# Model-Based Fault Detection of Inverter-Based Microgrids and a Mathematical Framework to Analyze and Avoid Nuisance Tripping and Blinding Scenarios

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods: Feeder Modelling and Fault Detection Approach

_{fB}, S

_{fC}, or S

_{fD}. To perform the aforementioned approach, the different models for the non-faulted and faulted microgrid cases are derived in the Appendix A. These models are provided in a state-space format, from which sets of system transfer functions can be obtained, and can be used in the simulation. Three different state-space representations for various fault points within the studied microgrid and one model describing the system under normal operation are derived in the Appendix A to show that there is a clear distinction in the matrices for faulted and non-faulted conditions.

_{fC}is closed, this configuration corresponds to the case of a fault at the beginning of the feeder. If switch S

_{fB}is closed, this configuration corresponds to a fault near the system load. If switch S

_{fD}is closed, this configuration corresponds to a fault in the middle of the cable. Finally, if all switches are open, this case refers to normal operation of the microgrid.

_{b}, which can be derived by setting ρ = 0 (see Appendix A) and solving

_{g}(s) is the transfer function resulting from the input v

_{g}. The parameters A, B, C and D are matrices corresponding to the state-space description derived in the Appendix A. Four different transfer functions, T

_{n}(s) (where n = 1, 2, 3 and 4) are derived for the four different state space descriptions (see Appendix A) which are used in the fault detection approach described in Figure 2.

## 3. Theory/Calculation: Analytical Analysis of the Communication-Free Approach for Determining System Constraints to Avoid Nuisance Tripping and Blinding

_{g}), filter and output connecter impedance at the end of the feeder. We ignore the source measurement (i

_{g}and v

_{g}) because the source and the microgrid side measurements (i

_{b}, and v

_{b}) are separated by cable with a specific length and would require the use of a communication channel to send the measurement information to the protection system if measurements on both ends are utilized. Hence, by ignoring the source, system fault identification can be communication-free. It is understood that the error between estimated system current and actual current will not be close to zero depending on the available i

_{g}. However, we propose the following:

**Proposition**

**1.**

_{g}for certain system constraints. This proposition is explained by the inequalities below:

- (1)
- For no fault (all fault switches are opened): |e
_{1}| < |e_{2}| and |e_{1}| < |e_{3}| and |e_{1}| < |e_{4}| - (2)
- For (S
_{fB}only closed, other switches opened): |e_{2}| < |e_{1}| and |e_{2}| < |e_{3}| and |e_{2}| < |e_{4}| - (3)
- For (S
_{fC}only closed, other switches opened): |e_{3}| < |e_{1}| and |e_{3}| < |e_{2}| and |e_{3}| < |e_{4}| - (4)
- For (S
_{fD}only closed, other switches opened): |e_{4}| < |e_{1}| and |e_{4}| < |e_{2}| and |e_{4}| < |e_{3}|

_{n}is the error resulting from the difference between the measured value and the estimated output of transfer function n which is derived from condition n (n = 1, 2, 3 and 4 for when all switches of Figure 3 are open, S

_{fB}is closed, S

_{fC}is closed, and S

_{fD}is closed, respectively). This error is calculated using the amplitude of the steady-state signal. In equalities 1–4 in Proposition 1 show that the error associated with the transfer function describing the true dynamics of the system must be lower than any of the errors resulting from the remaining transfer functions with incorrect system dynamics for the given microgrid fault scenario in order to successfully detect the fault.

_{fB}closed) only. Other conditions have not been explored because other conditions will give less restrictive constraints. The reason why condition (S

_{fB}closed) will give the most restrictive constraint is because the current contribution from the DER (i

_{g}) for other conditions (S

_{fC}closed or S

_{fD}closed) will be lower than the current for condition (S

_{fB}closed). This is because there is more impedance within the path for all faulted conditions compared to condition S

_{fB}. Additionally, the other fault conditions make a significant alteration to the state-space matrix which is evident by inspecting Equations (A3), (A8) and (A12).

_{a}and Z

_{g}. Hence, during normal operation (Figure 4) the actual measured value of current magnitude, |I

_{m}|, during a non-faulted condition is

_{b}is the impedance of the line and Z

_{L}is the load impedance. Note that V

_{m}, I

_{a}, Z

_{L}, Z

_{g}, Z

_{b}, are all complex phasors. Note that the magnitude of power phasors are RMS values. In order to prove Proposition 1 and find the constraints upon which the proposition remains valid, we need to find a relationship describing the estimated current for when the current contribution (I

_{a}) from the distributed generation is ignored. This condition is described by (4). Equation (4) is necessary in order to construct inequalities proving Proposition 1 and the reason for its necessity will be explained in further subsections:

_{E}

^{NF}is defined as the estimate of the measured current for a non-faulted condition, I

_{m}

^{NF}. This estimate of the non-faulted condition could be the output of the transfer function derived from Equations (A3), (A4), (A5) and (A6) if v

_{g}is treated as a disturbance (i.e., ρ = 0 in the relationships found in the Appendix A). Throughout this article, we will use the subscript E to denote the value estimated by the model function and subscript m to denote the actual measured value.

_{fB}closed in Figure 4) the current magnitude representing the actual measurement is defined by

_{eq}is a complex phasor, and Z

_{F}= R

_{F}. The current estimate of the faulted condition when S

_{fB}is closed is described by

#### 3.1. Theory/Calculation: Analytical Analysis of Masked Fault (Blinding) Scenarios

_{b}, Z

_{L}and Z

_{F}are positive. Therefore, Equation (9) is satisfied when Equation (10) is satisfied.

_{m}, and I

_{a}are complex quantities, and |I

_{m}| is a the RMS signal. Substituting Equation (4) and Equation (5) into Equation (13), simplifying and rearranging gives

**Constraint**

**1:**

_{m}in Constraint 1 is a constant quantity chosen above the backup protection value. Current I

_{a}should be as large as possible during faulted conditions which is typically 2 p.u. for inverters. Analyzing the microgrid according to Equation (15), the range of fault impedances (Z

_{F}) that the apparatus can detect can be determined. The constraint should be tested for the minimum and maximum possible values of Z

_{L}. Constraint 1 represented by Equation (15) provides the following physical insight: the ratio of the faulted system equivalent impedance to the non-faulted system equivalent impedance must be lower than the ratio of the voltage drop across the cable up to the fault point to the total voltage (V

_{m}).

**Constraint**

**2:**

_{F}based on Constraint 1 can be substituted into Constraint 2 to make sure that Equation (17) is true. If it is not, then Z

_{F}is lowered and the analysis process is repeated until Equation (17) is satisfied. At the end of the iterations, the maximum value of Z

_{F}that the communication-free method can detect will be obtained. An optimization problem can also be formulated to find the optimal values.

_{F}= 0 Ω gives us a condition where it is always satisfied. Therefore, we check the system parameters that allow the system to operate in this region if this region exists for a general Z

_{F}(all possible types of faults—low and high impedance). This region (LHS < 0 and RHS ≥ 0) is expressed by

#### 3.2. Theory/Calculation: Analytical Analysis of False Positive (Nuisance Tripping) Scenarios

_{F}, this gives us the value of the highest impedance for a fault on the system, which the apparatus can detect. However, in this section the analysis is for a false positive scenario, which means that there is no fault to begin with. Hence, the constraints here restrict the value of the assumed fault impedance in the model equations. However, when the range of the assumed fault resistance is decreased, this also decreases the range of faults that can be detected when faults occur. Hence, a trade-off between protection system dependability (guaranteeing the relay always operates for faults) and security (guaranteeing the relay will not trip when there is no fault) is made. In protection system design, this trade-off always exists [31] and the same is true for the proposed method. From Constraint 3, to make the protection system more secure, the fault impedance used in the model equations needs to be lowered. However, the more this impedance is lowered, the lower the range of fault impedances that can be detected in the communication-free approach (more security as opposed to more dependability). Hence, a thorough analysis has to be done by the designer to choose whether dependability versus security is valued most for a particular application. It is worth noting here that the communication-based approach is more robust in detecting a wide range of faults with a higher range of fault impedance.

**Constraint**

**3:**

_{m}is chosen (for Section 3.2 constraints) based on the largest possible stable normal operating value (i.e., V

_{m}= 1.10 p.u.) as opposed to the smallest possible faulted value for Section 3.1 constraints. This is because a nuisance trip means that we are operating normally as opposed to a faulted condition in a blinding scenario in Section 3.1. Also, the constraint is checked for the largest and smallest possible I

_{a}and Z

_{L}during normal operation.

_{F}to increase in order to increase dependability while keeping the same level of security.

**Constraint**

**4:**

## 4. Results

_{fB}closed), (S

_{fC}closed), and (S

_{fD}closed), respectively, which is clear from the zero error results. The bottom plots demonstrate the validation of the inequalities in Proposition 1. That is e

_{2}, e

_{3}, and e

_{4}should have the lowest errors for Figure 5, Figure 6 and Figure 7, respectively.

_{fC}closed) is applied later in the simulation. Figure 8 shows the results when a fault S

_{fD}is applied at 20% of cable length (from load side) with a different fault resistance. This figure shows that the model matches the one described by (S

_{fB}closed) because this fault model is closest to this condition compared to the other fault model options.

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Nominal Feeder Model (Non-Faulted)

_{b}= R

_{b1}+ R

_{b2}and L

_{b}= L

_{b1}+ L

_{b2}.

_{F}, occurs near the load within the microgrid depicted in Figure 3. This is when switch S

_{fB}is closed. The state-space matrices A, B, C and D for this faulted condition can be shown to be Equations (A4)–(A7), respectively.

#### Appendix A.2. Faulted Condition Model (Fault at Microgrid Side of Cable)

_{fC}is closed. The state-space description for this condition can be expressed by Equations (A8)–(A11):

#### Appendix A.3. Faulted Condition Model (Fault in the Middle of Cable)

_{fD}is closed. The state-space description for this scenario is described by Equations (A5), (A6) (A12) and (A13). This covers the wide range of fault locations on a feeder so that one set of dynamic relationships either matches one of the faulted conditions or is closest to one set compared to the others. For instance, when a fault occurs a distance away from the middle of the cable closer to the load, this condition should match the model described by Equation (A7) and neither the non-faulted model nor the other faulted models. A simulation case in Figure 8 has been shown to demonstrate that this is true.

## References

- Zamani, M.A.; Sidhu, T.S.; Yazdani, A. Investigations into the control and protection of an existing distribution network to operate as a microgrid: A case study. IEEE Trans. Ind. Electron.
**2014**, 61, 1904–1915. [Google Scholar] [CrossRef] - Mirsaeidi, S.; Said, D.M.; Mustafa, M.W.; Habibuddin, M.H. A protection strategy for micro-grids based on positive-sequence component. IET Renew. Power Gener.
**2015**, 9, 600–609. [Google Scholar] [CrossRef] - Oureilidis, K.O.; Demoulias, C.S. A fault clearing method in converter-dominated microgrids with conventional protection means. IEEE Trans. Power Electron.
**2016**, 31, 4628–4640. [Google Scholar] [CrossRef] - Mirsaeidi, S.; Said, D.M.; Mustafa, M.W.; Habibuddin, M.H.; Ghaffari, K. An analytical literature review of the available techniques for the protection of micro-grids. Int. J. Electr. Power Energy Syst.
**2014**, 58, 300–306. [Google Scholar] [CrossRef] - Lai, K.; Illindala, M.S.; Haj-ahmed, M.A. Comprehensive protection strategy for an islanded microgrid using intelligent relays. In Proceedings of the Industry Applications Society Annual Meeting, Dallas, TX, USA, 18–22 October 2015; pp. 1–11. [Google Scholar]
- Nikkhajoei, H.; Lasseter, R.H. Microgrid protection. In Proceedings of the 2007 IEEE Power Engineering Society General Meeting, Tampa, FL, USA, 24–28 June 2007; pp. 1–6. [Google Scholar]
- Sortomme, E.; Venkata, S.S.; Mitra, J. Microgrid protection using communication-assisted digital relays. IEEE Trans. Power Deliv.
**2010**, 25, 2789–2796. [Google Scholar] [CrossRef] - Najy, W.K.; Zeineldin, H.H.; Woon, W.L. Optimal protection coordination for microgrids with grid-connected and islanded capability. IEEE Trans. Ind. Electron.
**2013**, 60, 1668–1677. [Google Scholar] [CrossRef] - De Santis, M.; Noce, C.; Varilone, P.; Verde, P. Analysis of the origin of measured voltage sags in interconnected networks. Electr. Power Syst. Res.
**2018**, 154, 391–400. [Google Scholar] [CrossRef] - Elkhatib, M.E.; Ellis, A. Communication-assisted impedance-based microgrid protection scheme. In Proceedings of the 2017 IEEE Power & Energy Society General Meeting, Chicago, IL, USA, 16–20 July 2017; pp. 1–5. [Google Scholar]
- Sortomme, E.; Mapes, G.J.; Foster, B.A.; Venkata, S.S. Fault analysis and protection of a microgrid. In Proceedings of the 2008 40th North American Power Symposium, Calgary, AB, Canada, 8–30 September 2008; pp. 1–6. [Google Scholar]
- Brahma, S.M.; Girgis, A.A. Development of adaptive protection scheme for distribution systems with high penetration of distributed generation. IEEE Trans. Power Deliv.
**2004**, 19, 56–63. [Google Scholar] [CrossRef] - Wan, H.; Li, K.K.; Wong, K.P. An adaptive multiagent approach to protection relay coordination with distributed generators in industrial power distribution system. IEEE Trans. Ind. Appl.
**2010**, 46, 2118–2124. [Google Scholar] [CrossRef] - Ustun, T.S.; Ozansoy, C.; Ustun, A. Fault current coefficient and time delay assignment for microgrid protection system with central protection unit. IEEE Trans. Power Syst.
**2013**, 28, 598–606. [Google Scholar] [CrossRef] - Huang, W.; Nengling, T.; Zheng, X.; Fan, C.; Yang, X.; Kirby, B.J. An impedance protection scheme for feeders of active distribution networks. IEEE Trans. Power Deliv.
**2014**, 29, 1591–1602. [Google Scholar] [CrossRef] - Zamani, M.A.; Sidhu, T.S.; Yazdani, A. A protection strategy and microprocessor-based relay for low-voltage microgrids. IEEE Trans. Power Deliv.
**2011**, 26, 1873–1883. [Google Scholar] [CrossRef] - Casagrande, E.; Woon, W.L.; Zeineldin, H.H.; Svetinovic, D. A differential sequence component protection scheme for microgrids with inverter-based distributed generators. IEEE Trans. Smart Grid
**2014**, 5, 29–37. [Google Scholar] [CrossRef] - Li, X.; Dyśko, A.; Burt, G.M. Traveling wave-based protection scheme for inverter-dominated microgrid using mathematical morphology. IEEE Trans. Smart Grid
**2014**, 5, 2211–2218. [Google Scholar] [CrossRef] - Li, X.; Dysko, A.; Burt, G. Enhanced protection for inverter dominated microgrid using transient fault information. In Proceedings of the 11th IET International Conference on Developments in Power Systems Protection (DPSP 2012), Birmingham, UK, 23–26 April 2012; p. 20. [Google Scholar]
- Shi, S.X.; Jiang, B.; Dong, X.Z.; Bo, Z.Q. Protection of microgrid. In Proceedings of the 10th IET International Conference on Developments in Power System Protection (DPSP 2010), Manchester, UK, 29 March–1 April 2010; p. 11. [Google Scholar]
- Cardenas, J.; Muthukrishnan, V.; McGinn, D.; Hunt, R. A hybrid algorithm for fault locating in looped microgrids. In Proceedings of the 2016 IEEE Energy Conversion Congress and Exposition (ECCE), Milwaukee, WI, USA, 18–22 September 2016; pp. 1–6. [Google Scholar]
- Gururani, A.; Mohanty, S.R.; Mohanta, J.C. Microgrid protection using Hilbert–Huang transform based-differential scheme. IET Gener. Transm. Distrib.
**2016**, 10, 3707–3716. [Google Scholar] [CrossRef] - Abdulwahid, A.H.; Wang, S. A new differential protection scheme for microgrid using Hilbert space based power setting and fuzzy decision processes. In Proceedings of the 2016 IEEE 11th Conference on Industrial Electronics and Applications (ICIEA), Hefei, China, 5–7 June 2016; pp. 6–11. [Google Scholar]
- Ma, X.; Li, B.; Wang, Q.; Bo, Z.; Zhang, M.; Ma, X. Research on microgrid protection based on transient polarity comparison. In Proceedings of the 2016 China International Conference on Electricity Distribution (CICED), Xi’an, China, 10–12 August 2016; pp. 1–5. [Google Scholar]
- Tumilty, R.M.; Brucoli, M.; Burt, G.M.; Green, T.C. Approaches to network protection for inverter dominated electrical distribution systems. In Proceedings of the 3rd IET International Conference on Power Electronics, Machines and Drives (PEMD 2006), Dublin, Ireland, 4–6 April 2006; pp. 622–626. [Google Scholar]
- Jayawarna, N.; Jones, C.; Barnes, M.; Jenkins, N. Operating microgrid energy storage control during network faults. In Proceedings of the 2007 IEEE International Conference on System of Systems Engineering, San Antonio, TX, USA, 16–18 April 2007; pp. 1–7. [Google Scholar]
- Mirsaeidi, S.; Said, D.M.; Mustafa, M.W.; Habibuddin, M.H.; Ghaffari, K. Fault location and isolation in micro-grids using a digital central protection unit. Renew. Sustain. Energy Rev.
**2016**, 56, 1–17. [Google Scholar] [CrossRef] - Liu, P.; Wu, Y.; Su, Y.; Duan, B. Fault detection and location of microgrid based on distributed decision. In Proceedings of the IECON 2017—43rd Annual Conference of the IEEE the Industrial Electronics Society, Beijing, China, 29 October–1 November 2017; pp. 5054–5059. [Google Scholar]
- Al Hassan, H.A.; Fu, Q.; Bharavaju, V.; Yang, Y.; Grainger, B.M. High-speed algorithm for renewable energy based microgrid fault detection and protective coordination. In Proceedings of the 2017 IEEE Energy Conversion Congress and Exposition (ECCE), Cincinnati, OH, USA, 1–5 October 2017; pp. 519–525. [Google Scholar]
- Di Fazio, A.R.; Russo, M.; Valeri, S.; De Santis, M. Linear method for steady-state analysis of radial distribution systems. Int. J. Electr. Power Energy Syst.
**2018**, 99, 744–755. [Google Scholar] - Horowitz, S.H.; Phadke, A.G. Power System Relaying; John Wiley & Sons: Chichester, UK, 2008. [Google Scholar]
- Bidram, A.; Davoudi, A.; Lewis, F.L.; Guerrero, J.M. Distributed cooperative secondary control of microgrids using feedback linearization. IEEE Trans. Power Syst.
**2018**, 28, 3462–3470. [Google Scholar] [CrossRef]

**Figure 1.**Inverter-based microgrid under study. CB refers to a circuit breaker. The dotted blue line represents a microgrid feeder.

**Figure 3.**Microgrid feeder circuit during normal operation (switches OPEN) and during faulted conditions (specific switches CLOSED).

**Figure 4.**Equivalent impedance-based model for when all switches of Figure 3 are open (normal operation) and during a faulted condition between the load and cable (S

_{fB}closed).

**Figure 5.**Results during normal operation and when a fault S

_{fB}is applied at 0.3 s, RF = 0.01 Ω. The dotted blue line reaches approx. 82 p.u. before 0.3 s and approx. 40 p.u. after 0.3 s (not shown).

**Figure 6.**Results when load (RL = 12 Ω) is switched at t = 0.15 s when a fault S

_{fC}is applied at 0.3 s, RF = 0.01 Ω. The dotted blue line reaches approx. 82 p.u. before 0.3 s (not shown).

**Figure 7.**Results during normal operation and when a fault S

_{fD}is applied at 0.3 s, RF = 0.01 Ω. The dotted line reaches approx. 82 p.u. before 0.3 s and approx. 20 p.u. after 0.3 s (not shown).

**Figure 8.**Results when a fault S

_{fD}at 20% of cable length is applied at 0.3s with RF = 0.04 Ω after load (RL = 12) is switched at t = 0.15 s. The dotted blue line reaches approx. 82 p.u. before 0.3 s and approx. 40 p.u. after 0.3 s (not shown).

**Figure 9.**Results for the space where all the constraints are satisfied for fault impedances ≤300 Ω. The green pixels indicate that the constraints are satisfied (indicating certainty in avoiding blinding and nuisance tripping) and the white pixels indicate the conditions violating any of the constraints (indicating the possibility of nuisance tripping or blinding).

Relay Technique | Operating Principle | Robustness Against Distributed Load? | Communication System | Handling Converter Faults | Cost |
---|---|---|---|---|---|

Overcurrent | Current | No | Not Required | Weak | Low |

Symmetrical Component | Current | Fault Dependent | Not Required | Fair | Moderate |

Overvoltage | Voltage | No | Not Required | Fair | Low |

Differential | Current | No | Required | Excellent | High |

Distance | Impedance | No | Not Required | Fair | Moderate |

Adaptive | Current | Yes | Required | Excellent | High |

Traveling Wave | Current | Yes | Required | Good | High |

PMU | Bus Phase | Yes | Required | Excellent | High |

Phase-Based | Voltage & Current Phase | No | Not Required | Excellent | Moderate |

Model-Based (This work) | Impedance Model | Yes | Not Required | Excellent | Moderate |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Al Hassan, H.A.; Reiman, A.; Reed, G.F.; Mao, Z.-H.; Grainger, B.M. Model-Based Fault Detection of Inverter-Based Microgrids and a Mathematical Framework to Analyze and Avoid Nuisance Tripping and Blinding Scenarios. *Energies* **2018**, *11*, 2152.
https://doi.org/10.3390/en11082152

**AMA Style**

Al Hassan HA, Reiman A, Reed GF, Mao Z-H, Grainger BM. Model-Based Fault Detection of Inverter-Based Microgrids and a Mathematical Framework to Analyze and Avoid Nuisance Tripping and Blinding Scenarios. *Energies*. 2018; 11(8):2152.
https://doi.org/10.3390/en11082152

**Chicago/Turabian Style**

Al Hassan, Hashim A., Andrew Reiman, Gregory F. Reed, Zhi-Hong Mao, and Brandon M. Grainger. 2018. "Model-Based Fault Detection of Inverter-Based Microgrids and a Mathematical Framework to Analyze and Avoid Nuisance Tripping and Blinding Scenarios" *Energies* 11, no. 8: 2152.
https://doi.org/10.3390/en11082152