# Performance Analysis of Distributed Estimation for Data Fusion Using a Statistical Approach in Smart Grid Noisy Wireless Sensor Networks

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Review of Data Aggregation

#### 1.3. Diffusion Kalman Filtering for Comparison with Proposed Method

#### 1.4. Organization of the Paper

## 2. Materials and Methods

#### 2.1. Problem Formulation

#### 2.2. Statistical Information Fusion Function

#### 2.3. Accuracy Measurement Metric

#### 2.4. Computational Complexity Measurement Metric

#### 2.5. Energy Consumption Model

#### 2.6. Estimation Based Diffusion Kalman Filtering (EBDKF)

#### 2.7. Network Latency

#### 2.8. Information Fusion in Chain-Based IOT Platform using Proposed Method

#### 2.8.1. Background

#### 2.8.2. Method of Performance Evaluation

#### 2.9. Information Fusion in Tree-Based IOT Platform Using Proposed Method

#### 2.9.1. Background

#### 2.9.2. Method of Performance Evaluation

#### 2.10. Information Fusion in Randomly Placed Non-Hierarchical Bidirectional Graph Based IOT Platform Using Proposed Method

#### 2.10.1. Background

#### 2.10.2. Method of Performance Evaluation

## 3. Results and Discussion

#### 3.1. Simulations for Chain-Based Network

#### 3.1.1. Simulations for Data Accuracy of a Deterministic Constant (Slowly Varying) Parameter Having Bounded Sensor Noise and Measurements

#### 3.1.2. Simulations for Data Accuracy of a Deterministic Dynamic Parameter Having Bounded Sensor Noise

#### 3.1.3. Evaluation of Complexity of Information Fusion

#### 3.1.4. Network Lifetime

#### 3.1.5. Network Latency

#### 3.2. Simulations for Hierarchical Tree Structured Network

#### 3.2.1. Simulations for Data Accuracy of a Deterministic Constant (Slowly Varying) Parameter Having Bounded Sensor Noise

#### 3.2.2. Evaluation of Complexity of Information Fusion

#### 3.2.3. Network Lifetime

#### 3.2.4. Network Latency

#### 3.3. Simulation for Data Accuracy in Randomly Placed Non-Hierarchical Bidirectional Graph

#### 3.3.1. Simulations for Data Accuracy of a Deterministic Constant (Slowly Varying) Parameter Having Bounded Sensor Noise

#### 3.3.2. Evaluation of Complexity of Information Fusion

#### 3.3.3. Network Lifetime

#### 3.3.4. Network Latency

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of the Upper Bound (17)

## References

- Gungor, V.C.; Sahin, D.; Kocak, T.; Ergut, S.; Buccella, C.; Cecati, C.; Hancke, G. Smart Grid Technologies: Communication Technologies and Standards. IEEE Trans. Ind. Inf.
**2011**, 7, 529–539. [Google Scholar] [CrossRef] [Green Version] - Farhangi, H. The path of the smart grid. IEEE Power Energy Mag.
**2009**, 8, 18–28. [Google Scholar] [CrossRef] - Gungor, V.C.; Sahin, D.; Kocak, T.; Ergut, S.; Buccella, C.; Cecati, C.; Hancke, G. A Survey on Smart Grid Potential Applications and Communication Requirements. IEEE Trans. Ind. Inf.
**2012**, 9, 28–42. [Google Scholar] [CrossRef] [Green Version] - Ma, R.; Chen, H.-H.; Huang, Y.-R.; Meng, W. Smart Grid Communication: Its Challenges and Opportunities. IEEE Trans. Smart Grid
**2013**, 4, 36–46. [Google Scholar] [CrossRef] - Ahmed, E.; Yaqoob, I.; Imran, M.; Guizani, M.; Gani, A. Internet-of-things-based smart environments: State of the art, taxonomy, and open research challenges. IEEE Wirel. Commun.
**2016**, 23, 10–16. [Google Scholar] [CrossRef] - A Smarter Grid with the Internet of Things—Texas Instruments. Available online: http://www.ti.com/lit/ml/slyb214/slyb214.pdf (accessed on 9 April 2019).
- Wang, Y.; Ma, Y.; Lin, W.; Zhang, T. Research on application and security protection of internet of things in smart grid. In Proceedings of the IET International Conference on Information Science and Control Engineering (ICISCE), Taipei, Taiwan, 3–6 December 2012; pp. 1–5. [Google Scholar]
- Sehgal, A.; Perelman, V.; Kuryla, S.; Schonwalder, J. Management of resource constrained devices in the internet of things. IEEE Commun. Mag.
**2012**, 50, 144–149. [Google Scholar] [CrossRef] - Gubbi, J.; Buyya, R.; Marusic, S.; Palaniswami, M. Internet of Things (IoT): A vision, architectural elements, and future directions. Futur. Gener. Comput. Syst.
**2013**, 29, 1645–1660. [Google Scholar] [CrossRef] [Green Version] - Welbourne, E.; Battle, L.; Cole, G.; Gould, K.; Rector, K.; Raymer, S.; Balazinska, M.; Borriello, G. Building the Internet of Things Using RFID: The RFID Ecosystem Experience. IEEE Internet Comput.
**2009**, 13, 48–55. [Google Scholar] [CrossRef] - Henry, L.; Chatura, S. A novel statistical model for distributed estimation in wireless sensor network. IEEE Trans. Signal Process.
**2015**, 63, 3154–3164. [Google Scholar] - Gubner, J. Distributed estimation and quantization. IEEE Trans. Inf. Theory
**1993**, 39, 1456–1459. [Google Scholar] [CrossRef] [Green Version] - Khan, A.A.; Rehmani, M.H.; Reisslein, M. Cognitive Radio for Smart Grids: Survey of Architectures, Spectrum Sensing Mechanisms, and Networking Protocols. IEEE Commun. Surv. Tutor.
**2015**, 18, 860–898. [Google Scholar] [CrossRef] - Bi, Z.; Da Xu, L.; Wang, C. Internet of Things for Enterprise Systems of Modern Manufacturing. IEEE Trans. Ind. Inf.
**2014**, 10, 1537–1546. [Google Scholar] - Jamieson, K.; Balakrishnan, H. PPR: Partial packet recovery for wireless networks. ACM SIGCOMM Comput. Commun. Rev.
**2007**, 37, 409–420. [Google Scholar] [CrossRef] - Cooklev, T. Wireless Communication Standards; Park Avenue: New York, NY, USA, 2004. [Google Scholar]
- IEEE 802.11 p Working Group. IEEE Standard for Information Technology-Local and Metropolitan Area Networks-Specific Requirements-part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 6: Wireless Access in Vehicular Environments; IEEE Std 802; IEEE: Piscataway, NJ, USA, 2010. [Google Scholar]
- Gungor, V.C.; Lu, B.; Hancke, G.P. Opportunities and Challenges of Wireless Sensor Networks in Smart Grid. IEEE Trans. Ind. Electron.
**2010**, 57, 3557–3564. [Google Scholar] [CrossRef] [Green Version] - Shah, G.A.; Gungor, V.C.; Akan, O.B. A Cross-Layer QoS-Aware Communication Framework in Cognitive Radio Sensor Networks for Smart Grid Applications. IEEE Trans. Ind. Inf.
**2013**, 9, 1477–1485. [Google Scholar] [CrossRef] - Mohomad, D.; Xavier, F. On the Communication requirement for the smart grid. Energy Power Eng.
**2011**, 3, 53–60. [Google Scholar] - Chamberland, J.-F.; Veeravalli, V. Wireless Sensors in Distributed Detection Applications. IEEE Signal Process. Mag.
**2007**, 24, 16–25. [Google Scholar] [CrossRef] - Yin, L.; Wang, Y.; Yue, D.-W. Serial Distributed Detection Performance Analysis in Wireless Sensor Networks under Noisy Channel. In Proceedings of the 2009 5th International Conference on Wireless Communications, Networking and Mobile Computing, Beijing, China, 24–26 September 2009; pp. 1–4. [Google Scholar]
- Carvalho, H.; Heinzelman, W.; Murphy, A.; Coelho, C. A general data fusion architecture. In Proceedings of the Sixth International Conference of Information Fusion, Cairns, Australia, 8–11 July 2003; pp. 1465–1472. [Google Scholar]
- Brooks, R.; Iyengar, S. Multi-Sensor Fusion: Fundermentals and Application with Software; Prentice Hall PTR: Upper Saddle River, NJ, USA, 1997. [Google Scholar]
- Giridhar, A.; Kumar, P. Computing and communicating functions over sensor networks. IEEE J. Sel. Areas Commun.
**2005**, 23, 755–764. [Google Scholar] [CrossRef] - Rajagopalan, R.; Varshney, P.K. Data-aggregation techniques in sensor networks: A survey. IEEE Commun. Surv. Tutor.
**2006**, 8, 48–63. [Google Scholar] [CrossRef] [Green Version] - Nardelli, P.H.J.; Ramezanipour, I.; Alves, H.; De Lima, C.H.M.; Latva-Aho, M. Average Error Probability in Wireless Sensor Networks With Imperfect Sensing and Communication for Different Decision Rules. IEEE Sens. J.
**2016**, 16, 3948–3957. [Google Scholar] [CrossRef] [Green Version] - Javadi, S.H.; Peiravi, A. Distributed Detection in Serial and Parallel Configurations of Wireless Sensor Networks. In Proceedings of the 2018 Electric Electronics, Computer Science, Biomedical Engineerings’ Meeting (EBBT), Istanbul, Turkey, 18–19 April 2018; pp. 1–5. [Google Scholar]
- Jagyasi, B.G.; Merchant, S.N.; Chander, D.; Desai, U.B.; Dey, B.K. AdWAS: Adaptive Weighted Aggregation Scheme for Single-hop and Multi-hop Wireless Sensor Network. In Proceedings of the IEEE 2007 Second International Conference on Communiations and Networking in China, Shanghai, China, 22–24 August 2007; pp. 835–839. [Google Scholar]
- Niu, R.; Varshney, P.K.; Moore, M.H.; Klamer, D. Decision fusion in a wireless sensor network with a large number of sensors. In Proceedings of the 7th IEEE International Conference on Information Fusion (ICIF’04), Stockholm, Sweden, 28 June–1 July 2004. [Google Scholar]
- Liu, Y.; Zhang, J.; Li, M. Support degree and adaptive weighted spatial-temporal fusion algorithm of multi sensor. In Proceedings of the IEEE 2010 Chinese Control and Decision Conference, Xuzhou, China, 26–28 May 2010. [Google Scholar]
- Wang, C.T.; Wang, Z.; Zhu, Y.; Han, Z.H. The Application of Data-Level Fusion Algorithm Based on Adaptive-Weighted and Support Degree in Intelligent Household Greenhouse. In Innovative Techniques and Applications of Modelling, Identification and Control; Springer: Singapore, 2018; pp. 93–108. [Google Scholar]
- Yager, R.R. The Power Average Operato. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2001**, 31, 724–731. [Google Scholar] [CrossRef] - Chang, K.C.; Saha, R.K.; Bar-Shalom, Y. On optimal track-totrack fusion. IEEE Trans. Aero. Electron. Syst.
**1997**, 33, 1271–1276. [Google Scholar] [CrossRef] - Saha, R.; Chang, K. An efficient algorithm for multisensor track fusion. IEEE Trans. Aerosp. Electron. Syst.
**1998**, 34, 200–210. [Google Scholar] [CrossRef] - Mori, S.; Chong, W.H.B.C.Y.; Chang, K.C. Track association and track fusion with non-deterministic target dynamics. In Proceedings of the 2nd International Conference Information Fusion, Sunnyvale, CA, USA, 24–27 May 1999. [Google Scholar]
- Mori, S.; Chong, W.H.B.C.Y.; Chang, K.C. Track association and track fusion with non-deterministic target dynamics. IEEE Trans. Aero. Electron. Syst.
**2002**, 38, 659–668. [Google Scholar] [CrossRef] - Zhu, Y.; Li, X.R. Best linear unbiased estimation fusion. In Proceedings of the 2nd International Conference Information Fusion, Sunnyvale, CA, USA, 24–27 May 1999. [Google Scholar]
- Olfati-Saber, R.; Shamma, J.S. Consensus filters for sensor networks and distributed sensor fusion. In Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 15 December 2005. [Google Scholar]
- Olfati-Saber, R.; Fax, J.A.; Murray, R.M. Consensus and Cooperation in Networked Multi-Agent Systems. Proc. IEEE
**2007**, 95, 215–233. [Google Scholar] [CrossRef] - Olfati-Saber, R. Distributed Kalman filtering for sensor networks. In Proceedings of the 2007 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 12–14 December 2007. [Google Scholar]
- Kirti, S.; Scaglione, A. Scalable distributed Kalman filtering through consensus. In Proceedings of the 2008 IEEE International Conference on Acoustics, Speech and Signal Processing; Institute of Electrical and Electronics Engineers (IEEE), Las Vegas, NV, USA, 31 March–4 April 2008; pp. 2725–2728. [Google Scholar]
- Tu, S.-Y.; Sayed, A.H. Diffusion Strategies Outperform Consensus Strategies for Distributed Estimation Over Adaptive Networks. IEEE Trans. Signal Process.
**2012**, 60, 6217–6234. [Google Scholar] [CrossRef] [Green Version] - Cattivelli, F.S.; Sayed, A.H. Diffusion Strategies for Distributed Kalman Filtering and Smoothing. IEEE Trans. Autom. Control
**2010**, 55, 2069–2084. [Google Scholar] [CrossRef] - Hu, J.; Xie, L.; Zhang, C. Diffusion Kalman filtering based on covariance intersection. IEEE Trans. Signal Process.
**2011**, 60, 891–902. [Google Scholar] [CrossRef] - Wang, G.; Li, N.; Zhang, Y. Diffusion distributed Kalman filter over sensor networks without exchanging raw measurements. Signal Process.
**2017**, 132, 1–7. [Google Scholar] [CrossRef] - Chen, H. Performance-energy trade-offs for decentralized estimation in a multi-hop sensor network. IEEE Sens. J.
**2010**, 10, 1304–1310. [Google Scholar] [CrossRef] - Yang, S.; Huang, T.; Guan, J.; Xiong, Y.; Wang, M. Diffusion Kalman Filter Based on Local Estimate Exchanges. In Proceedings of the 2015 IEEE International Conference on Digital Signal Processing, Singapoore, 21–24 July 2015; pp. 828–832. [Google Scholar]
- Xiao, J.-J.; Cui, S.; Luo, Z.-Q.; Goldsmith, A. Power scheduling of universal decentralized estimation in sensor networks. IEEE Trans. Signal Process.
**2006**, 54, 413–422. [Google Scholar] [CrossRef] - Communication Requirement of the Smart Grid Technologies; Department of Energy: Washington, DC, USA, 2010.
- Vazifehdan, J.; Prasad, R.V.; Jacobsson, M.; Niemegeers, I. An Analytical Energy Consumption Model for Packet Transfer over Wireless Links. IEEE Commun. Lett.
**2011**, 16, 30–33. [Google Scholar] [CrossRef] - Kay, S. Fundamentals of Statistical Signal Processing: Estimation Theory; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 1993; p. 07458. [Google Scholar]
- Huang, Y.; Hua, Y. Multihop Progressive Decentralized Estimation in Wireless Sensor Networks. IEEE Signal Process. Lett.
**2007**, 14, 1004–1007. [Google Scholar] [CrossRef] - OMNET++ Simulation Manual. Available online: https://doc.omnetpp.org/omnetpp/manual/ (accessed on 27 March 2019).
- Xian, X.; Shi, W.; Huang, H. Comparison of OMNET++ and other simulator for WSN simulation. In Proceedings of the 2008 3rd IEEE Conference on Industrial Electronics and Applications, Singapore, 3–5 June 2008; pp. 1439–1443. [Google Scholar]
- Zuniga, M.; Bhaskar, K. Link Layer Models for Wireless Sensor Networks; USC: Edinburgh, UK, 2005; pp. 1–4. [Google Scholar]
- Li, B.; Leung, H.; Seneviratne, C. Distributed consensus in Noisy Wireless Sensor Networks. In Proceedings of the 19th International conference on Information Fusion, Heidelberg, Germany, 5–8 July 2016. [Google Scholar]

**Figure 2.**Binary tree-based network topology using data gathering with information fusion function $f(..)$.

**Figure 4.**Performance comparison among the proposed statistical, EBDKF, and AVG information fusion methods in a chain-based IoT platform for different BER.

**Figure 5.**Network Mean estimate and 95% confidence interval limits for a true state of 1.80 compared among the proposed statistical, EBDKF and AVG information fusion methods in a chain-based IoT platform for different BER.

**Figure 6.**Bit level error performance comparison among the proposed statistical, EBDKF, and AVG information fusion methods in a chain-based IoT platform for (

**a**) BER of 0.005 and (

**b**) BER of 0.05.

**Figure 7.**The number of nodes required to achieve a given percentage error using the proposed statistical method, average, and EBDKF data fusion for IoT platform.

**Figure 8.**The performance comparison among the proposed statistical, AVG, and EBDKF information fusion methods in a retransmission allowed chain IoT platform.

**Figure 9.**The total retransmission attempts used by AVG and EBDKF fusion methods in a retransmission allowed chain IoT platform.

**Figure 10.**The performance comparison among the proposed statistical, AVG and EBDKF for a dynamic system parameter having communication link with (

**a**) BER of 0.0001 and (

**b**) BER of 0.05.

**Figure 11.**Network lifetime comparison with the proposed statistical, EBDKF and AVG information fusion methods when BER = 0.05 in a chain-based IoT platform.

**Figure 12.**Accuracy comparison among the proposed statistical, AVG, and EBDKF information fusion methods in a tree-based IoT platform.

**Figure 13.**Network Mean estimate and 95% confidence interval limits for a true state of 1.80; compared among the proposed statistical, EBDKF, and AVG information fusion methods in a tree-based IoT platform for different number of sensor nodes.

**Figure 14.**Network MSE vs. the number of data gathering cycles for BER of 0.0003 and 0.05 among the proposed statistical, AVG and EBDKF information fusion methods in a tree-based IoT platform: (

**a**) for first 600 data gathering cycles and (

**b**) for first 50 data gathering cycles.

**Figure 15.**Bit level error performance comparison among the proposed statistical and the AVG information fusion method in a tree-based IoT platform when (

**a**) BER = 0.005 and (

**b**) BER = 0.05.

**Figure 16.**Network MSE comparison among the proposed statistical, AVG, and EBDKF information fusion methods in a retransmission allowed tree structured network when BER is 0.05.

**Figure 17.**Average number of fusion nodes and total child nodes in tree-based IoT platform using maximum reliable connection method for different fusion techniques.

**Figure 18.**Network lifetime comparison for 3 information fusion methods in a tree-based IoT platform having 38 nodes.

**Figure 19.**Accuracy comparison with Data gathering cycles for BER s of 0.0003 and 0.05 (

**a**): For first 600 Data gathering cycles (

**b**): For first 50 Data gathering cycles.

**Figure 21.**Network Lifetime comparison of three different fusion techniques for an ad hoc-type network.

**Table 1.**Network MSE for a dynamic system parameter for different fusion functions under different BER.

Fusion Function | MSE for BER = 0.00001 | MSE for BER = 0.05 |
---|---|---|

EBDKF | 0.0428 | 266.5 |

AVG | 0.1162 | 2.0107 |

Proposed method | 0.0702 | 0.3019 |

Fusion Function | CPU Cycles | Time ((${\mathit{T}}_{\mathit{A}}$)($\mathsf{\mu}$s)) |
---|---|---|

Average | 36 | 2.25 |

Proposed Method | 128 | 8 |

EBDKF | 252 | 15.75 |

Fusion Function | Latency for Zig Bee (ms) | Latency for WCDMA (ms) |
---|---|---|

Average | 32.004(N−1) | 6.97(N−1) |

Proposed Method | 8.008(N−1) | 1.74(N−1) |

EBDKF | 98.42 | 23.316 |

Fusion Function | CPU Cycles | Time ((${\mathit{T}}_{\mathit{A}}$)($\mathsf{\mu}$s)) |
---|---|---|

Average | 48 | 3 |

Proposed method | 156 | 9.75 |

EBDKF | 252 | 15.75 |

**Table 5.**Average fusion complexity per node for different information fusion functions in a bidirectional graph.

j | Fusion Function | Complexity of Fusion |
---|---|---|

1 | Average | 2 Additions, 1 division |

1 | Proposed method | 2 Additions, 5 divisions, 1 multiplication |

1 | EBDKF | 6 Additions, 4 divisions, 8 multiplications |

2 | Average | 3 Additions, 1 division |

2 | Proposed method | 4 Additions, 7 divisions, 1 multiplication |

2 | EBDKF | 10 Additions, 4 divisions, 8 multiplications |

3 | Average | 4 Additions, 1 division |

3 | Proposed Method | 6 Additions, 9 divisions, 1 multiplication |

3 | EBDKF | 14 Additions, 4 divisions, 8 multiplications |

n | Average | $(n+1)$ Additions, 1 division |

n | Proposed Method | 2n Additions, $3+2n$ divisions, 1 multiplication |

n | EBDKF | $(4n+2)$ Additions, 4 divisions, 8 multiplications |

**Table 6.**CPU overhead of different information fusion functions for randomly placed non-hierarchical graph.

Fusion Function | CPU Cycles | Time ((${\mathit{T}}_{\mathit{A}}$)($\mathsf{\mu}$s)) |
---|---|---|

Average | 60 | 3.75 |

Proposed method | 180 | 11.25 |

EBDKF | 1387 | 86.6875 |

**Table 7.**Network latency of different information fusion functions for non-hierarchical sensor network.

Fusion Function | Latency for Zig Bee (ms) | Latency for WCDMA (ms) |
---|---|---|

Average | 8.034(K + 1) + 0.00375 | 1.775(K + 1) + 0.00375 |

Proposed Method | 8.034(K + 1) + 0.01125 | 1.775(K + 1) + 0.01125 |

EBDKF | 49.202(K + 1) + 0.0867 | 11.65(K + 1) + 0.0867 |

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**MDPI and ACS Style**

Seneviratne, C.; Wijesekara, P.A.D.S.N.; Leung, H.
Performance Analysis of Distributed Estimation for Data Fusion Using a Statistical Approach in Smart Grid Noisy Wireless Sensor Networks. *Sensors* **2020**, *20*, 567.
https://doi.org/10.3390/s20020567

**AMA Style**

Seneviratne C, Wijesekara PADSN, Leung H.
Performance Analysis of Distributed Estimation for Data Fusion Using a Statistical Approach in Smart Grid Noisy Wireless Sensor Networks. *Sensors*. 2020; 20(2):567.
https://doi.org/10.3390/s20020567

**Chicago/Turabian Style**

Seneviratne, Chatura, Patikiri Arachchige Don Shehan Nilmantha Wijesekara, and Henry Leung.
2020. "Performance Analysis of Distributed Estimation for Data Fusion Using a Statistical Approach in Smart Grid Noisy Wireless Sensor Networks" *Sensors* 20, no. 2: 567.
https://doi.org/10.3390/s20020567