Appendix A. Three-Phase Power Flow Notation
Appendix A documents the notation required to define the fundamentals of the mathematical formulations. The notation discussed here is used in the ac unbalanced three-phase power flow. These equations are used as the core power flow formulation dictating the physics of power flow in distribution feeders with networkable microgrids. It should be noted that not all of the following parameters and variables are used in the formulation presented hererin, but are included for a general formulation that could be used in power flow and optimal power flow formulations.
Sets:
—set of complex numbers
—set of buses (nodes), indexed by i
—set of edges (lines and transformers), indexed by ; each edge is assigned an arbitrary direction from a bus i to a bus j and is uniquely identified with k
—set of three-phase transformers indexed by
—set of generator, indexed by g
—set of generator connected to bus , indexed by g
—set of batteries, indexed by
—set of batteries connected to bus , indexed by
—set of outgoing edges from i, indexed by
—set of incoming edges to i, indexed by
—phase set
—phase set for bus i
—phase set for line
Parameters:
—symmetric complex admittance matrix for edge
—symmetric complex impedance matrix for edge
—admittance matrix of three-phase transformer
—admittance of transformer
—tap ratio of transformer
—connection matrix of three-phase transformer
Variables:
—vector of complex current generation at
—vector of complex current provided (injected) at
—vector of complex current consumed at
—vector of complex current flow on line
—vector of complex voltage on bus
—vector of complex power generation at
—vector of complex power provided (injected) at
—vector of complex power consumed at
—vector of complex power flow on line
Appendix B. Short-Circuit Formulation Notation
Appendix B documents the notation required to define the fundamentals of the mathematical formulations. The notation discussed here is used to model short-circuit analysis.
Sets:
—set of faults, indexed by f
—set of grid-following inverters, indexed by g
—set of grid-forming inverters, indexed by g
Parameters:
—the bus of fault f
—admittance matrix of fault f
—complex parameter used for calculating sequence components
—virtual impedance real part upper limit, indexed by g
—virtual impedance imaginary part upper limit, indexed by g
—scaling for voltage upper and lower limit, indexed by g
—reference voltage of inverter, indexed by g
Variables:
—vector of complex fault current generation at for fault f
—vector of complex fault current at for fault f
—vector of complex load current at for fault f
—vector of complex fault current flow on line for fault f
—vector of complex voltage on bus for fault f
—virtual resistance of inverter g
—virtual reactance inverter g
—voltage set point of inverter g
Appendix C. Protection Formulation Notation
Appendix C documents the notation required to define the fundamentals of the mathematical formulations. The notation discussed here is used to model protection.
Sets:
—set of protection zones, indexed by
—set of protective devices, indexed by p
—set of protective devices in zone , indexed by p
—set of branches with protective devices, indexed by p
Parameters:
—operating area for ground fault admittance relay
—operating area for line fault admittance relay
—time-overcurrent relay time-dial scaling
—time-overcurrent relay curve shape
s—coordination time interval for time-overcurrent relays
s—maximum allowable trip time for time-overcurrent relays
Variables:
—time-overcurrent relay time-dial setting, indexed by
—time-overcurrent relay pickup current, indexed by
—time-overcurrent relay trip time, indexed by
—1st order statistic of time-overcurrent relay trip times
—Measured positive-sequence impedance between relay and fault for line-ground faults
—Measured positive-sequence impedance between relay and fault for line-line faults
—Measured positive-sequence impedance at relay
—Measured zero-sequence impedance at relay
—Residual compensation factor for ground-fault relay
Appendix D. Restoration Notation
Appendix D documents the notation required to define the fundamentals of the mathematical formulations. The notation discussed here is used to model common engineering limits and constraints, which restrict the operations of distribution feeder restoration with microgrids.
Parameters:
—vector of complex power demand at bus during restoration
Variables:
—vector of complex current generation at during restoration
—vector of complex current consumed at during restoration
—vector of complex current flow on line during restoration
—vector of complex voltage on bus during restoration
—vector of complex power generation at during restoration
—vector of complex power consumed at during restoration
—vector of complex power flow on line during restoration —open and close variable for edge ; denotes an open edge, denotes a closed edge
—on and off variable for the load at bus i during restoration; denotes the load is not energized, denotes the load is energized
—on and off variable for the generator g during restoration; denotes the generator is off, denotes the generator is on
Figure 1.
Star-mesh transformation for a three-phase ungrounded fault.
Figure 1.
Star-mesh transformation for a three-phase ungrounded fault.
Figure 2.
Star-mesh transformation for a line-line-ground fault.
Figure 2.
Star-mesh transformation for a line-line-ground fault.
Figure 3.
Star-mesh transformation for a three-phase to ground fault.
Figure 3.
Star-mesh transformation for a three-phase to ground fault.
Figure 4.
Sub-figures (
a–
c) present two adjacent three-, two-, and single-phase buses that are connected via a distribution line. Sub-figure (
d) presents two adjacent three-phase buses that are connected via an ideal transformer [
64].
Figure 4.
Sub-figures (
a–
c) present two adjacent three-, two-, and single-phase buses that are connected via a distribution line. Sub-figure (
d) presents two adjacent three-phase buses that are connected via an ideal transformer [
64].
Figure 5.
The IEEE 123-Node System [
66].
Figure 5.
The IEEE 123-Node System [
66].
Figure 6.
Graphical illustration of constraints for a grid-following inverter.
Figure 6.
Graphical illustration of constraints for a grid-following inverter.
Figure 7.
Graphical illustration of constraints for a grid-forming inverter.
Figure 7.
Graphical illustration of constraints for a grid-forming inverter.
Figure 8.
Oneline diagram for an ideal protection scheme.
Figure 8.
Oneline diagram for an ideal protection scheme.
Figure 9.
Oneline diagram for a DCB scheme.
Figure 9.
Oneline diagram for a DCB scheme.
Figure 10.
Schematic for one relay in a directional comparison blocking (DCB) scheme.
Figure 10.
Schematic for one relay in a directional comparison blocking (DCB) scheme.
Figure 11.
Examples of time-overcurrent relay curves [
20].
Figure 11.
Examples of time-overcurrent relay curves [
20].
Figure 12.
Oneline diagram of the Three-Node Islanded Microgrid System.
Figure 12.
Oneline diagram of the Three-Node Islanded Microgrid System.
Figure 13.
Transient microgrid model.
Figure 13.
Transient microgrid model.
Table 1.
Comparison of results for short-circuit flow formulation.
Table 1.
Comparison of results for short-circuit flow formulation.
Fault Node | Fault Phasing | IEEE PES Fault Current (A) | OpenDSS Fault Current (A) | IEEE-DSS Difference (%) | PMsP Fault Current (A) | IEEE-PMsP Difference (%) |
---|
13 | LG | 4400.1 | 4422.0 | +0.50 | 4444.9 | +1.02 |
13 | LL | 4886.6 | 4736.0 | −3.08 | 4954.2 | +1.38 |
13 | 3P | 5435.3 | 5305.0 | −2.40 | 5517.3 | +1.51 |
67 | LG | 2339.5 | 2392.0 | +2.24 | 2362.2 | +0.97 |
67 | LL | 2890.0 | 2879.0 | −0.38 | 2933.1 | +1.49 |
67 | 3P | 3236.6 | 3251.0 | +0.44 | 3285.6 | +1.51 |
113 | LG | 1382.6 | 1390.0 | +0.54 | 1392.2 | +0.69 |
Table 2.
Summary of case study configurations.
Table 2.
Summary of case study configurations.
Configuration | Generation Type | Generation Placement | Islanding State |
---|
Base-GC | none | – | grid-connected |
SG-M-GC | synchronous | multiple | grid-connected |
SG-M-I | synchronous | multiple | islanded |
GflI-M-GC | grid-following | multiple | grid-connected |
GfrI-S-I | grid-forming | single | islanded |
GfrI-M-I | grid-forming | multiple | islanded |
GflI-GfrI-M-I | grid-following and grid-forming | multiple | islanded |
Table 3.
Comparison of results for configuration GflI-M-GC of the IEEE 123-Node System.
Table 3.
Comparison of results for configuration GflI-M-GC of the IEEE 123-Node System.
Fault Node | Fault Phasing | OpenDSS Fault Current (A) | PMsP Fault Current (A) | Difference (%) |
---|
13 | LG | 4385.2 | 4490.6 | +2.40 |
13 | LL | 5108.1 | 4995.3 | −2.21 |
13 | 3P | 5315.0 | 5578.8 | +4.96 |
67 | LG | 2478.3 | 2495.2 | +0.68 |
67 | LL | 3173.0 | 3132.2 | −1.29 |
67 | 3P | 3433.5 | 3538.0 | +3.04 |
113 | LG | 1532.0 | 1545.9 | +0.91 |
Table 4.
Inverter controller parameters.
Table 4.
Inverter controller parameters.
Subsystem | Symbol | Value |
---|
Voltage loop | kpv | 0.35 |
Voltage loop | krv | 400 |
Voltage loop | kvh5 | 4 |
Voltage loop | kvh7 | 20 |
Voltage loop | kvh11 | 11 |
Current loop | kpi | 0.7 |
Current loop | kri | 400 |
Current loop | kih5 | 30 |
Current loop | kih7 | 30 |
Current loop | kih11 | 30 |
Table 5.
Hardware parameters.
Table 5.
Hardware parameters.
Name | Symbol | Value | Unit |
---|
Fundamental frequency | f | 60 | Hz |
Line-line voltage | V | 480 | V |
Inverter rated power | P | 50 | kW |
DC-bus voltage | Vdc | 1800 | V |
Output filter inductance | L | 18 | F |
Output filter capacitance | C | 250 | nF |
Maximum rms output current | Imax | 70 | A |
Cable resistance | Rc | 39 | m |
Cable inductance | Lc | 70.8 | H |
Load real power | Pd | 25 | kW |
Load reactive power | Qd | 12.5 | kW |
Table 6.
Comparison of results for configurations Base-GC, SG-M-GC and SG-M-I of the IEEE 123-Node System.
Table 6.
Comparison of results for configurations Base-GC, SG-M-GC and SG-M-I of the IEEE 123-Node System.
Fault Node | Fault Phasing | Base-GC Fault Current (A) | SG-M-GC Fault Current (A) | SG-M-I Fault Current (A) |
---|
13 | LG | 4444.9 | 5914.654 | 3969.269 |
13 | LL | 4954.2 | 6211.437 | 3914.959 |
13 | 3P | 5517.3 | 7116.605 | 4559.597 |
67 | LG | 2362.2 | 4788.279 | 3896.403 |
67 | LL | 2933.1 | 5301.483 | 3969.117 |
67 | 3P | 3285.6 | 6153.95 | 4598.621 |
113 | LG | 1392.2 | 2987.6 | 2802.407 |
Table 7.
Comparison of results for configurations GfrI-S-I, GfrI-M-I, and GflI-GfrI-M-I of the IEEE 123-Node System, and 20 fault resistance.
Table 7.
Comparison of results for configurations GfrI-S-I, GfrI-M-I, and GflI-GfrI-M-I of the IEEE 123-Node System, and 20 fault resistance.
Fault Node | Fault Phasing | GfrI-S-I Fault Current (A) | GfrI-M-I Fault Current (A) | GflI-GfrI-M-I Fault Current (A) |
---|
13 | LG | 119.1 | 99.4 | 114.5 |
13 | LL | 206.0 | 138.9 | 148.7 |
13 | 3P | 355.4 | 229.0 | 171.9 |
67 | LG | 118.0 | 73.6 | 107.7 |
67 | LL | 203.3 | 177.5 | 168.0 |
67 | 3P | 355.4 | 236.8 | 170.0 |
113 | LG | 115.2 | 74.4 | 62.7 |
Table 8.
Comparison of results for configurations GfrI-S-I, GfrI-M-I, and GflI-GfrI-M-I of the IEEE 123-Node System, and 20 fault resistance.
Table 8.
Comparison of results for configurations GfrI-S-I, GfrI-M-I, and GflI-GfrI-M-I of the IEEE 123-Node System, and 20 fault resistance.
Fault Node | Fault Phasing | GfrI-S-I Fault Current (A) | GfrI-M-I Fault Current (A) | GflI-GfrI-M-I Fault Current (A) |
---|
13 | LG | 1165.7 | 816.6 | 861.8 |
13 | LL | 1165.7 | 892.6 | 883.5 |
13 | 3P | 1165.8 | 1082.1 | 1087.2 |
67 | LG | 1165.7 | 1l023.4 | 1008.8 |
67 | LL | 1165.7 | 839.7 | 963.6 |
67 | 3P | 1165.8 | 1080.3 | 1091.0 |
113 | LG | 1165.7 | 446.1 | 475.1 |
Table 9.
Comparison of results for a grid-forming inverter on the microgrid of
Figure 12.
Table 9.
Comparison of results for a grid-forming inverter on the microgrid of
Figure 12.
Fault Resistance () | Fault Phasing | Simulink Fault Current (A) | PMsP Fault Current (A) | Difference (%) |
---|
20.0 | LG | 13.79 | 13.73 | −0.44 |
20.0 | LL | 11.94 | 11.89 | −0.42 |
20.0 | 3P | 13.79 | 13.73 | −0.44 |
5.0 m | LG | 103.4 | 114.6 | −10.8 |
5.0 m | LL | 60.55 | 57.29 | −5.38 |
5.0 m | 3P | 69.93 | 66.15 | −5.41 |