology database specifications that have been reported in [1] for the database preparation are assumed, namely the fault location at each existing branch of the real indicative OV LV BPL urban case A ranges from 0.1 m to the end of the corresponding branch with a step of 0.1m. Note that the faulty branch termination is again assumed to be open-circuit.
As the iSHM footprint of the real OV LV BPL topologies with one branch line fault that are based on the real indicative OV LV BPL urban case A is concerned, the iSHM footprint of the real OV LV BPL topologies with one branch line fault is illustrated as superimposed white areas on class maps in Fig. 5 with reference to the iSHM class map of Fig. 1. Similarly to iSHM footprint of Fig. 5, the mSHM footprint of the real OV LV BPL topologies with one branch line fault that are based on the real indicative OV LV BPL urban case A is shown in Fig. 6.

HS-DET Method Database Footprint on iSHM and mSHM Class Maps for the Default Operation Settings
In this Section, the iSHM and mSHM footprints of the real OV LV BPL topologies with a single hook for energy theft are first illustrated. For given real OV LV BPL topology, say, the real indicative OV LV BPL suburban case, one hook, which is treated as a new branch [21], [29], [30], is hung on the existing real indicative OV LV BPL suburban case. With reference to HS-DET method database [21], real OV LV BPL topologies with a single hook for energy theft that are based on the real indicative OV LV BPL suburban case with their respective theoretical coupling scheme channel transfer functions can be retrieved. Similarly to the TIM and FIIM database specifications, appropriate HS-DET method database specifications that have been reported in [20] for the database preparation are assumed, namely the location of the hook for energy theft that ranges from 0.1 m to 1000 m with a step of 10 m and the length of the hook that ranges from 0.1 m to 200 m with a step of 10 m. Note that the branch termination, which is used for the simulation of the hook of the energy theft, is again assumed to be opencircuit.
As the iSHM footprint of the real OV LV BPL topologies with a single hook for energy theft that are based on the real indicative OV LV BPL suburban case is concerned, the iSHM footprint of the real OV LV BPL topologies with a single energy theft hook is illustrated as superimposed white areas on class maps in Fig. 7 with reference to the iSHM class map of Fig. 1. Similarly to iSHM footprint of Fig. 7, the mSHM footprint of the real OV LV BPL topologies with a single hook for energy theft that are based on the real indicative OV LV BPL suburban case is shown in Fig. 8.  From Figs. 7 and 8, it is evident that the proposed technique concerning the detection of branch line faults through iSHM and mSHM footprints can be easily extended in order to detect hooks for energy theft. Indeed, the insertion of a single hook, which is treated by the DHM as a new branch insertion to the existing real indicative OV LV BPL suburban case, has as result the shift of the iSHM and mSHM footprints right inside the class area of urban case A with few exceptions that remain close to the right border of the class area of suburban case; the later exceptions are real OV LV BPL topologies that derive from the real indicative OV LV BPL suburban case while short hooks for energy thefts are deployed across them. Since the relative location of iSHM and mSHM footprints is more sensitive to the number of long branches rather than the multipath environment aggravation due to the short branches, the detection of the hooks for energy theft becomes easier in comparison with the detection of the branch line faults due to the visible violation of class area borders. Here, it should be mentioned that the detection of long hooks for energy thefts that is offered by the examination of the iSHM footprints may act as supplementary technique to the HS-DET method of [21], [29]- [31] that detects more easily the short hooks for energy theft.

Conclusions
In this second paper, the numerical results concerning the iSHM and mSHM footprints of the OV LV BPL networks have been demonstrated as well as a new technique, which acts supplementary to the existing FIIM and HS-DET method. From iSHM and mSHM footprints of the OV LV BPL topologies it has been verified that: (i) Depending on the number of branches, the real OV LV BPL topologies are classified in the existing class areas; (ii) As the length of the branches of the real OV LV BPL topologies increases, their iSHM footprint tends to be located at the upper right regions of the corresponding class area; (iii) Branch line faults and hook style energy theft can be easily identified by studying the relative location in the corresponding class area given the real OV LV BPL topology of the basis; (iv) Hook style energy theft can be more easily detected in comparison with the branch line faults; and (v) Complimentary to FIIM and HS-DET method, which better operate with short branches, the study of iSHM and mSHM footprints can act as a useful supplementary technique for identifying branch line faults of short branches and hook style energy thefts of long hooks.

CONFLICTS OF INTEREST
The author declares that there is no conflict of interests regarding the publication of this paper.