HSOFDMbased TimeDomain Reflectometry for Power Line Sensing: Characteristics and Limitations
Abstract
This study discusses key characteristics and limitations of timedomain reflectometry (TDR) systems based on the Hermitian symmetric orthogonal frequencydivision multiplexing (HSOFDM) scheme for power line sensing. In this sense, a system model with a power line modem injecting signals and capturing raising reflections for sensing a power distribution network is outlined. Next, pulse compression and channel estimation reflectogram processing approaches are carefully described and the effects of system parametrization and multiple access schemes on the HSOFDMbased TDR system performance are addressed. Finally, numerical results covering a comparison between pulse compression and channel estimation, system limitations based on parametrization considering typical European underground lowvoltage and US overhead mediumvoltage (MV) scenarios and narrowband (NB) power line communication (PLC) regulatory constraints, and comparison among multiple access techniques in a Brazilian MV overhead scenario are presented for supporting the carried out discussion. Based on the attained results, it is shown that channel estimation outperforms the pulse compression in terms of computational complexity and sidelobe level. Also, it is shown that the NBPLC frequency range provides fair range resolution and maximum unambiguous range values. Finally, it is seen that the use of the frequencydivision multiple access multiple access schemes presents different signaltointerferenceplusnoise ratio (SINR) performance among different power line modems (PLMs) connected to a power distribution grid, while the use of timedivision multiple access and codedivision multiple access schemes results in fair SINR performance among the PLMs at the cost of obtaining less reflectograms over time due to time multiplexing and spreading processes, respectively.
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HSOFDMbased TimeDomain Reflectometry for Power Line Sensing: Characteristics and Limitations
Lucas Giroto de Oliveira, Student Member, IEEE, Mateus de L. Filomeno,
and Moisés V. Ribeiro, Senior Member, IEEE
^{0}^{0}footnotetext: Manuscript received MM DD, YYYY; revised MM DD, YYYY. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior  Brasil (CAPES)  Finance Code 001, Copel Distribuição LTD  PD 28660420/2015 , CNPq, FAPEMIG, INERGE and Smarti9 LTD.^{0}^{0}footnotetext: Lucas Giroto de Oliveira and Mateus de L. Filomeno are with the Electrical Engineering Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora, Brazil (email: lgiroto@ieee.org, mateus.lima@engenharia.ufjf.br).^{0}^{0}footnotetext: Moisés V. Ribeiro is with the Electrical Engineering Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora, Brazil, and Smarti9 Ltd, Brazil (email: mribeiro@ieee.org).
Index Terms
Timedomain reflectometry, orthogonal frequencydivision multiplexing, multiple access, power line communication.
I Introduction
Today, electrical energy is a key resource in the entire world. The need to connect power generating units to final consumers has led to a significant expansion of power transmission and distribution infrastructures. Such expansion implies in greater subjectivity of the latter to faults along with both infrastructures. In order to minimize the interruption time as well as the damages resulting from faults, several protection schemes have been developed.
Conventional protection schemes for power line networks, however, fail in tasks such as detection and location of hard faults, e.g. high impedance faults or distant low impedance faults, and soft faults, eg. cable degradation. Among more efficient alternatives are analysis harmonic currents and voltages [1, 2], and the use of traveling waves for detecting and locating impedance discontinuities along the network. The latter, which has long been subject of study for sensing of wired networks in scenarios such as aircraft [3], has been gaining attention for sensing of power lines. In this context, the adaptation of power line modems for network sensing [4] as well as techniques that exploit characteristics of traveling waves propagating back and forward the power line for applications such as topology inference [5] or fault detection and location [6] have been investigated.
Potential techniques for wired network sensing based on traveling waves are impedance spectroscopy, transferometry, and reflectometry [3, 7]. Variations of the latter are timedomain reflectometry (TDR), frequencydomain reflectometry (FDR) [8], and joint timefrequency domain reflectometry (JTFDR) [9], which consist of injecting signals into the analyzed network and capturing the raised reflections, which are respectively processed in time, frequency, or both domains. The use of TDR principles for fault sensing in power lines, subject of study in the past [10, 11], has been recently revisited in the literature [12] due to low computational complexity associated with the postprocessing of obtained reflectograms. Such aspect of TDRbased techniques allows online obtaining of reflectograms [13], therefore enabling efficient sensing of power line networks at both symbol and mains levels [4].
Effectiveness in terms of digital signal processing and electromagnetic compatibility (EMC) without the inherent drawback of poor spectrum control in classical TDR techniques can be achieved by multicarrier timedomain reflectometry (MCTDR) [14, 15] and its timedomain version [16], which only differ in the domain where reflectogram processing takes place. For enhanced compatibility with existing power line communication (PLC) systems an optimized digital signal processing, an interesting approach is the orthogonal multitone timedomain reflectometry (OMTDR) [13], which is variation of the MCTDR based on orthogonal frequencydivision multiplexing (OFDM).
Although TDR based on OFDM or its baseband version Hermitian symmetric OFDM (HSOFDM) seems a good candidate for power line sensing, aspects such as measurement range and number of reflectograms obtained over time become more relevant due to the long distances in power distribution networks and the linear periodically time variant (LPTV) behavior of transfer functions experienced by signals injected by PLMs into the network. To the best of the author’s knowledge, the use of channel estimation, performed for communication purposes in PLC systems [17], rather than the usual pulse compression performed for reflectometric sensing in wired networks [18, 13] has also not been addressed in the literature. For an efficient distributed sensing, the effect of multiple access on TDR measurements, already considered for pulse compression TDR in some aspects [19, 13], must also be taken into account for HSOFDMbased TDR systems relying on channel estimation. Given this context, the present study investigates the influences of the parametrization of an HSOFDMbased TDR system as well as limitations imposed by multiple access schemes on its performance for power line sensing.
The main contributions of this study are summarized as follows.

Description of an HSOFDMbased TDR system sensing a power distribution network. Based on this formulation, we discuss the pulse compression and channel estimation reflectogram processing approaches and analyze the effect of the TDR system parametrization on its performance using peaktosidelobe level ratio (PSLR), integratedsidelobe level ratio (ISLR), range resolution, and maximum unambiguous range as metrics taking into account regulatory constraints and considering typical European underground low voltage (LV) and US overhead medium voltage (MV) scenarios.

Introduction of timedivision multiple access (TDMA), frequencydivision multiple access (FDMA), and codedivision multiple access (CDMA) multiple access schemes for an HSOFDMbased TDR system for power line sensing based on the channel estimation procedure and comparative analysis among them in terms of number of obtained reflectograms and signaltointerferenceplusnoise ratio (SINR) considering a Brazilian overhead MV scenario.
Our major findings are as follows:

Channel estimation outperforms the pulse compression procedure in terms of both computational complexity and sidelobe level, being therefore a more attractive reflectogram processing approach for HSOFDMbased TDR systems.

HSOFDMbased TDR systems operating in the narrowband (NB)PLC frequency range are suitable for the sensing of LV and MV power distribution networks sections, offering fair range resolution and maximum unambiguous range values.

The use of TDMA and CDMA multiple access schemes results in a smaller number of obtained reflectograms over time respectively due to time multiplexing and spreading processes and in a fair SINR level among the multiple PLMs connected to the power distribution network, being the SINR higher in the CDMA case due to signaltonoise ratio (SNR) gain provided by this scheme. On the other hand, the use of FDMA results in a higher number of obtained reflectograms over time in comparison with TDMA and CDMA and in higher SINR to PLMs associated with subcarriers in higher frequnecy bins due to the exponentially decreasing additive noise power spectral density (PSD).
The remainder of this paper is organized as follows. Section II describes the HSOFDMbased TDR system for sensing of a power distribution network. Section III discusses the pulse compression and channel estimation procedures for reflectogram processing. Next, Section IV addresses the effect of system parametrization on its performance and Section V discusses the use of multiple access schemes for power line sensing with multiple PLMs. A numerical analysis for supporting the carried out discussion is carried out in Section VI. Finally, concluding remarks are placed in Section VII.
Notation
Throughout the paper, and indicate the transpose and Hermitian transpose operators, respectively; is the convolution operator; and denote the Hadamard product and division, which respectively perform elementwise multiplication and division of two equalsized matrices; represents the expectation operator; the size discrete Fourier transform (DFT) matrix is denoted by ; and denote size matrices respectively composed of zeros and ones.
Ii System Model
Let a baseband TDR system be consisted by a fullduplex PLM connected to a power distribution network, which, at a single given point, injects signals and captures reflections that travel at a phase velocity . Assuming that the injection and subsequent capture of reflections of signals takes place within a coherence time , in which variations in loads or any other element of the network are irrelevant, one can consider the power distribution network as \@iaciLTI linear timeinvariant (LTI) system. The reflections captured by the PLM from such power distribution network, which are raised by impedance discontinuities along the path traveled by the injected signal, are therefore the output of a reflection channel with impulse response .
Such reflections are quantified via the input reflection coefficient between the PLM of output impedance denoted in the continuous frequencydomain as and the power distribution network of input impedance , being expressed as [20]
(1) 
From , one finally obtains the impulse response of the reflection channel via the inverse Fourier Transform, i.e.,
(2) 
The reflectometric sensing of the power distribution network can therefore be performed via an analysis of the reflections raised by the impedance discontinuities. For this purpose, the considered TDR system is used for obtaining a reflectogram, which is an estimate of the reflection channel impulse response. In this paper, it is assumed that such procedure is performed by an HSOFDM system that is bandlimited to a bandwidth and has sampling frequency .
The considered HSOFDMbased TDR system is depicted in Fig. 1, starting with a complex vector , such that . This vector is inputted to the function that represents, in a condensed form, the digital processing performed at the transmitter side, which is composed by three processing stages. The first one is the Hermitian symmetric mapping [21] that transforms into the length discretefrequency domain HSOFDM symbol such that . Next, the inverse discrete Fourier transform (IDFT) transforms into the discretetime domain vector , which is realvalued due to the Hermitian symmetry of . This is expressed as . Finally, an length cyclic prefix is appended to , resulting in the vector . Considering that the reflection channel impulse response has an length discretetime domain representation , no intersymbol interference (ISI) is experienced if the constraint is satisfied.
Back to Fig. 1, the discretetime vector undergoes an digitaltoanalog conversion, being converted into the continuoustime signal that is inputted to the reflection channel of impulse response . It is worth highlighting that, in order for the LTI assumption of the reflection channel to hold, the duration of , i.e., the HSOFDM symbol in the continuoustime domain, must satisfy the constraint . The first term of the previous expression accounts for the number of samples of the discretetime vector , while is the sampling period. The resulting signal from the convolution between the transmit signal and the channel is . To this signal is added the noise , which is a zeromean widesense stationary (WSS) random process, resulting in the received signal . Note that, due to the noise presence, is also a WSS random process.
At the receiver side, the signal passes through an analogtodigital converter, originating the discretetime domain vector . Next, is inputted to the function , which synthesizes the digital processing at the receiver side of the HSOFDMbased TDR system. The performed processing by this function starts with cyclic prefix removal from , which originates the discretetime domain vector .
The operations ranging from the transmitter processing on to ultimately obtain can be equivalently expressed in the discretefrequency domain as
(3) 
where , , is the discretefrequency domain representation of ; , , is the length zeropadded version of the length discretefrequency equivalent of the reflection channel impulse response ; and , is a vector composed by proper Gaussian random variables and is a length discretetime domain window of the addtive noise .
Alternatively, the vector can be represented as . Each these variables have equal mean , , and different variances , being uncorrelated in the frequency domain, i.e., for and such that , . Assuming that the twosided additive noise PSD is flat within each subband, one can represent it as the vector , , where is the additive noise PSD at the subchannel and is the subcarrier frequency spacing/bandwidth.
The processing performed by is completed with the obtaining of an length reflectogram that is denoted in the discretefrequency domain as , . This is performed via processing on the discretefrequency domain vector , in such a way that the influence of the transmit vector on it is minimized or eliminated. A proper reflectogram must be a good estimate of the reflection channel frequency response and present a high SNR in order for noise effect to be negligible. The processing performed by is finally completed with a IDFT on , resulting in the discretetime domain reflectogram that is actually used for analyzing the reflections along the network.
After a reflectogram has been obtained, analog counterpart can be yielded in order to reduce the temporal granularity and, as a consequence, the spatial granularity of the reflectogram and the impedance discontinuity location accuracy. This can be performed via sinc interpolation in the time domain by a reconstruction filter [22]. Alternatively, zeropadding can be performed on the discretefrequency domain vector before transforming it to the time domain [23]. Both these alternatives yield valid reflectograms under the condition that the signal has been sampled at sampling rate of at least . A third alternative would be oversampling at a sampling frequency [13], which would ultimately yield a reflectogram , , , with finer temporal and spatial granularities for higher values. Although this is a valid alternative, it would yield significantly higher computational complexity, as all processing stages at the receiver would be performed on longer vectors.
The quality of the obtained reflectogram will ultimately depend not only on the SNR level, but also on the processing performed on . Given this context, Section III carries out a careful description of two approaches for the obtaining of the reflectogram from the received vector performed by .
Iii Reflectogram Processing Approaches
A conventional procedure for obtaining reflectograms in TDR systems is the pulse compression, which is based on the correlation between the transmit and received signals without cyclic prefix, i.e., and , and has already been addressed in the literature [13, 24]. An alternative processing, usually resorted to in radar systems, consists of obtaining a reflectogram that is an adequate estimate of by removing the effect of the transmit signal on via processing in the frequency domain [25]. These two approaches are described in Subsections IIIA and IIIB, respectively.
Iiia Pulse compression
The pulse compression consists of correlating the received vector with a copy of the transmit vector so that the obtained reflectogram is equivalent to the one obtained via the transmission of a narrower pulse, which corresponds to the autocorrelation of [8, 13, 26].
Such procedure is performed via a linear convolution between the received vector and a matched filter to , denoted by . This linear convolution can be implemented by means of a circular convolution between length zeropadded versions of and , demanding computational complexity [27, 22]. Alternatively, it can be implemented in the discretefrequency domain via multiplication of the length zeropadded versions of and , , preceded by an DFT and followed by an IDFT [27, 24]. If fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) are used, this approach becomes more efficient and faster than the circular convolution in the discretetime domain [27], imposing the constraint that the zeropadded versions of the aforementioned vectors present length and demanding a reduced computational complexity of .
The length zeropadded versions of and are, respecively, , , and , . The resulting reflectogram can therefore be represented in the discretefrequency domain as
(4)  
where presents length , i.e., ; , , represents the autocorrelation of in the discretefrequency domain; , , is the length zeropadded version of ; , , is the length zeropadded version of ; and , , is the length zeropadded version of the length noise discretetime domain window in the discretefrequency domain.
A closer examination of (4) reveals that the obtained reflectogram is biased by the autocorrelation of the transmit signal . The resulting reconstructed analog reflectogram obtained by the pulsecompression TDR system will be therefore equivalent to a reflectogram obtained by a conventional TDR system that performs injection of a pulse into the network, being the continuoustime domain reconstructed version of the discretefrequency domain autocorrelation of , i.e., . If presents high PSLR and ISLR values [28, 26], the reflectogram will present significant clutter and the performance of the fault detection and location procedures will therefore be compromised. Such distortion can be minimized by signal processing techniques such as windowing, which reduces the level of sidelobes at the cost of main lobe broadening and consequent resolution loss [29, 30, 13], or designing pulses with desired autocorrelations. The latter approach, however, may limit the use of the HSOFDM signal to reflectometric sensing, possibly inhibiting its use for communication purposes.
IiiB Channel estimation
In this approach, the reflectogram is obtained by performing a DFT on , followed by a elementwise division of the resulting vector by and an IDFT [25]. The resulting operation performed by is therefore equivalent to a channel estimation procedure [17]. The obtained reflectogram can be expressed in the discretefrequency domain as
(5)  
with presenting length , i.e., .
Finally, the discretetime domain reflectogram is then simply obtained via a IDFT on . If DFT and IDFT are performed by their fast counterparts, i.e., FFT and IFFT, the computational complexity associated with this procedure is due to the two aforementioned stages plus the Hadamard division, being therefore significantly smaller than in the pulse compression case.
Unlike the pulse compression case, the reflectogram from (5) is unbiased. The only factors limiting the reflectogram quality will therefore be SNR level and system bandwidth. The band limiting of the HSOFDMbased TDR system to a bandwidth results in a reconstructed analog reflectogram will be equivalent to the one obtained by the injection of , , into the network by a conventional TDR system, which is due to the resulting flat, unitary spectral content from the Hadamard division that undergoes Hadamard product with in (5). Although clutter level may not be as relevant as in the pulse compression case, it can still be reduced by windowing [31].
Iv System Parametrization and Limitations
For the system described in Section II, which provides a reflectogram via pulse compression and channel equalization approaches described in Section III, a proper parametrization must be performed in order for the resulting reflectogram to be representative of the network, besides achieving desired maximum unambiguous range and range resolution values. This is mainly done by adopting HSOFDM symbol length , cyclic prefix length , and occupied frequency bandwidth values that satisfy the aforementioned constraints, as described in the following subsections.
Iva Channel coherence bandwidth
In order for to be appropriately reconstructed from , a further constraint must be satisfied besides the one related to the channel coherence time mentioned in Section II. This constraint is that the frequency resolution, or subcarrier frequency spacing/bandwidth, must be smaller than the reflection channel coherence bandwidth , within which the reflection channel frequency response can be considered flat. The reflection channel coherence bandwidth is expressed by [32]
(6) 
where is a threshold and is the frequency correlation function expressed by
(7) 
The frequency resolution, i.e., subcarrier frequency spacing/bandwidth, must therefore be set such that . As and , it holds the constraint
(8) 
IvB Maximum unambiguous range
The maximum unambiguous range is limited by the length of the received vector [33]. This is due to the fact that, in both pulse compression and channel equalization approaches, the reflectogram originates from the removal or minimization of the effect of on . After this is done, the information on the reflection channel impulse response will be contained in a length vector. One must therefore have in order for no ambiguities to occur in the reflectogram. This holds even for the pulse compression case, where the reflectogram length is only longer due to zero padding.
Additionally, if the cyclic prefix is not long enough, there may be ambiguities in the reflectogram due to ISI [33]. In order to avoid such effect, the cyclic prefix length must satisfy . Due to the aforementioned limitations, the maximum unambiguous range in both pulse compression and channel equalization approaches is
(9) 
i.e., the minimum length between and , multiplied by the sampling period and the phase velocity and divided by in order to account for the round trip time of the reflections [26]. Therefore, for a given maximum unambiguous range , one has the constraints
(10) 
and
(11) 
IvC Range resolution
V Multiple Access Schemes
If PLMs are to operate over a single power distribution network, therefore constituting a distributed HSOFDMbased TDR system, multiple access schemes must be adopted. The most usual ones are TDMA, FDMA, and CDMA [34, 33].
For the pulse compression processing approach, TDMA, FDMA, and CDMA schemes have already been addressed in the literature [34, 19]. Out of this reason and due to the evident advantages of the channel estimation procedure over the pulse compression pointed out in Section III, the discussion on multiple access schemes for an HSOFDMbased TDR system presented in the following subsections will be focused on the channel estimation procedure.
Va Timedivision multiple access
TDMA is a widely used multipleinput multipleoutput (MIMO) scheme in applications such as radar [33] and communications [35]. For PLMs transmitting and receiving length HSOFDM symbols, each one of them accessing the same power distribution network in different time slots, there will be no interference among reflectograms of the different PLMs. Furthermore, each of these reflectograms will have the same maximum unambiguous range and range resolution exactly as defined in Section IV, being also subject to the channel coherence bandwidth constraint.
In this scheme, there is a number of
(13) 
obtained measurements per PLM per second. If intended, the PLMs that are not transmitting in the current time slot can also use the received signal from the transmitting PLM for obtaining a transferogram (i.e., channel impulse response between two PLMs), which can be used for data fusion among the PLMs [7]. This would result in a total number of measurements per PLM per second, which is constituted by reflectograms and transferograms.
VB Frequencydivision multiple access
In the case where different PLMs are to transmit simultaneously, an alternative is the use of the FDMA principle. For better exploiting information on the network, an interleaved FDMA scheme can be used rather than a localized one that allocated adjacent subcarriers to each PLM [36, 13]. In this approach, interleaved subcarriers are allocated to each PLM in an HSOFDMbased TDR system with HSOFDM symbol length . The resulting reflectogram for each PLM is therefore obtained by discarding the subcarriers allocated to the remaining PLMs and, after obtaining the discretefrequency domain reflectogram , performing a size IDFT on it in order to obtain the discretetime domain reflectogram . Finally, the obtained length reflectogram undergoes a reconstruction process that will originate as described in Section II.
In the considered Hermitian symmetric orthogonal frequencydivision multiple access (HSOFDMA) scheme, the vector , , from Section II is generated via a premapping on a vector , , whose sample will be ultimately transmitted at the continuousfrequency bin Hz after all processing and digitaltoanalog conversion has been performed at the transmitter side. This premapping is expressed as
(14) 
where . In order for interferencefree operation of the PLMs to take place, a set , , of interleaved samples of are allocated to the PLM. Also, all PLMs place null values at the samples that have not been allocated to it.
Once the discretefrequency domain reflectogram has been obtained at the receiver side, the PLM performs a channel estimation processing on the corresponding samples to the frequency bins allocated to it. Next, is transformed into a modified reflectogram according to the mapping
(15) 
in which is the sample of the modified reflectogram , . Such mapping is performed in order to map a length Hermitian symmertric vector into the length vector with no loss of spectral content.
Finally, the PLM obtains an length discretetime domain reflectogram , , , whose sample is expressed as
(16)  
which, for the singlePLM case, is exactly the same result obtained by performing an IDFT on .
The obtained reflectograms in the proposed HSOFDMA scheme will have maximum unambiguous range experienced by each PLM equal to . The range resolution for each PLM in its turn will be , with being the effective bandwidth experienced by each PLM. As subcarriers are allocated to each PLM, the effective subcarrier separation experienced by the latter will be . Consequently, , and it holds that the range resolution is exactly as defined in Section IV for the singleuser case, i.e., . Although the range resolution is maintained, the information on the network contained in the subcarriers discarded by the PLM will cause distortion on the obtained reflectograms. The tradeoff between number of PLMs connected to the network and reflectogram quality must therefore be observed in order for fault detection and location procedures not to be negatively affected.
As a consequence, one obtains a number of
(17) 
reflectograms per PLM per second. If transferograms are also to be obtained, one would have a total number of measurements per PLM per second, encompassing reflectograms and transferograms.
VC Codedivision multiple access
Unlike the TDMA and FDMA schemes, the use of CDMA allows the PLMs to occupy the entire bandwidth over all time slots [19, 34, 33]. The implementation of this scheme consists of adding an encoding process on the discretefrequency domain vector .
In this context, let us consider an size Hadamard matrix whose rows are constituted by codewords that are orthogonal among each other. The codeword of the row of this matrix, , is represented by the vector , , which is in its turn allocated to the PLM. Due to the orthogonality among the codewords, the mutual interference among the PLMs is minimized.
The encoding process performed by the PLM is then performed via a multiplication between the codeword vector by the transpose counterpart of , resulting in the spread symbol matrix
(18) 
for which holds . The rows of this matrix are then sequentially fed to the Hermitian symmetric mapping and further processing, being finally transmitted by the PLM. Thus, the resulting transmission time is times longer than the seconds of the singlePLM case.
After cyclic prefix removal and DFT are performed on the received symbols at the receiver side, a reception matrix is formed, being its rows associated with the rows of .
(19) 
where is a transmission matrix whose rows result from the Hermitian symmetric mapping on the respective rows of ; is an equivalent reflection channel matrix, expressed as ; is a noise matrix whose rows are length discretefrequency domain windows of the additive noise ; and is the interference noise matrix, which accounts for the mutual interference among the PLMs. The PLM then performs a decoding process on , which generates the vector and is expressed as
(20) 
where is the cardinality of the codeword vector . The discretefrequency domain vector can be alternatively expressed as
(21) 
in which is the resulting additive noise from the decoding process, whose PSD is expressed as , , with representing the resulting additive noise PSD at the subchannel; and is the resulting interference noise. The fact that the resulting additive noise has variance and therefore PSD times smaller than the ones of the original additive noise is due to the averaging process that happens at the decoding stage of the CDMA scheme in (20) and ultimately results in an SNR gain [19].
After this stage, a channel estimation procedure is performed on as described in Subsection IIIB for the singlePLM case. The resulting discretefrequency domain reflectogram finally undergoes an IDFT, originating the discretetime domain reflectogram . Although the effect of the transmit symbol on the reflectogram is properly removed, the use of CDMA introduces mutual interference among the PLMs, whose reflectograms consist of the reflections raised by the injected signal by the own PLM plus a term that accounts for the decoded symbols transmitted by the remaining PLMs as well as additive noise. Despite being significantly attenuated by the decoding process, this interference term might distort the obtained reflectogram [33].
In spite of the undesired interference, the obtained reflectograms in the CDMA scheme yield the same maximum unambiguous range and range resolution as in the singleuser case from Section IV, whose channel coherence bandwidth constraint still holds. Due to the extended transmission time, a number
(22) 
of reflectograms are obtained per PLM per second. In the case where transferograms are also obtained, the total number of measurements per PLM per second would be measurements, which encompasses reflectograms and transferograms.
Vi Numerical Analysis
In this section, the carried out discussion on reflectogram processing approaches and multiple access schemes throughout the paper is validated via numerical results. Given this context, Subsection VIA compares the pulse compression and channel estimation procedures, while Subsection VIB addresses system limitations due to regulatory constraints. Finally, the discussed multiple access schemes are analyzed in Subsection VIC.
Via Pulse compression and channel estimation comparison
In order to corroborate with the claim that the channel estimation procedure outperforms the classical pulse compression for TDR, Fig. 2 presents a comparison of both approaches in terms of computational complexity. Based on the expressions provided in Subsections IIIA and IIIB, a far higher computational complexity is demanded by the pulse compression approach as it is performed on a longer vector.
Furthermore, the claim that the quality of obtained reflectograms is higher in the channel estimation procedure is endorsed by the results from Figs. 3 and 4. In these figures, PSLR and ISLR values are respectively shown for the equivalent transmit pulse obtained via pulse compression and the equivalent transmit pulse for the channel estimation procedure. In this figure, typical modulations schemes for PLC systems, namely binary phaseshift keying (BPSK), quadrature phaseshift keying (QPSK), and eight phaseshift keying (8PSK) [37], are considered for the pulse compression approach, being the results for the channel estimation approach independent of the adopted scheme.
The attained results show that the PSLR decreases with the modulation order for the pulse compression approach. As increases, the difference among the PSLR values yielded by either pulse compression or channel estimation becomes negligible. Finally, it is observed that PSLR levels decrease inversely along with .
Regarding ISLR, the attained values also decrease with the modulation order for the pulse compression procedure, being the difference between QPSK and 8PSK more subtle. The best ISLR values are attained by the channel estimation procedures, which significantly outperforms the pulse compression procedure by about dB if QPSK and 8PSK are assumed for the latter, and by dB if pulse compression is performed on a signal belonging to a BPSK modulation. Unlike the PSLR case, there is a slight increase of ISLR values along with . This is explained by the fact that, given a fixed , the side lobe level is increased, while the main lobe duration of seconds and therefore the main lobe level are maintained for longer HSOFDM symbols.
ViB System limitations
Although an adequate parametrization of the HSOFDMbased TDR system can yield valid reflectograms, regulatory constraints limit the PSLR and ISLR values, as well as achievable range resolution and maximum unambiguous range. The latter two parameters are also influenced by characteristics of the cable where the injected signal propagates, which define the phase velocity . An awareness of such limitations is therefore paramount for performing adequate sensing of the power distribution network.
In this context, two scenarios are addressed, namely an European underground lowvoltage power distribution network and an US overhead mediumvoltage power distribution network in a rural area. For the lowvoltage scenario, it is considered a power supply cable NAYY150SE with resistance , inductance , conductance , and capacitance per unit length calculated as in [38], whereas for the mediumvoltage cable, the power supply cable with distributed parameters listed in [37] is adopted. Based on these parameters, the phase velocity is calculated by [20], resulting in for the considered LV cable, and for the considered MV cable.
Fig. 5 shows the range resolution as a function of the occupied frequency bandwidth for the considered LV and MV scenarios. The achieved values range from tens of thousands of kilometers for low values to a few meters for higher values, with a ratio of between the resolution in the MV and LV and scenarios due to their different phase velocity. The presented results indicate that values in the NBPLC frequency range, i.e., kHz, result in fair range resolution values, i.e., m, and therefore a fair capability of resolving close impedance discontinuities for typical distances covered by PLC signaling in LV and MV power distribution networks, which are about km in MV scenarios and shorter in LV scenarios [4, 39].
For typical PLC systems, one observes . As a consequence, only limits according to the relation in (9). Based on this assumption and considering typical sampling frequency values for NBPLC, namely MHz and MHz, Fig. 6 shows the maximum unambiguous range as a function of the cyclic prefix length in the considered LV and MV scenarios. In this figure, one observes a minimum value of about km for , which is experienced with MHz in the LV scenario. For , it holds km for both considered sampling frequencies in both LV and MV scenarios. As the typical spacing between PLMs in a power distribution network is considerably shorter than that, these results show that typical cyclic prefix lengths for NBPLC systems, i.e., and , are enough for providing fair maximum unambiguous range values.
Focusing on regulatory constraints, the considered regulations in this paper are Federal Communications Commission (FCC), Association of Radio Industries and Businesses (ARIB), and European Committee for Electrotechnical Standardization (Comité Europén de Normalisation Électrotechnique in French) (CENELEC), which are addressed in the IEEE 1901.2 Standard [37] for NBPLC. The frequency range, occupied frequency bandwidth , sampling frequency , FFT/IFFT size, range and number of active subcarriers , and cyclic prefix length associated with these regulations are listed in Table I. In this table, the a number of ative subcarriers is adopted for both FCC and ARIB regulations for covering the whole frequency band covered by these regulations described in [37]. Also, the whole CENELEC band is considered, i.e., from the lowest frequency of the CENELEC A band to the highest frequency of the CENELEC D band, being the adopted number of active subcarriers equal to the necessary for covering this whole frequency bandwidth. For HSOFDMbased TDR systems parametrized according to these three regulations, Table II lists the achieved PSLR and ISLR values, as well as range resolution and maximum unambiguous range for the considered LV and MV scenarios. Based on the data of this table, one can conclude that FCC and ARIB regulations allow a more appropriate sensing of shorter power distribution network sections than CENELEC, providing finer range resolution and shorter maximum unambiguous range. Also, adopting the long of both FCC and ARIB regulations results in a significant increase of at the cost of obtaining less reflectograms over time due to the transmission of more cyclic prefix samples.
Regulation  FCC  ARIB  CENELEC  

Frequency range (kHz)  
(kHz)  
(MHz)  
FFT/IFFT size  
Active subcarriers  
Standard  
Long   
ViC Channel estimation with multiple access
For a comparison among the three multiple access schemes described in Section V that allow the simultaneous obtaining of reflectograms by multiple PLMs connected to the same power distribution network, an HSOFDMbased TDR system parametrized according to the FCC regulation is considered in this subsection. Additionally, BPSK modulation scheme was adopted.
In this context, simulations were carried out based on an multiconductor transmission line (MTL)based model [20, 40] of a real MV power distribution network section in the city of Curitiba, Brazil, depicted in Fig. 7. This section is consisted by a feeder, which is an high voltage (HV)/MV transformer, followed by km of line and a delta load. The input impedance of the HV/MV transformer at the MV side, as well as the input impedance of the network after the delta load are assumed to be much higher than the characteristic impedance of the line, thus behaving as an open circuit. Two PLMs are connected to this network right after the feeder, being one connected between phases A and B and the other between phases C and B. After an km line section, other PLMs are connected to the network between the same pairs of phases. Thus, there is a line section with length of km between the latter PLMs and the delta load. Regarding the additive noise, the adopted model is the one reported in [41, 42], which presents onesided PSD in the continuousfrequency domain equal to dBm/Hz.
Regulation  FCC  ARIB  CENELEC  

(m)  LV  
MV  
(km)  Std.  LV  
MV  
Long  LV    
MV    
Sidelobe level  PSLR (dB)  
ISLR (dB) 
The number of reflectograms obtained over time , which varies due to the different procedures adopted by the multiple access schemes for providing orthogonality among the signals of the multiple PLMs, is shown in Fig. 8 as a function of the number of PLMs for both standard and short cyclic prefix lengths. In this figure, one observes decreasing values along with for both TDMA and CDMA schemes, while the FDMA scheme presents constant regardless of the number of PLMs. This happens due to the long time interval during which a PLM does not transmit signals in the TDMA scheme. The lower values in the CDMA scheme are on the other hand due to the spreading of HSOFDM symbols, which results in longer effective transmission time for HSOFDM symbols in this scheme. It is worth highlighting that, for a given , the number of obtained reflectograms is the same for both TDMA and CDMA schemes.
As previously mentioned, the use of FDMA results in a fixed . This is shown in Fig. 8 based on the assumption that , i.e., the number of subcarriers is not less than the number of PLMs. Nevertheless, the obtained reflectograms by each PLM will be distorted if the number of subcarriers ( in the FCC case) assigned to each of them does not satisfy both the reflection channel coherence bandwidth constraint from (8) and the maximum unambiguous range constraint from (10), being the latter only relevant if due to the relationship from (9).
A further analysis is a comparison of the TDMA, FDMA, and CDMA multiple access schemes considering an onesided transmit signal PSD of dBm/Hz. Consequently, the total transmission power allocated to all active subcarriers of each PLMs considering the FDMA scheme is therefore dBm, while a total of dBm are allocated to each PLM in the TDMA and CDMA schemes. In order to assess the performance of the aforementioned schemes, the SINR of the discretefrequency domain received vector was adopted as a metric. For TDMA and FDMA schemes, the SINR for each PLMs is equal to the ratio between the total received signal power and the total additive noise power, being equivalent the overall SNR considering the allocated subcarriers to them. The CDMA case, on the other hand, has also an interference noise term resulting from the decoding process from (20) and pointed out in (21). As a consequence, the SINR at the four PLMs is equal to the ratio between their associated total received signal power and the total additive noise plus interference noise power.
The attained SINR values are listed in Table III. The lower SINR experienced by the PLMs in the TDMA scheme is due to the fact that the whole additive noise spectral content impairs the captured signal from the reflection channel. Meanwhile, the subcarrier hopping in the FDMA scheme results in higher SINR. This is due to the fact that greater part of the additive noise power is concentrated in lower frequencies as its PSD decreases exponentially. As a consequence, the subcarrier hopping makes the PLMs experience less effective noise power and therefore increases the SINR in comparison to the TDMA scheme. The increasing SINR along with the PLM index is due to the fact that subcarriers belonging to the set allocated to the PLM are associated to higher frequency bins as increases, which is described in Subsection VB and results in lower effective additive noise power for higher . Finally, the higher average SINR among the PLMs in the CDMA scheme is due to the SNR gain due to noise averaging in the decoding process, as described in Subsection VC.
PLM index  1  2  3  4 

TDMA  dB  dB  dB  dB 
FDMA  dB  dB  dB  dB 
CDMA  dB  dB  dB  dB 
Vii Conclusion
This study has discussed the main aspects of an HSOFDMbased TDR system for power line sensing. In summary, a system model covering the injection of HSOFDM signals into the power distribution grid and subsequent reception of raised reflections has been outlined and pulse compression and channel estimation approaches for obtaining reflectograms have been discussed. Also, limitations of the TDR system based on its parametrization and multiple access schemes have been addressed.
The carried out discussion has been supported by numerical results, which has shown the superiority of the channel estimation procedure over the pulse compression. Furthermore, the influence of the TDR system parametrization on sidelobe level, range resolution, and maximum unambiguous range. The attained results show that the NBPLC frequency range is suitable for the sensing of LV and MV power distribution networks, with the appropriate frequency bandwidth for monitoring a power distribution network section being inversely proportional to its length. For the latter analysis, typical European underground LV and US overhead MV scenarios have been considered and compliance to FCC, ARIB, and CENELEC NBPLC regulations has also been addressed.
Finally, a comparison among multiple access schemes has been performed considering a Brazilian overhead MV scenario. It has been shown that TDMA offers a fair SINR level for all PLMs sensing a power distribution network, while the use of FDMA results in higher SINR to PLMs associated with subcarriers in higher frequnecy bins due to the exponentially decreasing additive noise PSD. The attained SINR levels by the PLMs in CDMA scheme is higher than in the other schemes and fair among the PLMs, which is due to the averaging process that lowers the additive noise PSD in the decoding stage. Additionally, a higher number of reflectograms, which may be distorted if the reflection channel coherence bandwidth constraint is not observed, is obtained over time in the FDMA scheme. TDMA and CDMA schemes, however, obtain less but nondistorted reflectograms due to time multiplexing and spreading processes.
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