This paper investigates the efficiency and accuracy of the best
Electrical power industry at the beginning of the
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 IPbased power network supported by a plethora of SG applications [
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 [
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 [
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
The overhead MV and LV distribution lines, which are examined in this paper, are shown in Figures
Typical MTL structures [
The ground is considered as the reference conductor with conductivity
The underground MV and LV lines, which are examined in this paper, are presented in Figures
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 singlecore 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 [
In accordance with [
Endtoend distribution BPL topology with
With reference to Figure
Similar to overhead distribution BPL case, five indicative underground distribution BPL topologies of average path length, which are reported in Table II of [
With reference to Figure
Analytically described in [
In order to describe the propagation of modes across distribution BPL networks, the hybrid method, which is analytically demonstrated in [
In accordance with [
Based on (
Although a plethora of experimental results and derived theoretical analyses validate the theoretical accuracy of the aforementioned modeling approach [
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.
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 [
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
With reference to (
Based on the specifications of Fortran software package of [
As it has already been mentioned in Section
With respect to (
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 flatfading subchannel frequency spacing are assumed to be equal to 1–30 MHz and 1 MHz, respectively. Therefore, the number of subchannels
To simplify the analysis without, however, harming its generality, in the case of overhead and underground MV/BPL networks, only
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
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)
Same as in Figures
In Figures
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)
Same as in Figures
Observing Figures
Already mentioned in [
The presence of branches along the endtoend 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.
Based on the behavior of spectral notches in the coupling transfer functions, distribution BPL topologies can be categorized into three channel classes as follows [
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:
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).
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.
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
Best L1PMA successfully achieves representing either the local extrema or the tail of coupling transfer functions. In contrast with wavelet or spline approximations [
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
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.
From Figures
Optimal number of monotonic sections and PES for distribution BPL networks (OV: overhead; UN: underground).
Urban case A  Urban case B  Suburban case  Rural case  “LOS” case  

Number of monotonic sections 
PES 
Number of monotonic sections 
PES 
Number of monotonic sections 
PES 
Number of monotonic sections 
PES 
Number of monotonic sections 
PES  
OVMV 
WtG^{1}  12  1.2  22  0.9  20  0.7  6  0.7  6  0.7 
WtW 
14  1.3  22  1.0  20  1.5  5  0.3  5  0.5  


OVLV  WtG^{1}  10  1.0  21  0.7  20  0.6  2  1.2  2  1.4 
WtW 
14  1.5  21  1.1  20  1.4  1  0.4  1  0.4  


UNMV  StP^{1}  19  0.9  20  0.7  22  1.0  5  1.1  1  1.1 
PtP 
19  0.7  20  0.7  22  1.1  5  1.1  1  1.0  


UNLV  StP^{1}  16  1.0  20  1.1  18  1.4  1  0.9  1  0.8 
PtP 
16  1.3  20  1.1  18  1.2  1  1.4  1  1.5 
From Table
As the optimal number of monotonic sections is concerned, the following remarks are pointed out:
In 15% of the cases examined, best L1PMA has achieved fitting coupling transfer functions with optimal number of monotonic sections greater than 20.
In 25% of the cases examined, best L1PMA has fitted coupling transfer functions with optimal number of monotonic sections lower than 5.
In general terms, the average PES of best L1PMA is equal to 0.99
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.
As it has already been reported in Section
To examine the impact of faults on the determination of coupling transfer function of distribution BPL networks, in Figures
Theoretical, measured, and best L1PMA coupling transfer function of the five indicative topologies of overhead MV/BPL networks for two indicative fault distributions (CUD and NU) when different coupling schemes are applied. (a)–(e)
Same as in Figures
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)
Same as in Figures
From Figures
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 [
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 [
Indicative processing time of best L1PMA for distribution BPL networks (OV: overhead; UN: underground).
Urban case A  Urban case B  Suburban case  Rural case  “LOS” case  

Processing time (s)  Figure  Processing time (s)  Figure  Processing time (s)  Figure  Processing time (s)  Figure  Processing time (s)  Figure  
OVMV  WtG^{1}  2.230  Figure 
2.477  Figure 
2.291  Figure 
2.407  Figure 
2.569  Figure 
OVLV  WtG^{1}  2.653  Figure 
3.107  Figure 
2.539  Figure 
2.677  Figure 
2.677  Figure 
UNMV  StP^{1}  2.644  Figure 
2.572  Figure 
2.501  Figure 
2.624  Figure 
2.943  Figure 
UNLV  StP^{1}  2.420  Figure 
2.774  Figure 
2.592  Figure 
2.648  Figure 
2.792  Figure 
Apart from the visual result of Figures
PES and

Urban case A  Urban case B  Suburban case  Rural case  “LOS” case  

PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  
OVMV  WtG^{1}  3 dB  11.3  11.3 

8.8  8.8 

13.1  13.1 

32.6  41.5 

36.2  46.0 

6 dB  21.4  22.0 

17.1  17.1 

35.6  35.6 

68.9  93.4 

63.8  101.0 


9 dB 
29.1 
30.1 

25.3 
25.3 

54.5 
54.5 

64.7 
105.7 

127.3 
128.2 
 
WtW 
3 dB  6.7  6.7 

6.3  6.3 

11.5  11.5 

16.5  21.6 

12.9  20.1 


6 dB  18.0  18.0 

13.6  13.6 

22.9  22.9 

19.6  35.5 

41.5  44.2 


9 dB 
25.7 
25.8 

22.5 
22.5 

28.9 
29.1 

45.4 
54.8 

61.4 
65.8 
 


OVLV  WtG^{1}  3 dB  10.7  10.8 

8.0  8.0 

18.8  18.8 

61.6  79.5 

41.2  79.9 

6 dB  25.8  28.3 

13.4  13.4 

41.7  41.7 

43.0  103.9 

102.6  156.1 


9 dB 
36.1 
36.8 

26.1 
26.1 

73.4 
73.4 

136.9 
195.4 

42.2 
200.0 
 
WtW 
3 dB  7.9  7.9 

7.3  7.3 

11.6  11.5 

6.9  25.2 

16.6  21.4 


6 dB  15.6  15.6 

14.1  14.1 

22.6  22.8 

11.4  45.8 

25.9  36.0 


9 dB 
25.6 
25.8 

23.4 
23.4 

31.3 
31.3 

53.4 
68.2 

43.8 
60.3 
 


UNMV  StP^{1}  3 dB  4.9  4.9 

4.2  4.2 

6.8  6.8 

6.7  7.6 

7.0  10.1 

6 dB  10.4  10.4 

8.3  8.3 

13.6  13.6 

11.6  16.7 

14.2  24.1 


9 dB 
15.4 
15.4 

12.4 
12.4 

18.1 
18.1 

25.7 
25.5 

15.3 
27.9 
 
PtP 
3 dB  4.2  4.2 

4.2  4.2 

5.2  5.2 

5.5  6.7 

5.2  6.9 


6 dB  7.8  7.8 

8.7  8.7 

11.0  11.0 

10.9  11.7 

8.6  12.4 


9 dB 
16.2 
16.2 

10.5 
10.5 

13.7 
13.7 

13.6 
14.9 

9.7 
15.7 
 


UNLV  StP^{1}  3 dB  12.2  12.2 

6.8  6.8 

13.0  13.0 

12.9  28.9 

24.4  44.3 

6 dB  22.7  22.7 

13.3  13.3 

33.7  33.7 

24.7  59.6 

58.9  99.8 


9 dB 
26.1 
26.0 

17.3 
17.3 

44.4 
44.4 

47.5 
87.0 

72.3 
148.1 
 
PtP 
3 dB  6.9  7.1 

6.2  6.2 

9.6  9.6 

7.6  11.0 

10.8  15.3 


6 dB  11.6  12.1 

10.1  10.1 

18.9  18.9 

18.8  24.3 

11.9  28.6 


9 dB 
19.5 
20.2 

16.2 
16.2 

28.0 
28.4 

16.3 
38.1 

21.6 
50.0 

PES and

Urban case A  Urban case B  Suburban case  Rural case  “LOS” case  

PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  PES (%) 

Diff. (%)  
OVMV  WtG^{1}  2 dB  11.6  11.6 

10.2  10.2 

18.5  18.5 

36.6  42.0 

43.3  48.6 

6 dB  34.9  35.8 

23.1  23.1 

56.8  56.8 

80.3  102.8 

100.5  119.2 


9 dB 
49.1 
51.0 

41.4 
41.4 

96.5 
96.5 

214.6 
269.3 

147.2 
179.9 
 
WtW 
2 dB  9.0  9.2 

7.4  7.4 

18.0  18.0 

20.9  28.7 

13.3  21.1 


6 dB  26.1  27.3 

25.0  25.0 

43.2  43.2 

49.1  62.9 

60.5  67.1 


9 dB 
37.7 
36.6 
− 
32.6 
32.6 

61.2 
61.2 

63.7 
77.8 

94.2 
102.8 
 


OVLV  WtG^{1}  2 dB  12.4  12.9 

8.0  8.0 

17.2  17.2 

28.7  61.6 

41.4  84.4 

6 dB  34.2  36.7 

27.5  27.5 

48.8  48.8 

75.6  181.5 

111.2  254.3 


9 dB 
54.6 
58.1 

36.5 
36.5 

79.9 
79.9 

232.3 
393.9 

147.1 
353.9 
 
WtW 
2 dB  8.1  8.5 

6.8  6.8 

12.6  12.6 

5.2  20.1 

15.4  26.5 


6 dB  31.7  30.8  − 
19.1  19.1 

36.4  26.8 

14.9  75.9 

27.3  67.3 


9 dB 
40.0 
40.0 

27.4 
27.4 

53.4 
53.4 

48.1 
98.0 

62.9 
118.4 
 


UNMV  StP^{1}  2 dB  4.0  4.0 

5.8  5.9 

6.7  6.7 

8.9  10.7 

5.5  8.8 

6 dB  21.3  21.3 

10.7  10.5  − 
23.0  23.0 

17.4  22.5 

12.2  25.1 


9 dB 
23.8 
23.8 

23.2 
23.2 

35.4 
35.4 

24.2 
34.4 

25.1 
38.1 
 
PtP 
2 dB  4.1  4.1 

3.9  3.9 

5.6  5.6 

7.2  7.2 

4.6  7.1 


6 dB  18.4  18.4 

13.1  12.9  − 
13.2  13.2 

15.3  16.0 

14.9  20.1 


9 dB 
21.3 
21.3 

19.5 
19.5 

20.5 
20.5 

30.3 
34.3 

13.1 
29.0 
 


UNLV  StP^{1}  2 dB  10.2  10.6 

9.0  9.0 

18.0  18.0 

15.7  29.0 

23.3  59.6 

6 dB  30.2  30.6 

20.4  20.4 

44.0  44.0 

29.0  74.8 

25.5  105.1 


9 dB 
58.8 
62.2 

31.1 
31.6 

69.4 
66.4 
− 
66.9 
150.1 

58.7 
224.0 
 
PtP 
2 dB  8.4  8.4 

5.6  5.6 

9.0  9.0 

7.6  13.4 

7.9  14.6 


6 dB  20.5  20.6 

15.8  15.8 

29.1  29.1 

16.4  42.3 

19.1  35.5 


9 dB 
34.4 
34.4 

23.6 
23.6 

46.2 
46.5 

32.2 
71.2 

25.5 
70.2 

In Tables
From Tables
Further analyzing the data of Tables
Now, as concerns the behavior of best L1PMA against gross errors, best L1PMA is characterized by its remarkable property to ignore the presence as well as the magnitude of these gross errors during the determination of the best fit in all the cases examined. More specifically, best L1PMA achieves average PES improvement equal to 13.8% and 26.4% when the gross faults that occurred follow CUD (i.e.,
Synoptically, it is evident that the fault magnitude affects the approximation efficacy of best L1PMA to the coupling transfer functions of distribution BPL networks. In order to highlight this factor influence, PES is plotted versus the maximum value
PES of distribution BPL topologies when various coupling schemes are applied and different fault CUDs (variable maximum value
Similar to Figures
PES of distribution BPL topologies when various coupling schemes are applied and different fault NDs (mean
From Figures
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 to the fact that even if coupling transfer functions of rural and “LOS” topologies maintain relatively stable inclination due to their “LOS” attenuation, a plethora of faults can abrupt the best fit of best L1PMA by adding a fixed difference between theoretical and approximated coupling transfer functions across the examined frequency band. Consequently, the high optimal number of monotonic sections locally isolates the effect of the faults that occurred resulting in a biased best fit result.
Moreover, comparing the PES performance among different coupling schemes, WtW and PtP coupling schemes present lower PES than the respective WtG and StP ones. This is due to the fact that WtW and PtP coupling transfer functions present lower attenuation values in comparison with the respective WtG and StP ones further affecting their PES performance.
Concluding the analysis of Section
Among the wide range of applications that best L1PMA is relevant to, an application of best L1PMA to approximate and to mitigate faults during the determination of transfer functions of distribution BPL networks has been proposed in this paper. To assess the performance of best L1PMA, PES metric has been used allowing (i) the evaluation of the approximation accuracy of the best L1PMA to the theoretical coupling transfer functions and (ii) the evaluation of the restoration of the theoretical coupling transfer functions when faults occur.
Despite the large number and the extent of the notches (local extrema) due to the multipath environment, the steady inclination due to the “LOS” attenuation, and the burst mode of the assumed faults during the measurements, best L1PMA has successfully mitigated all the previous deficiencies in all the examined distribution BPL networks. More analytically, in the case of transfer functions of distribution BPL networks with no faults, best L1PMA has achieved average PES equal to
Taking into consideration the diverse nature of distribution BPL topologies and the performance of best L1PMA as a fault counteracting technique, best L1PMA can operate as the necessary intermediate antifault method between the measurement phase and the theoretical data processing.
Mediumvoltage
Lowvoltage
Smart grid
Broadband over power lines
Internet protocol
Multiconductor transmission line
Best L1 piecewise monotonic data approximation
Percent error sum
Aluminium conductor steel reinforced
Paperinsulated lead cable
Crosslinked polyethylene
Line of Sight
Transmission line
Continuous uniform distribution
Normal distribution.
The author declares that there are no competing interests.