 Original research
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Protection and control of microgrids using dynamic state estimation
Protection and Control of Modern Power Systems volume 3, Article number: 31 (2018)
Abstract
High penetration of Converter Interfaced Generations (CIGs) presents challenges in both microgrid (μGrid) circuit and other system with CIG resources, such as wind farms and PV plants. Specifically, protection challenges are mainly brought by the insufficient separation between fault and load currents, especially for μGrids in islanded operation, and the short connection length in μGrids. In addition, CIG resources exhibit limited inertia and weak coupling to any rotating machinery, which can result in large transients during disturbances. To address the above challenges, this paper proposes a Dynamic State Estimation (DSE) based algorithm for protection and control of systems with substantial CIG resources such as a μGrid. It requires a highfidelity dynamic model and time domain (sampled value) measurements. For μGrid circuit protection, the algorithm dependably and securely detects internal faults by checking the consistency between the circuit model and available measurements. For CIG control, the algorithm estimates the frequency at other parts of a μGrid using CIG local information only and then utilizes it to provide supplementary feedback control. Simulation results prove that DSE based protection algorithm detects internal faults faster, ignores external faults and has improved sensitivity towards high impedance faults when compared to conventional protection methods. DSE based CIG control scheme also minimizes output oscillation and transient during system disturbances.
Introduction
There exists increasing penetration of renewables in modern power systems. The renewable generating units, including wind turbines, photovoltaic cells, etc., are commonly connected to the grids via converter interfaces, and thus are referred as Converter Interfaced Generations (CIGs). Large numbers of CIGs can be integrated within μGrids to minimize the impact on the main grid [1,2,3]. However, this brings challenges on the protection of μGrids against faults and the control of CIGs. Details of the above two challenges are described in the following sections.
Protection of μGrid circuits
The protection challenge is brought by the following characteristics of the μGrids with CIGs: (a) bidirectional power flow introduced by CIGs; (b) significant difference between the fault current characteristics in gridconnected mode and in islanded mode; (c) short connection length under protection; and (d) frequent output change from the intermittent generations.
Specifically, legacy protection schemes, e.g. overcurrent protection, distance protection, and differential protection, will encounter different limitations in protecting μGrids.
For overcurrent protection, directional elements are required considering the bidirectional power flow of μGrids. However, it is almost impossible to provide required selectivity because: (a) fault current may change due to connectivity changes of CIGs; (b) fault current variation within a μGrid is limited; and (c) fault current is comparable to load current when the fault current is contributed only by CIGs.
For distance protection, the limitation comes from the short length of the connection lines, resulting in uncertain characteristics of the relay. The relay can easily fail to operate due to the calculated impedance error brought by (a) the sequence model approximation of distance relays, (b) significant difference between the fault current contributed by the main grid and the CIGs during gridconnected mode, and (c) the impact of grounding impedance on the relay algorithm.
Differential protection is one of the most effective schemes in protecting μGrid circuits, in which the zero sum of currents at both terminals of the line is tested. The sensitivity could be further improved with the use of negative or zero sequence currents. However, it still has limitations in detecting high impedance faults.
To overcome above limitations, researchers have also proposed several new methods. In [4], an adaptive protection scheme with μGrid central protection unit is proposed. The method updates settings of overcurrent relays and adjusts coordination among relays according to operating conditions of CIGs and the topology of the μGrid. However, the method does not consider variable fault current level due to output change of CIGs. In addition, it needs a sophisticated communication network between the central protection unit and all relays/CIGs. In [5], a protection strategy using microprocessorbased relays for a lowvoltage μGrid is proposed, which does not require communication nor adaptive protection. However, its sensitivity is poor when applied to short lines and it may trip internal faults with a long delay. In [6], differential protection and the comparative voltage protection are utilized as the main and backup protection functions. However, the method may not be able to detect high impedance faults where the differential current is less than 10% of the nominal current. In [7], a differential energy based protection scheme is proposed. However, the effectiveness of the scheme highly depends on userselected settings.
Control of CIGs
Conventional power systems are mainly powered by directly connected synchronous generators, where the load/generation balance is ensured by controlling the frequency of the synchronous generator. Also, the variations of frequency and phase angle are relatively small and slow due to high moment of inertia. For a μGrid with high penetration of CIGs, the frequency is irrelevant to the load/generation balance [8], and CIGs do not naturally exhibit inertia [9]. Therefore during disturbances, the output of CIGs may severely oscillate, causing damage or shutdown of the converters.
To solve the above problems, researchers have proposed control schemes to mimic frequency and inertial response for CIGs [10, 11]. There are limitations to these schemes as the equivalent torques and inertia are greatly related to electric currents which are limited by the converters. Thus, these schemes may be weakened from the fact that the fault current level of CIGs (typically less than 160% of rated current for short durations and less than 120% for longer durations [12]) is much smaller compared to the fault current level of conventional synchronous generators (500% ~ 1000% of rated current). Researchers have also proposed some additional control schemes based on local information without the need for communication, including the twoaxis and threeaxis based converter control schemes [13,14,15]. However, they may not be able to follow the frequency variation of the rest of the μGrid resulting in large output oscillation. Therefore, it is necessary to develop an algorithm that is based on local information only but also has the capability to access the information of the other parts of the μGrid without communication channels.
Proposed method
A Dynamic State Estimation (DSE) based algorithm [16,17,18] is proposed in this paper to solve the above protection and control problems. The algorithm estimates the states of the components of interest using available sampled measurements and a highfidelity dynamic model, so that the electrical transients can be accurately captured. This feature of the DSE algorithm enables more dependable and secure protection and control of μGrid components. Specifically, for protection application, the estimated states are utilized to check consistency between the measurements and the model, and a trip signal is issued if any inconsistency is detected. For control application, the estimated states are utilized to first calculate the remote side frequency information and then this information is adopted to provide a supplementary feedback control to the CIG controller.
The rest of the paper is arranged as follows. Section 2 introduces the DSE based algorithm and its application on protection and control of μGrids. Section 3 provides the simulation results for μGrid protection and Section 4 demonstrates the results for CIG control. Finally, Section 5 concludes the paper.
Methods
DSE based algorithm and its applications
The DSE based algorithm requires a high fidelity dynamic model of the components of interest. To build an objectoriented algorithm that works for all components, a standard Algebraic Quadratic Companion Form (AQCF) syntax is briefly introduced (details can be found in [16]). The DSE algorithm is then introduced with the model in AQCF syntax. Finally, the application details on μGrid protection and CIG control are provided.
AQCF syntax
The physical laws that the components of interest obey can be described via a set of algebraic and differential equations. These equations are further quadratized by introducing additional variables to make sure that the highest order of nonlinearities is less than or equal to two. For any device, the equations can be written in the following device Quadratized Dynamic Model (QDM) syntax, where x(t) and i(t) represent the state vector and terminal current vector of the component:
An example of a μGrid circuit model (cable) in QDM syntax is presented in Appendix.
Equations (1) is then integrated using the quadratic integration method [19] to convert the differential equations into algebraic equations, where the system dynamics are represented in terms of past history terms. This generates the model in device AQCF syntax, where t_{m} = t − Δt and Δt is the sampling interval:
A number of measurements are typically available for the components of interest. Each available measurement can be expressed in terms of the states in (2). This generates the measurement model in AQCF syntax, where z(t, t_{m}) is the measurement vector that includes measurements at time t and t_{m},
DSE algorithm
The DSE algorithm executes every two sampling intervals and estimates the states x(t, t_{m}) from measurements z(t, t_{m}). Here the unconstraint optimization method is utilized,
where \( \mathbf{W}=\operatorname{diag}\left\{1/{\sigma}_1^2,1/{\sigma}_2^2,\cdots \right\} \), and σ_{i} is the standard deviation of measurement i.
The best estimation of states \( \widehat{\mathbf{x}}\left(t,{t}_m\right) \) can be calculated by the iterative algorithm until convergence,
where the Jacobian matrix is H = ∂h(x)/∂x.
Application details of μGrid circuit protection
The DSE Based Protection (EBP) determines an internal fault by checking consistency between the measurements and the model. This consistency is evaluated by the confidence level P_{conf}(t), which can be calculated by,
where P(ζ(t), m_{v}) is the probability of χ^{2} distribution given χ^{2} ≤ ζ(t) with the dimension difference between the measurement and the model as the degrees of freedom, \( \widehat{\mathbf{r}}\left(t,{t}_m\right) \) is the vector of residuals and \( \widehat{\mathbf{s}}\left(t,{t}_m\right) \) is the vector of normalized residuals.
The μGrid circuit needs to be tripped if there exists an internal fault which is manifested with an inconsistency between the measurements and the model, i.e. the confidence level is low. The EBP relay asserts an internal fault condition when the confidence level which is simply the probability of “lack of fault in the protection zone” goes below 10%, (the 10% is a user selected cutoff value and has proved to work well). To ensure dependable and secure protection, a user defined intentional delay τ_{delay} is introduced to determine the trip decision as:
Application details of CIG control
The states of the line connecting the CIG and the rest of the μGrid (remote side) are accurately estimated via dynamic state estimator located at the CIG site where only local information is needed. The dynamic state estimation provides the estimated remote side voltages and currents since the circuit AQCF model is part of the dynamic estimation model. The remote side frequency and the rate of frequency change are subsequently calculated from the remote side voltages and currents. Finally, a supplementary control scheme is utilized, where the estimated frequency information from the remote side is introduced as inputs. Note that this DSE scheme is based on sampled values so it can accurately capture electrical transients including frequency information. This enables the controller to utilize remote side frequency information without any telemetered data (communications).
The detailed control diagram is shown in Fig. 1. The control input signals (at time t_{k}) include the frequencies and rates of change of frequency at the local side (f_{local} and df_{local}/dt) and the remote side (f_{remote} and df_{remote}/dt), and the real power (P_{m}) and the reactive power (Q_{m}). The control output signals (at time t_{k + 1}) include the frequency f_{ctrl}, phase angle θ_{ctrl} and modulation index m_{ctrl}, corresponding to the control of the frequency, phase angle and amplitude of the output voltage, respectively. These three output signals are utilized to generate switching sequences as detailed below.

(a)
The aim of the frequency control is to make the frequency output f_{ctrl}(t_{k + 1}) track the remote side frequency f_{remote}(t_{k + 1}), to minimize the oscillation of the angle difference between the two sides. The process first calculates Δδ_{remote}(t_{k + 1}) − Δδ_{local}(t_{k + 1}), i.e. the change of the angle difference during the period [t_{k}, t_{k + 1}], and proportional and integral (PI) control is then utilized to calculate the frequency output f_{ctrl}(t_{k + 1}).

(b)
The aim of the phase angle control is to make the real power output P_{m}(t_{k + 1}) track the reference power P_{ref}(t_{k + 1}). This process is implemented via a PI controller.

(c)
The aim of the modulation index control is to make the reactive power output Q_{m}(t_{k + 1}) track the reference power Q_{ref}(t_{k + 1}). This process is also implemented via a PI controller.
Results
μGrid circuit protection
The example test system is a 480 V μGrid illustrated in Fig. 2. The μGrid is connected to the main grid with a static switch (SS) at the point of common coupling and a 13.8 kV/480 V distribution transformer. There are two distributed generations inside the μGrid, including a small wind turbine system and a PV system. Note that both distributed generations are connected to the μGrid via converters.
The line under protection is A1A2, with a length of 330 ft, and its cable structure is shown in Fig. 3. The sequence parameters are shown in Table 1, and the series resistance, series reactance and shunt capacitance matrices per unit length are shown in Table 2. Threephase currents and voltages are measured at both sides of line A1A2, and the performance of the proposed EBP technique for the protection of the μGrid are demonstrated via several events. Note that two legacy relays are considered for comparison, i.e. a distance protection relay (installed at side A2) and a line differential protection relay.
Settings of legacy relays
Distance relay settings
Zone 1, 2 and 3 of the distance relay are selected as 80%, 125% and 260% of the positive sequence impedance, respectively. Detail settings are shown in Table 3.
Line differential relay settings
Zerosequence current line differential protection with alpha plane method [20] is used. The restraint region is between 1/6 to 6, with the total angular extent of 195°. The thresholds are: phase current 144 A, 3I_{0} (zerosequence current) 12 A, and 3I_{2} (negativesequence current) 12 A. The relay trips if the current phasor ratio falls outside the restraint region and at least one of the thresholds is exceeded, with a delay of 1 cycle.
EBP settings
For consistency, the intentional delay for the EBP relay is selected as τ_{delay}= 1 cycle and the reset time is τ_{reset}= 2 cycles.
Simulation results
The performance of the proposed EBP scheme and two legacy schemes (distance and line differential protection) is investigated using the following 4 events. Events 1 to 3 are faults when the μGrid is in gridconnected mode, whereas event 4 is a fault when the μGrid is in islanded mode.
Event 1: Internal bolted fault, gridconnected mode
A bolted (0.01 Ω) phase A to ground fault occurs at 1.4 s in the middle of line A1A2. In this event, the μGrid is connected to the main grid and the location of the fault is shown as F_{1} in Fig. 2. Threephase currents and voltages at both sides of A1A2 are provided in Fig. 4.
Distance relay
The calculated impedance trace, superimposed on the relay characteristics, is shown in Fig. 5. As can be seen, the calculated impedance falls outside of the tripping zone. This is mainly due to the large fault current from the grid side and the short length of the cable. Therefore, the distance relay fails to detect this internal fault.
Line differential relay
The trace of the zerosequence current phasor ratio, superimposed on the relay characteristics, is shown in Fig. 6. Along the trace, the characters ‘x’ and ‘o’ show whether any of the thresholds is exceeded. Prior to the fault, the ratio stays near (− 1, 0) with no threshold exceeded. During the fault, the ratio enters the trip region one sample after the fault occurrence (at 1.4002 s), with the thresholds exceeded. Therefore, the line differential protection trips the line at 1.4169 s (1.4002 s + 1 cycle of 0.0167 s).
EBP relay
The performance of EBP relay is depicted in Fig. 7. The first two channels demonstrate the residuals and the normalized residuals of the threephase currents at side A2. It can be observed that the residuals are small before the fault but become extremely large during the fault. The confidence level in the third channel drops to zero one sample (208 μs) after the fault occurrence. The fault is detected at 1.4002 s and the circuit is tripped (in the fourth channel) at 1.4169 s.
Summary of event 1
For this internal bolted phase A to ground fault during the gridconnected mode, the distance relay fails to detect this internal fault, whereas both the line differential relay and the EBP relay detect the fault at 1.4002 s and trip the fault at 1.4169 s.
Event 2: Internal high impedance fault, gridconnected mode
A high impedance (50 Ω) phase A to ground fault occurs at the middle of A1A2 at 1.4 s with the μGrid connected to the main grid. The location of the fault is F_{1} as shown in Fig. 2. Threephase currents and voltages at both sides of A1A2 are provided in Fig. 8.
Distance relay
The calculated impedance trace, superimposed on the relay characteristics, is shown in Fig. 9. As can be seen, the calculated impedance during fault stays close to the value before the fault and falls outside of the tripping zone. This is due to the high fault impedance. Therefore, the distance relay fails to detect this internal fault.
Line differential relay
The trace of the zerosequence current phasor ratio, superimposed on the relay characteristics, is shown in Fig. 10. Prior to the fault, the ratio stays close to (− 1, 0) with no threshold exceeded. During the fault, the ratio enters the trip region at 1.4161 s, though still no threshold is exceeded. In fact the calculated ‘3I_{0}’ during the fault is less than 4.5 Amps, which is far less than the threshold setting of 12 Amps. Therefore, the line differential protection also fails to detect this internal fault.
EBP relay
The performance of the EBP relay is depicted in Fig. 11. The confidence level drops to zero one sample after fault occurrence and oscillates during the fault, indicating abnormality of the circuit. The fault is detected at 1.4002 s and the circuit is tripped at 1.4223 s.
Summary of event 2
For this internal high impedance phase A to ground fault during gridconnected mode, both the distance relay and the line differential relay fail to detect this internal fault, whereas the EBP relay detects the fault at 1.4002 s and the fault is tripped at 1.4223 s.
Event 3: External bolted fault, gridconnected mode
With the μGrid connected to the main grid, a bolted (0.01 Ω) phase A to ground fault occurs at the secondary side of the transformer at 1.4 s. The location of the fault is shown as F_{2} in Fig. 2. Threephase currents and voltages at both sides of A1A2 are provided in Fig. 12.
Distance relay
The calculated impedance trace, superimposed on the relay characteristics, is shown in Fig. 13. As seen, the calculated impedance correctly falls outside of the tripping zone, and therefore, the distance relay will correctly ignore this external fault.
Line differential relay
The trace of the zerosequence current phasor ratio, superimposed on the relay characteristics, is shown in Fig. 14. Prior to the fault, the ratio stays close to (− 1, 0) with no threshold exceeded. During the fault, although the thresholds are exceeded, the ratio still stays close to (− 1, 0) inside the restraint zone. Therefore, the line differential protection will also correctly ignore this external fault.
EBP relay
The performance of the EBP relay is depicted in Fig. 15. The residuals increase slightly at the beginning of the fault and gradually decrease to the values of normal operating condition. The confidence level remains high during the fault, which indicates that the line under protection is healthy. Therefore, the EBP relay also correctly ignores this external fault.
Summary of event 3
For this external bolted phase A to ground fault during the gridconnected mode, the two legacy relays (distance and line differential) as well as the EBP relay all correctly ignore this external fault.
Event 4: Internal bolted fault, islanded mode
With the μGrid in islanded mode, a bolted (0.01 Ω) phase A to ground fault occurs at 1.4 s at the middle of A1A2, i.e. F_{1} as shown in Fig. 2. Threephase currents and voltages at both sides of A1A2 are provided in Fig. 16.
Distance relay
The calculated impedance trace, superimposed on the relay characteristics, is shown in Fig. 17. The calculated impedance falls into the tripping zone 2 at 1.420 s, though the impedance should be theoretically inside zone 1 (this bolted fault is within 80% of the line). This is mainly due to the sequence model approximation of the line and its short length. Therefore, the distance relay trips this internal fault with delay at 1.5696 s (1.4196 s + 0.15 s).
Line differential relay
The trace of the zerosequence current phasor ratio, superimposed on the relay characteristics, is shown in Fig. 18. Prior to the fault, the ratio stays close to (− 1, 0) with no threshold exceeded. During the fault, the ratio enters the trip region at 1.4014 s, with thresholds exceeded. Therefore, the line differential protection trips the circuit at 1.4181 s (1.4014 s + 1 cycle of 0.0167 s).
EBP relay
The performance of the EBP relay is depicted in Fig. 19. The first two channels demonstrate the residuals and the normalized residuals of the threephase currents at side A2, respectively. It can be observed that the residuals are small before the fault but become extremely large during the fault. The confidence level in the third channel drops to zero one sample after the fault occurrence. The fault is thus detected at 1.4002 s and the circuit is tripped (in the fourth channel) at 1.4169 s.
Summary of event 4
For this internal bolted phase A to ground fault during the islanded mode, the distance relay detects the fault at 1.4196 s and trips the circuit at 1.5696 s, whereas the line differential relay detects the fault at 1.4014 s and trips the circuit at 1.4181 s. For the EBP relay, the fault is detected at 1.4002 s and the circuit is tripped faster than both legacy relays at 1.4169 s.
Event 5: Internal high impedance fault, gridconnected mode, with 1% random error
A high impedance (50 Ω) phase A to ground fault occurs in the middle of A1A2 at 1.4 s. This event is the same as event 2 except that the threephase currents and voltages at both sides of line A1A2 are added with 1% random error. The threephase currents and voltages at both sides of A1A2 are similar to those in Fig. 8.
Distance relay
The calculated impedance trace is similar to that in Fig. 9 and the distance relay fails to detect this internal fault.
Line differential relay
The trace of the zerosequence current phasor ratio is similar to that in Fig. 10 and the line differential protection also fails to detect this internal fault.
EBP relay
The performance of the EBP relay is depicted in Fig. 20. Prior to the fault, the confidence level remains at a relatively high level. During the fault, the confidence level drops to zero one sample after fault occurrence and oscillates during the fault, indicating abnormality of the circuit. The fault is detected at 1.4002 s and the circuit is tripped at 1.4183 s.
Summary of event 5
For this internal high impedance phase A to ground fault during the gridconnected mode with added 1% random error, both the distance relay and line differential relay fail to detect this internal fault, whereas the EBP relay detects the fault at 1.4002 s and trips the line at 1.4183 s.
CIG control
The example test system is a 34.5 kV μGrid provided in Fig. 21. The μGrid is connected to the main grid with a static switch at the point of common coupling and a 230 kV/34.5 kV transformer. The series resistance, series reactance and shunt capacitance matrices per unit length of the μGrid circuit are shown in Table 4. There are three CIGs inside the μGrid, including two PV systems and a Wind Turbine System (WTS). To demonstrate the performance of the proposed CIG control scheme via the WTS, its controller ensures the local side phase angle (side L of the 34.5 kV transmission line LR) to follow the change of the remote phase angle (side R of the 34.5 kV transmission line LR). To achieve the above goal, the DSE algorithm based on AQCF syntax is utilized to accurately estimate the remote side frequency and rate of frequency change based on local information only. Note that in this simulation, intentional disturbances are introduced to generate frequency oscillations inside the μGrid.
Remote side frequency estimation
The simulation results of the remote side frequency estimation with local information only are provided in Fig. 22. The measurements are the threephase currents and voltages at the local side (side L). Note that the length of the 34.5 kV line LR is 2.5 miles. As shown in Fig. 22, the first two channels depict the actual and estimated threephase voltages at the remote side of the line LR. The third and fourth channels provide the actual and estimated frequencies, and the estimation error, respectively. It can be observed that the estimation error of the frequency is within −9.464 × 10^{−5}~2.82 × 10^{−4}Hz. The actual and estimated rates of frequency change, and the estimation error are shown in the fifth and sixth channels, respectively. As seen, the estimation error of the rate of the frequency change is within −1.906 × 10^{−3}~1.689 × 10^{−3}Hz/s.
To further validate the accuracy of the frequency estimation algorithm, further studies are conducted with different lengths of line LR and the results are shown in Table 5. It can be seen that the frequency estimation error is higher for longer lines though the estimation is still very accurate (maximum absolute error of 4.51 × 10^{−4}Hz for frequency and 2.896 × 10^{−3}Hz/s for the rate of frequency change).
CIG control
The estimate frequency and the rate of frequency change are utilized as feedback signals to control the output of the CIG. Traditional local information based control scheme utilizes Pf and QV droop control, however, the proposed DSE based control scheme is based on PQ mode and utilizes the estimated remote side frequency as the frequency input. The references of the real and reactive power are 2 MW and 0.5 MVar, respectively. Figure 23 shows the output of the CIG with the traditional local information based control scheme and the proposed DSE based control scheme. The first two channels depict the real power outputs from both schemes, while the third and fourth channels provide the reactive power outputs from both schemes. The fifth and sixth channels show the local and remote phase angles of both schemes.
With the traditional local information based control scheme, the real and reactive power outputs, the power factor and the phase angle difference oscillate. In this case, the CIG may exceed the desirable operation constraints potentially causing damage or leading to the shutdown of the CIG. With the proposed control scheme, the oscillation of the real and reactive power is significantly reduced. In addition, the power factor is kept within the operation constraint and the phase angle difference is stable. Therefore, undesirable transients are successfully minimized by the proposed control scheme.
Conclusion
Converter Interfaced Generations (CIGs) bring significant challenges in μGrid protection and CIG control. This paper proposes a Dynamic State Estimation (DSE) based algorithm using highfidelity dynamic models and sampled value measurements for dependable and secure protection and control of μGrid systems. For the protection application, the DSE based protection algorithm detects internal faults by checking any inconsistency between the measurements and the model. The advantages of the proposed protection scheme include: (a) detection of internal faults faster than legacy schemes; (b) improved sensitivity towards high impedance faults; (c) satisfactory performance in μGrid protection in both gridconnected mode and islanded mode; and (d) working correctly during nonideal conditions such as measurements with added 1% random error. For CIG control application, the DSE provides accurate estimation of the remote side frequency as well as the rate of frequency change from local information only, which can then be used to provide supplementary feedback control to the converters of the CIG. The advantages of the proposed control scheme include: (a) no need for communication from remote sides; (b) accurate estimation of the remote side frequency and rate of frequency change based on local information; and (c) minimizing the CIG output oscillation during system disturbances.
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Funding
This work is supported by Electric Power Research Institute (EPRI). Its support is greatly appreciated.
Availability of data and materials
Please contact author for data requests.
Author information
Affiliations
Contributions
YL carried out the dynamic state estimation based protection scheme, the dynamic state estimation based converter control scheme and drafted the manuscript. APM participated in the design and coordination of the study and helped to draft the manuscript. LS participated in the implementation of both the protection and the converter control schemes, and helped to draft the manuscript. SC participated in the implementation of the converter control scheme and helped to draft the manuscript. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Y. Liu.
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Competing interests
The authors declare that they have no competing interests.
Appendix
Appendix
μGrid circuit model (cable)
This Appendix describes the model of μGrid circuit (cable) in QDM syntax. Note that here a multisection model is adopted, where each section is a πequivalent model of the cable. The length of each section is selected to make sure that it is comparable to the traveling length of electromagnetic wave during one sampling interval. The overall QDM of any μGrid circuit can be obtained by mathematically combining QDMs of all sections. Figure 24 shows a πequivalent model of a circuit section with the cable structure in Fig. 3.
The model of the above section in QDM syntax is,
all other vectors / matrices are null;
where i_{a1}(t), i_{b1}(t), i_{c1}(t),i_{n1}(t), i_{a2}(t), i_{b2}(t), i_{c2}(t) and i_{n2}(t) are the threephase and neutral currents at both terminals of the section; v_{a1}(t), v_{b1}(t), v_{c1}(t),v_{n1}(t), v_{a2}(t), v_{b2}(t), v_{c2}(t) and v_{n2}(t) are the threephase and neutral voltages at both terminals of the section; i_{aL}(t), i_{bL}(t), i_{cL}(t) and i_{nL}(t) are the threephase and neutral currents through inductors; R, L and C are series resistance, series inductance, and shunt capacitance matrices of the section. Note that matrices R, L and C are accurately calculated from the physical structure and environmental condition of the cables.
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Liu, Y., Meliopoulos, A.P., Sun, L. et al. Protection and control of microgrids using dynamic state estimation. Prot Control Mod Power Syst 3, 31 (2018). https://doi.org/10.1186/s4160101801042
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Keywords
 Converter interfaced generation (CIG)
 Dynamic state estimation (DSE)
 μGrid protection