Fault study formulations for PowerModels and PowerModelsDistribution
Author lanl-ansi
3 Stars
Updated Last
1 Year Ago
Started In
February 2020


CI Documentation

Fault study for PowerModels and PowerModelsDistribution

In the future this will also include optimal protection coordation formulations and possibly also protection coordination constraints for optimal switching problems

Modeling Assumptions

  • Generators are modeled as voltage sources behind an impedance. For synchronous generation, this is the subtransient reactance $X_d''$. For inverters, this is currently a virtual resistance. A more accurate model for inverters will take into account their current limits
  • Loads are neglected
  • Faults are modeled as an admittance matrix

Usage Example (Balanced Case)

using PowerModels, PowerModelsProtection, Ipopt
net = PowerModels.parse_file("case5.raw", import_all=true)
solver = JuMP.with_optimizer(Ipopt.Optimizer)

net["fault"] = Dict()
net["fault"]["1"] = Dict("bus"=>2, "r"=>0.0001)

results = PowerModelsProtection.run_fault_study(net, solver )


TODO section has moved to issues

LLG Fault Model

Wye & Delta Load Configurations Unbalanced Wye to Delta Admittance Conversion

Inverter Fault Models

Grid-Following Inverter


  1. k*p = vr*cr - vi*ci
  2. k*q = vr*ci + vi*cr

where k is a decision variable ranging from 0 to 1


An inverter under unbalanced conditions will operate at a fixed power factor averaged across all phases while injecting only positive-sequence current. This sounds ugly, but the constraints don't appear to be too bad.


  1. a = ar + ai = exp(j*2*pi/3) = -1/2 + j*sqrt(3/2
  2. a^2 = a2r + j*a2i = exp(j*4*pi/3) = -1/2 - j*sqrt(3/2

Positive-sequence current constraints:

  1. car = c1r
  2. cai = c1i
  3. cbr = a2r*c1r + a2i*c1i
  4. cbi = a2r*c1i + a2i*c2r
  5. ccr = ar*c1r + ai*c1i
  6. cci = ar*c1i + ai*c1i

Constant power factor constraints:

  1. k*p = var*car + vbr*cbr + vcr*ccr - vai*cai - vbi*cbi - vci*cci
  2. k*q = var*cai + vbr*cbi + vcr*cci + vai*car + vbi*cbr + vci*ccr

Grid-Forming Inverter

Virtual Resistance Model

vr, vi set from inverter node voltage base power flow rs = 0.8 pu, gives 1.3 pu current into a short xs = 0 pu

Current Limiting Model

This model assumes that vm, va set from inverter node voltage from base power flow

vr[c] = kg[c]*vm[c]*cos(va[c])
vi[c] = kg[c]*vm[c]*sin(va[c])
-cmax <= crg[c] <= cmax for c in cnds
-cmax <= cig[c] <= cmax for c in cnds

Objective is sum( sum((vr[c] - vm[c]*cos(va[c]))^2 + (vi[c] - vm[c]*sin(va[c]))^2 for c in cnd) for g in inverter_gens)

Current Limiting Model with Droop


Vg0 = V0 + zg*Ig0
V = kg*Vg0 - Z*Ig
|Ig| <= Igmax

Objective is sum( (vg - vg0)^2 for g in inverter_gens)

This code is provided under a BSD license as part of the Multi-Infrastructure Control and Optimization Toolkit (MICOT) project, LA-CC-13-108.

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