Distance protection of transmission lines

The 138 kV line from Lincoln Road to Hollis is 40 km (25 miles) long. Overcurrent relays are a poor fit for it: the current for a fault at the far end depends heavily on how many generators and lines are in service, so a relay set for the strongest system would be blind under the weakest, and one set for the weakest would overreach into the next line under the strongest. Distance relays avoid that problem by measuring the impedance between the relay and the fault, computed from the voltage and current at the relay location. Because line impedance is proportional to length, the measured impedance tells the relay how far away the fault is, almost independently of the source strength. Distance protection, device 21, is the workhorse of transmission line protection across the United States.

Measuring impedance

For a three-phase or phase-to-phase fault, a distance element divides the phase-to-phase voltage by the difference of the corresponding phase currents: for a BC fault, Z = (V_B − V_C)/(I_B − I_C). That ratio equals the positive-sequence impedance from the relay to the fault, whatever the fault current. Six measuring loops cover all faults: three phase-to-phase loops and three phase-to-ground loops. Numerical relays evaluate all six continuously.

Ground faults need a correction, because the fault loop includes the return path through ground, whose impedance per kilometer differs from the positive-sequence impedance. A phase-to-ground element uses the phase voltage and the phase current plus a compensated residual current:

With the right zero-sequence compensation factor k₀, the ground element measures the same positive-sequence impedance to the fault as the phase elements, so one set of reach settings serves both. The Hollis line has Z₁ = 3.2 + j16.0 Ω and Z₀ = 12 + j48 Ω, which give k₀ = 0.678 at −4.1°.

Stepped zones

A distance relay cannot set its reach to exactly 100 % of the line. Measurement errors in CTs, VTs and the relay, uncertainty in the line impedance data, CVT transients and the effects of fault resistance all add up to perhaps 10 to 20 %. A relay set at 100 % could overreach into the next line and trip for a fault beyond the remote bus. Distance protection therefore uses stepped zones.

import cmath, math
ZL  = complex(3.2, 16.0)      # Lincoln Road - Hollis, 40 km, positive sequence, primary ohms
Z0L = complex(12.0, 48.0)     # zero sequence
ZN  = complex(4.8, 24.0)      # Hollis - Grafton, 60 km (the next line)
k = (800/5)/(138000/115)      # CT ratio / VT ratio converts primary to secondary ohms
pol = lambda z: f"{abs(z):6.2f} ohm at {math.degrees(cmath.phase(z)):5.1f} deg"
print("line impedance:       ", pol(ZL), f"  secondary {abs(ZL)*k:.2f} ohm")
zones = {'Zone 1 (85 % of line, no delay)': 0.85*abs(ZL),
         'Zone 2 (120 % of line, 0.3 s)': 1.20*abs(ZL),
         'Zone 3 (line + 120 % of next, 1.0 s)': abs(ZL) + 1.20*abs(ZN)}
for n, r in zones.items(): print(f"{n:38s} {r:6.2f} ohm primary, {r*k:5.2f} ohm secondary")
print("Zone 2 limit, line + 50 % of next line:", f"{abs(ZL) + 0.5*abs(ZN):.2f} ohm")
k0 = (Z0L - ZL)/(3*ZL)
print("zero-sequence compensation k0 =", f"{abs(k0):.3f} at {math.degrees(cmath.phase(k0)):.1f} deg")
# Infeed at Hollis: for a fault 50 % along the next line, Hollis adds current from its own source
for ratio in (0.0, 1.0, 2.0):
    zapp = ZL + (1 + ratio)*0.5*ZN
    print(f"infeed ratio {ratio:.1f}: apparent impedance {abs(zapp):6.2f} ohm  (true {abs(ZL + 0.5*ZN):.2f})")
def in_mho(z, reach, ang=math.degrees(cmath.phase(ZL))):
    zr = cmath.rect(reach, math.radians(ang)); return abs(z - zr/2) < abs(zr)/2
# Loadability check in the style of NERC PRC-023: 150 % of a 900 A rating at 0.85 pu voltage and 30 degrees
I = 1.5*900; V = 0.85*138e3/math.sqrt(3)
zload = cmath.rect(V/I, math.radians(30))
print("\nloadability point:", pol(zload))
for n, r in list(zones.items()) + [('Zone 3 extended for infeed', 60.0)]:
    print(f"  {n.split(' (')[0]:28s} reach {r:5.1f} ohm: load point inside? {in_mho(zload, r)}")
r = 10
while not in_mho(zload, r): r += 0.5
print(f"  a mho circle at the line angle first reaches the load point at about {r:.1f} ohm")

line impedance:         16.32 ohm at  78.7 deg   secondary 2.18 ohm
Zone 1 (85 % of line, no delay)         13.87 ohm primary,  1.85 ohm secondary
Zone 2 (120 % of line, 0.3 s)           19.58 ohm primary,  2.61 ohm secondary
Zone 3 (line + 120 % of next, 1.0 s)    45.69 ohm primary,  6.09 ohm secondary
Zone 2 limit, line + 50 % of next line: 28.55 ohm
zero-sequence compensation k0 = 0.678 at -4.1 deg
infeed ratio 0.0: apparent impedance  28.55 ohm  (true 28.55)
infeed ratio 1.0: apparent impedance  40.79 ohm  (true 28.55)
infeed ratio 2.0: apparent impedance  53.03 ohm  (true 28.55)

loadability point:  50.17 ohm at  30.0 deg
  Zone 1                       reach  13.9 ohm: load point inside? False
  Zone 2                       reach  19.6 ohm: load point inside? False
  Zone 3                       reach  45.7 ohm: load point inside? False
  Zone 3 extended for infeed   reach  60.0 ohm: load point inside? False
  a mho circle at the line angle first reaches the load point at about 76.0 ohm

The program converts the zone reaches to secondary ohms, which is what the relay's settings use. With 800:5 CTs (ratio 160) and 138 kV:115 V VTs (ratio 1200), secondary impedance is primary impedance times 160/1200, or 0.133. The 16.32 Ω line is 2.18 Ω secondary. Zone 1 at 85 % is 13.87 Ω primary, 1.85 Ω secondary; Zone 2 at 120 % is 19.58 Ω, 2.61 Ω secondary, comfortably below the limit of the line plus half the next line, 28.55 Ω, so it will not reach far into the Hollis–Grafton line. Zone 3 at the line plus 120 % of the 60 km Hollis–Grafton line is 45.69 Ω, 6.09 Ω secondary.

Mho and quadrilateral characteristics

Each zone's shape on the R–X diagram is its characteristic. The mho characteristic is a circle through the origin whose diameter is the reach, drawn along the line angle. It is inherently directional, because it passes through the origin, and it naturally keeps its distance from the load region near the R axis. Its weakness is limited coverage of fault resistance, especially for short lines, where the circle is small. The quadrilateral characteristic is a polygon with independent reach settings in the reactive and resistive directions; it can cover much more fault resistance and is favored for ground elements and short lines, at the cost of more settings and greater care near the load region.

Numerical relays usually polarize their mho elements with positive-sequence memory voltage, which makes the circle expand toward the source during a fault and improves its coverage of fault resistance without changing its reach along the line. That dynamic expansion is one reason mho elements perform better in service than their static circles suggest.

Infeed and outfeed

Zone 3 at Lincoln Road is meant to back up the Hollis–Grafton line. But for a fault on that line, the Hollis station adds its own fault current, called infeed. The relay at Lincoln Road measures its own voltage and current, and its voltage includes the drop caused by the Hollis current flowing in the faulted part of the next line, so the impedance it measures is larger than the true impedance to the fault:

Here Z_x is the impedance from Hollis to the fault, I_H the infeed from Hollis and I_LR the current from Lincoln Road. The program shows the consequence for a fault halfway along the next line, whose true impedance is 28.55 Ω: with an infeed ratio of 1 the relay sees 40.79 Ω, and with a ratio of 2 it sees 53.03 Ω, outside the 45.69 Ω Zone 3. Infeed makes distance relays underreach. Nadine has Leo rerun the study with realistic infeed from the planning model and extend Zone 3 to about 60 Ω so that it still covers the end of the Hollis–Grafton line under normal infeed. Outfeed, where some current leaves the faulted path, has the opposite effect and can make a relay overreach; it occurs with tapped loads and some parallel-line configurations.

Load encroachment and loadability

Under heavy load, the impedance a distance relay sees is the voltage divided by the load current, which falls as load rises. If a long-reaching Zone 3 extends far enough toward the R axis, a heavy load during a stressed system condition can look like a fault. That happened during several major North American blackouts, when Zone 3 relays tripped heavily loaded but unfaulted lines and spread the cascade. NERC's relay loadability standard, PRC-023, now requires transmission line relays to stay quiet at 150 % of the line's highest seasonal rating, evaluated at 0.85 pu voltage and a 30° load angle, for lines at 200 kV and above and for lower-voltage lines selected by the planning coordinator.

Larchmont applies the same test to its 138 kV lines as good practice. The Hollis line is rated 900 A, so the test current is 1350 A at 0.85 pu voltage, an impedance of 50.2 Ω at 30°. The program checks every zone, including the extended 60 Ω Zone 3, and finds the load point outside them all: a mho circle at the line angle would not reach it until its reach exceeded about 76 Ω. Where a needed reach would encroach on load, relays offer load-encroachment logic, a region around the load angle in which the distance elements are blocked unless the measured impedance moves out of it, or quadrilateral elements with limited resistive reach.

Power swings and other complications

After a disturbance, generators in different parts of the system swing against each other, and the impedance seen by a distance relay moves slowly across the R–X plane. A stable swing can pass through a distance zone and cause an unwanted trip. Power-swing blocking measures how fast the impedance moves: a fault makes it jump almost instantly, a swing makes it travel at a finite rate, so a slow transit between two concentric characteristics identifies a swing and blocks the distance elements. Out-of-step tripping logic, applied at selected locations, deliberately separates the system at chosen points when a swing is unstable.

Other complications include series-compensated lines, where series capacitors upset the proportionality of impedance and distance, multi-terminal lines, where infeed from a tap changes every zone, and mutual coupling between parallel lines, which affects ground elements. Each has well-developed solutions in relay guides, and each is a reason that transmission protection settings are reviewed by a second engineer before they are applied.