# Industrial PLC Network Modeling and Parameter Identification Using Sensitivity Analysis and Mean Field Variational Inference

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PLC Network Model

#### 2.1. Network Topology

#### 2.2. Cable Model

#### 2.3. Load Models

#### 2.4. Network Generation

#### 2.5. Transmitter and Receiver Models

#### 2.6. Model Solver

## 3. Mean Field Variational Inference

## 4. Results

#### 4.1. Demonstration on Single Network

#### 4.2. Demonstration on Multiple Networks

#### 4.3. Inferring Load Types

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BB | Broadband |

ELBO | Evidence lower bound |

MIMO | Multiple input multiple output |

NB | Narrowband |

PLC | Power line communications |

RLC | Resistor-inductor-capacitor |

TMCMC | Transisional Markov chain Monte Carlo |

TL | Transmission line |

UNB | Ultra narrowband |

## Appendix A. Transfer Function Computation

#### Appendix A.1. Receiver Load

#### Appendix A.2. Transmitter Computation

## References

- Galli, S.; Scaglione, A.; Wang, Z. For the grid and through the grid: The role of power line communications in the smart grid. Proc. IEEE
**2011**, 99, 998–1027. [Google Scholar] [CrossRef] [Green Version] - Nassar, M.; Lin, J.; Mortazavi, Y.; Dabak, A.; Kim, I.H.; Evans, B.L. Local utility power line communications in the 3–500 kHz band: Channel impairments, noise, and standards. IEEE Signal Process. Mag.
**2012**, 29, 116–127. [Google Scholar] [CrossRef] - Sonmez, M.A.; Zehir, M.A.; Bagriyanik, M.; Nak, O. Impulsive noise survey on power line communication networks up to 125 kHz for smart metering infrastructure in systems with solar inverters in Turkey. In Proceedings of the 2013 International Conference on Renewable Energy Research and Applications (ICRERA), Madrid, Spain, 20–23 October 2013; pp. 705–710. [Google Scholar]
- Coelho, P.; Gomes, M.; Moreira, C. Smart metering technology. In Microgrids Design and Implementation; Springer: Berlin/Heidelberg, Germany, 2019; pp. 97–137. [Google Scholar]
- Van de Kaa, G.; Fens, T.; Rezaei, J.; Kaynak, D.; Hatun, Z.; Tsilimeni-Archangelidi, A. Realizing smart meter connectivity: Analyzing the competing technologies power line communication, mobile telephony, and radio frequency using the best worst method. Renew. Sustain. Energy Rev.
**2019**, 103, 320–327. [Google Scholar] [CrossRef] - Mlỳnek, P.; Rusz, M.; Benešl, L.; Sláčik, J.; Musil, P. Possibilities of broadband power line communications for smart home and smart building applications. Sensors
**2021**, 21, 240. [Google Scholar] [CrossRef] - Masood, B.; Khan, M.A.; Baig, S.; Song, G.; Rehman, A.U.; Rehman, S.U.; Asif, R.M.; Rasheed, M.B. Investigation of deterministic, statistical and parametric NB-PLC channel modeling techniques for advanced metering infrastructure. Energies
**2020**, 13, 3098. [Google Scholar] [CrossRef] - Canete, F.J.; Cortes, J.A.; Diez, L.; Entrambasaguas, J.T. A channel model proposal for indoor power line communications. IEEE Commun. Mag.
**2011**, 49, 166–174. [Google Scholar] [CrossRef] - Tonello, A.M.; Versolatto, F. Bottom-up statistical PLC channel modeling—Part I: Random topology model and efficient transfer function computation. IEEE Trans. Power Deliv.
**2011**, 26, 891–898. [Google Scholar] [CrossRef] - Paul, C.R. Analysis of Multiconductor Transmission Lines; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Zhu, W.; Zhu, X.; Lim, E.; Huang, Y. State-of-art power line communications channel modelling. Procedia Comput. Sci.
**2013**, 17, 563–570. [Google Scholar] [CrossRef] [Green Version] - Marrocco, G.; Statovci, D.; Trautmann, S. A PLC broadband channel simulator for indoor communications. In Proceedings of the 2013 IEEE 17th International Symposium on Power Line Communications and Its Applications, Johannesburg, South Africa, 24–27 March 2013; pp. 321–326. [Google Scholar]
- Fang, X.; Wang, N.; Gulliver, T.A. A PLC channel model for home area networks. Energies
**2018**, 11, 3344. [Google Scholar] [CrossRef] [Green Version] - Aalamifar, F.; Schlögl, A.; Harris, D.; Lampe, L. Modelling power line communication using network simulator-3. In Proceedings of the 2013 IEEE Global Communications Conference (GLOBECOM), Atlanta, GA, USA, 9–13 December 2013; pp. 2969–2974. [Google Scholar]
- Passerini, F.; Righini, D.; Tonello, A.M. A bottom-up PLC channel model that includes radiation effects. In Proceedings of the 2018 IEEE International Symposium on Power Line Communications and its Applications (ISPLC), Manchester, UK, 8–11 April 2018; pp. 1–6. [Google Scholar]
- Tonello, A.M.; Versolatto, F.; Pittolo, A. In-home power line communication channel: Statistical characterization. IEEE Trans. Commun.
**2014**, 62, 2096–2106. [Google Scholar] [CrossRef] - Versolatto, F.; Tonello, A.M. A MIMO PLC random channel generator and capacity analysis. In Proceedings of the 2011 IEEE International Symposium on Power Line Communications and Its Applications, Udine, Italy, 3–6 April 2011; pp. 66–71. [Google Scholar]
- Banwell, T.; Galli, S. A novel approach to the modeling of the indoor power line channel part I: Circuit analysis and companion model. IEEE Trans. Power Deliv.
**2005**, 20, 655–663. [Google Scholar] [CrossRef] - Galli, S.; Banwell, T. A novel approach to the modeling of the indoor power line channel-Part II: Transfer function and its properties. IEEE Trans. Power Deliv.
**2005**, 20, 1869–1878. [Google Scholar] [CrossRef] - Corchado, J.A.; Cortés, J.A.; Canete, F.J.; Díez, L. An MTL-based channel model for indoor broadband MIMO power line communications. IEEE J. Sel. Areas Commun.
**2016**, 34, 2045–2055. [Google Scholar] [CrossRef] - Anatory, J.; Theethayi, N.; Thottappillil, R. Power-line communication channel model for interconnected networks—Part I: Two-conductor system. IEEE Trans. Power Deliv.
**2008**, 24, 118–123. [Google Scholar] [CrossRef] - Anatory, J.; Theethayi, N.; Thottappillil, R. Power-line communication channel model for interconnected networks—Part II: Multiconductor system. IEEE Trans. Power Deliv.
**2008**, 24, 124–128. [Google Scholar] [CrossRef] - Franek, L.; Fiedler, P. A multiconductor model of power line communication in medium-voltage lines. Energies
**2017**, 10, 816. [Google Scholar] [CrossRef] [Green Version] - Versolatto, F.; Tonello, A.M. An MTL theory approach for the simulation of MIMO power-line communication channels. IEEE Trans. Power Deliv.
**2011**, 26, 1710–1717. [Google Scholar] [CrossRef] - Zhai, M.Y. Transmission characteristics of low-voltage distribution networks in China under the smart grids environment. IEEE Trans. Power Deliv.
**2010**, 26, 173–180. [Google Scholar] [CrossRef] - Tomasoni, A.; Riva, R.; Bellini, S. Spatial correlation analysis and model for in-home MIMO power line channels. In Proceedings of the 2012 IEEE International Symposium on Power Line Communications and Its Applications, Beijing, China, 27–30 March 2012; pp. 286–291. [Google Scholar]
- Berger, L.T.; Schwager, A.; Pagani, P.; Schneider, D.M. MIMO power line communications. IEEE Commun. Surv. Tutor.
**2014**, 17, 106–124. [Google Scholar] [CrossRef] - Hao, L.; Guo, J. A MIMO-OFDM scheme over coupled multi-conductor power-line communication channel. In Proceedings of the 2007 IEEE International Symposium on Power Line Communications and Its Applications, Pisa, Italy, 26–28 March 2007; pp. 198–203. [Google Scholar]
- Ching, D.S.; Safta, C.; Reichardt, T.A. Sensitivity-informed bayesian inference for home PLC network models with unknown parameters. Energies
**2021**, 14, 2402. [Google Scholar] [CrossRef] - Ferreira, H.C.; Lampe, L.; Newbury, J.; Swart, T.G. Power Line Communications: Theory and Applications for Narrowband and Broadband Communications Over Power Lines; John Wiley & Sons: Hoboken, NJ, USA, 2010. [Google Scholar]
- Antoniali, M.; Tonello, A.M. Measurement and characterization of load impedances in home power line grids. IEEE Trans. Instrum. Meas.
**2013**, 63, 548–556. [Google Scholar] [CrossRef] - Kharraz, M.A.O.; Lavenu, C.; Jensen, P.; Picard, D.; Serhir, M. Characterization of the input impedance of household appliances in the FCC frequency band. In Proceedings of the 2017 IEEE International Symposium on Power Line Communications and Its Applications (ISPLC), Madrid, Spain, 3–5 April 2017; pp. 1–6. [Google Scholar]
- Pasdar, A.M.; Cavdar, I.H.; Sozer, Y. Power-line impedance estimation at FCC band based on intelligent home appliances status detection algorithm through their individual energy and impedance signatures. IEEE Trans. Power Deliv.
**2014**, 29, 1407–1416. [Google Scholar] [CrossRef] - Sudiarto, B.; Widyanto, A.N.; Hirsch, H. The mutual influence of appliances on the disturbance in the frequency range of 9–150 kHz produced by household appliances in the low voltage network. In Proceedings of the 2017 International Symposium on Electromagnetic Compatibility-EMC EUROPE, Angers, France, 4–7 September 2017; pp. 1–6. [Google Scholar]
- Moreno, Y.; Almandoz, G.; Egea, A.; Arribas, B.; Urdangarin, A. Analysis of Permanent Magnet Motors in High Frequency—A Review. Appl. Sci.
**2021**, 11, 6334. [Google Scholar] [CrossRef] - Boglietti, A.; Carpaneto, E. Induction motor high frequency model. In Proceedings of the Conference Record of the 1999 IEEE Industry Applications Conference. Thirty-Forth IAS Annual Meeting (Cat. No. 99CH36370), Phoenix, AZ, USA, 3–7 October 1999; Volume 3, pp. 1551–1558. [Google Scholar]
- Boglietti, A.; Cavagnino, A.; Lazzari, M. Experimental high-frequency parameter identification of AC electrical motors. IEEE Trans. Ind. Appl.
**2007**, 43, 23–29. [Google Scholar] [CrossRef] [Green Version] - Miloudi, H.; Bendaoud, A.; Miloudi, M.; Dickmann, S.; Schenke, S. Common mode and differential mode characteristics of AC motor for EMC analysis. In Proceedings of the 2016 International Symposium on Electromagnetic Compatibility-EMC EUROPE, Wroclaw, Poland, 25–29 July 2016; pp. 765–769. [Google Scholar]
- Miloudi, H.; Bendaoud, A.; Miloudi, M. A method for modeling a common-mode impedance for the AC motor. Elektrotehniski Vestn.
**2017**, 84, 241–246. [Google Scholar] - Miloudi, H.; Miloudi, M.; Gourbi, A.; Bermaki, M.; Bendaoud, A.; Zeghoudi, A. A high-frequency modeling of AC motor in a frequency range from 40 Hz to 110 MHz. Electr. Eng. Electromech.
**2022**, 6, 3–7. [Google Scholar] [CrossRef] - Blei, D.M.; Kucukelbir, A.; McAuliffe, J.D. Variational Inference: A Review for Statisticians. J. Am. Stat. Assoc.
**2017**, 112, 859–877. [Google Scholar] [CrossRef] [Green Version] - Salimans, T.; Kingma, D.P.; Welling, M. Markov Chain Monte Carlo and Variational Inference: Bridging the Gap. In Proceedings of the International Conference on Machine Learning, Lille, France, 6–11 July 2015. [Google Scholar]
- Ranganath, R.; Gerrish, S.; Blei, D.M. Black Box Variational Inference. In Proceedings of the 17th International Conference on Artificial Intelligence and Statistics, Reykjavik, Iceland, 22–25 April 2014. [Google Scholar]
- Mnih, A.; Gregor, K. Neural Variational Inference and Learning in Belief Networks. In Proceedings of the International Conference on Machine Learning, PMLR, Beijing, China, 21–26 June 2014. [Google Scholar]
- Bingham, E.; Chen, J.P.; Jankowiak, M.; Obermeyer, F.; Pradhan, N.; Karaletsos, T.; Singh, R.; Szerlip, P.; Horsfall, P.; Goodman, N.D. Pyro: Deep Universal Probabilistic Programming. J. Mach. Learn. Res.
**2018**, 20, 973–978. [Google Scholar] - Phan, D.; Pradhan, N.; Jankowiak, M. Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro. arXiv
**2019**, arXiv:1912.11554. [Google Scholar] - Duchi, J.; Hazan, E.; Singer, Y. Adaptive subgradient methods for online learning and stochastic optimization. J. Mach. Learn. Res.
**2011**, 12, 2121–2159. [Google Scholar]

**Figure 9.**Mean value of parameters vs iteration (

**left**) and variance vs iteration (

**right**). Each line corresponds to a different parameter.

**Figure 10.**True transfer function (without added noise) and mean of transfer function computed from posterior with 95% confidence bounds.

**Figure 11.**Transfer function from the original network and network modified by powering offloads in two rooms.

**Figure 12.**True transfer function component ${H}_{1,1}\left(f\right)$ and inferred transfer function for 10 different realizations of the network. Each color corresponds to a different realization. The dots are the true transfer function and lines are the inferred transfer function.

Parameter | Minimum | Maximum | Distribution Type |
---|---|---|---|

${r}_{w}$ [mm] connected to service panel | 1.03 | 2.06 | uniform |

${r}_{w}$ [mm] in rooms | 0.81 | 1.29 | uniform |

${l}_{w}$ [m] | 2 | 20 | uniform |

${R}_{const}$ [$\mathsf{\Omega}$] | 10 | 200 | log-uniform |

${C}_{c,leak}$ [nF] | 0.1 | 2.0 | uniform |

${R}_{s}$ [$\mathsf{\Omega}$] | 10 | 3000 | log-uniform |

${\omega}_{0s}$ [Mrad/s] | 0 | 30 | uniform |

${\zeta}_{s}$ | 0.1 | 2 | uniform |

${R}_{p}$ [$\mathsf{\Omega}$] | 10 | 3000 | log-uniform |

${\omega}_{0p}$ [Mrad/s] | 0 | 30 | uniform |

${\zeta}_{p}$ | 0.1 | 2 | uniform |

${\Delta}_{1}$ | −0.1 | 0.1 | uniform |

${\Delta}_{2}$ | −0.1 | 0.1 | uniform |

${C}_{d,leak}$ [F] | 0.1 | 2.0 | uniform |

${C}_{m}$ [nF] | 0.1 | 1.0 | uniform |

${L}_{m}$ [mH] | 5 | 20 | uniform |

${R}_{m1}$ [$\mathsf{\Omega}$] | 2000 | 15,000 | uniform |

${C}_{m,leak}$ [nF] | 0.2 | 5.0 | uniform |

Parameter | Transmitter 1 | Transmitter 2 |
---|---|---|

${Z}_{T0}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{TG1}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{TG2}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{TG3}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{T12}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{T13}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{T23}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

Parameter | Receiver 1 | Receiver 2 |
---|---|---|

${Z}_{RG}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{R1}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{R2}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

${Z}_{R3}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

**Table 4.**Mean parameter values from posterior in order of decreasing sensitivity, ordered left to right, then down. The true values are 0.25.

0.259 | 0.251 | 0.250 | 0.245 | 0.256 | 0.202 |

0.253 | 0.252 | 0.256 | 0.342 | 0.174 | 0.249 |

0.236 | 0.261 | 0.249 | 0.240 | 0.284 | 0.233 |

0.264 | 0.239 | 0.199 | 0.242 | 0.243 | 0.308 |

0.247 | 0.267 | 0.244 | 0.259 | 0.257 | 0.243 |

0.253 | 0.245 | 0.247 | 0.251 | 0.344 | 0.257 |

0.251 | 0.237 | 0.252 | 0.259 | 0.241 | 0.113 |

0.260 | 0.233 | 0.253 | 0.209 | 0.272 | 0.249 |

0.237 | 0.295 | 0.163 | 0.161 | 0.245 | 0.119 |

0.414 | 0.403 | 0.199 | 0.744 | 0.455 | 0.367 |

0.254 | 0.183 | 0.207 | 0.154 | 0.352 | 0.098 |

0.596 | 0.507 | 0.136 | 0.117 | 0.246 | 0.140 |

**Table 5.**Error magnitude of inferred parameters averaged across 10 realizations and the parameters in the given range. Parameters are ordered with decreasing sensitivity.

Parameters | Average Error Magnitude |
---|---|

1–10 | 0.029 |

10–20 | 0.080 |

20–40 | 0.088 |

40–60 | 0.176 |

**Table 6.**True and predicted load types. The load names are room number and outlet number. The * indicates that the constant impedance load was used as the representative member during the computations.

Load | True | Predicted |
---|---|---|

R1-O2 | Constant | {Constant *, Double RLC} |

R1-O3 | Motor | Motor |

R1-O4 | Constant | {Constant *, Double RLC} |

R2-O1 | Motor | Motor |

R2-O2 | Motor | Motor |

R2-O3 | Motor | Motor |

R2-O4 | Motor | Motor |

R3-O1 | Constant | {Constant *, Double RLC} |

R3-O2 | Constant | {Constant *, Double RLC} |

R3-O3 | Constant | {Constant *, Double RLC} |

R3-O4 | Motor | Motor |

R4-O1 | Double RLC | {Constant *, Double RLC} |

R4-O2 | Motor | Motor |

R4-O3 | Motor | Motor |

R4-O4 | Double RLC | {Constant *, Double RLC} |

R5-O1 | Motor | Motor |

R5-O2 | Constant | {Constant *, Double RLC} |

R5-O3 | Double RLC | {Constant *, Double RLC} |

R5-O4 | Motor | Motor |

R6-O1 | Constant | {Constant *, Double RLC} |

R6-O2 | Motor | Motor |

R6-O3 | Motor | Motor |

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## Share and Cite

**MDPI and ACS Style**

Wonnacott, R.; Ching, D.S.; Chilleri, J.; Safta, C.; Rashkin, L.; Reichardt, T.A.
Industrial PLC Network Modeling and Parameter Identification Using Sensitivity Analysis and Mean Field Variational Inference. *Sensors* **2023**, *23*, 2416.
https://doi.org/10.3390/s23052416

**AMA Style**

Wonnacott R, Ching DS, Chilleri J, Safta C, Rashkin L, Reichardt TA.
Industrial PLC Network Modeling and Parameter Identification Using Sensitivity Analysis and Mean Field Variational Inference. *Sensors*. 2023; 23(5):2416.
https://doi.org/10.3390/s23052416

**Chicago/Turabian Style**

Wonnacott, Raelynn, David S. Ching, John Chilleri, Cosmin Safta, Lee Rashkin, and Thomas A. Reichardt.
2023. "Industrial PLC Network Modeling and Parameter Identification Using Sensitivity Analysis and Mean Field Variational Inference" *Sensors* 23, no. 5: 2416.
https://doi.org/10.3390/s23052416