Notebooks · lib/power/shortcircuit

Three-phase short-circuit current by IEC 60909

IEC 60909-0:2016 computes the initial symmetrical short-circuit current Ik'' at a bus without a load flow: an equivalent voltage source c Un / sqrt(3) at the fault drives the positive-sequence network seen from the fault, so

Ik'' = c Un / (sqrt(3) |Zk|)

with Zk the Thevenin impedance at the bus, Zbus[k, k] of the network's short-circuit admittance matrix. In that network:

lines and cables are their series impedance; line charging, shunts and loads are left out (sections 3.6.1 and 4.4);
transformers are at their rated ratio (per unit 1 on the bus bases) and their impedance is multiplied by K_T = 0.95 cmax / (1 + 0.6 x_T), x_T in per unit of the system base (section 6.3.3, eq. 12a);
each generator in service is a shunt of impedance K_G (R_G + j X''d), K_G = cmax / (1 + X''d sin(phi_rG)) for a generator at its bus's nominal voltage (section 6.6.1, eq. 18), with R_G + j X''d on the machine base (PSS/E's ZSORCE) moved to the system base;
c = cmax = 1.1 for networks above 1 kV (Table 1).

A branch is a transformer when its tap ratio is set (not 0) or it shifts phase, MATPOWER's convention. The admittance matrix is complex; each column of Zbus needed is one sparse solve of its real form [[G, -B]; [B, G]] [zr; zi] = [e_k; 0].

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[1]lines 31–166

from power.case import BR_R, BR_STATUS, BR_X, BUS_I, Case, F_BUS, GEN_BUS, GEN_STATUS, SHIFT, T_BUS, TAP
from power.contingency import csc_from_triplets
from power.splu import lu_solve, mindeg_order, splu

#: IEC 60909-0 Table 1, maximum currents above 1 kV
CMAX: Num = 1.1
KT_NUM: Num = 0.95
KT_X: Num = 0.6
PIVOT: Num = 0.001
#: the generator machine table's columns (power.raw's Raw.machine)
M_MBASE: Int = 0
M_ZR: Int = 1
M_ZX: Int = 2
BASE_KV: Int = 9
MBASE: Int = 6
#: MATPOWER's bus and branch width, and a gen matrix through PMIN
BUS_COLS: Int = 13
GEN_COLS: Int = 10

record Fault {
    #: per faulted bus: Ik'' (kA), Sk'' = sqrt(3) Un Ik'' (MVA), Zk (per unit, system base)
    ikss_ka: Arr
    skss_mva: Arr
    rk: Arr
    xk: Arr
}

#| Ik'' at each bus row in `buses`, for generators with source impedance
#| machine[g] = (MBASE, R, X''d) on their own base and rated power factor
#| cos_phi; with_kt multiplies transformer impedances by K_T.
def ikss(c: Case, machine: Arr, cos_phi: Num, with_kt: Bool, buses: [Int]) -> Fault {
    nb: Int = int(shape(c.bus)[0])
    nl: Int = int(shape(c.branch)[0])
    ng: Int = int(shape(c.gen)[0])
    row = map()
    for i to nb {
        row[c.bus[i, BUS_I]] = i
    }
    #: the real form of Y: G at (i, j) and (n + i, n + j), -B at (i, n + j), B at (n + i, j)
    ri: [Int] = []
    ci: [Int] = []
    vals = []
    for k to nl {
        if c.branch[k, BR_STATUS] > 0.0 {
            f: Int = int(row[c.branch[k, F_BUS]])
            t: Int = int(row[c.branch[k, T_BUS]])
            z = c.branch[k, BR_R] + c.branch[k, BR_X] * 1.0i
            if c.branch[k, TAP] != 0.0 or c.branch[k, SHIFT] != 0.0 {
                z = z * (KT_NUM * CMAX / (1.0 + KT_X * c.branch[k, BR_X]))
            }
            ys = 1.0 / z
            ri, ci, vals = _put(ri, ci, vals, nb, f, f, ys)
            ri, ci, vals = _put(ri, ci, vals, nb, t, t, ys)
            ri, ci, vals = _put(ri, ci, vals, nb, f, t, -ys)
            ri, ci, vals = _put(ri, ci, vals, nb, t, f, -ys)
        }
    }
    sin_phi = sqrt(max(1.0 - cos_phi * cos_phi, 0.0))
    for g to ng {
        if c.gen[g, GEN_STATUS] > 0.0 {
            b: Int = int(row[c.gen[g, GEN_BUS]])
            xd = machine[g, M_ZX]
            kg = CMAX / (1.0 + xd * sin_phi)
            zg = kg * (machine[g, M_ZR] + xd * 1.0i) * c.base_mva / machine[g, M_MBASE]
            ri, ci, vals = _put(ri, ci, vals, nb, b, b, 1.0 / zg)
        }
    }
    m: Int = 2 * nb
    cp, rr, vv = csc_from_triplets(m, ri, ci, vals)
    lu = splu(m, cp, rr, vv, mindeg_order(m, cp, rr), PIVOT)
    nf: Int = len(buses)
    ik = zeros(nf)
    sk = zeros(nf)
    rk = zeros(nf)
    xk = zeros(nf)
    for q to nf {
        k = buses[q]
        e = zeros(m)
        e[k] = 1.0
        col = lu_solve(lu, e)
        rk[q] = col[k]
        xk[q] = col[nb + k]
        un = c.bus[k, BASE_KV]
        #: per unit current c / |Zk| times the base current baseMVA / (sqrt(3) Un), kA
        ik[q] = CMAX / hypot(rk[q], xk[q]) * c.base_mva / (sqrt(3.0) * un)
        sk[q] = sqrt(3.0) * un * ik[q]
    }
    return Fault { ikss_ka: ik, skss_mva: sk, rk: rk, xk: xk }
}

def _put(ri: [Int], ci: [Int], vals: Arr, n: Int, i: Int, j: Int, y: Complex) {
    ri = append(ri, i)
    ci = append(ci, j)
    vals = append(vals, re(y))
    ri = append(ri, n + i)
    ci = append(ci, n + j)
    vals = append(vals, re(y))
    ri = append(ri, i)
    ci = append(ci, n + j)
    vals = append(vals, -im(y))
    ri = append(ri, n + i)
    ci = append(ci, j)
    vals = append(vals, im(y))
    return ri, ci, vals
}

test "a generator behind a line: Ik'' is c / |K_G Z_G + Z_line| in base current" {
    bus = zeros(2, BUS_COLS)
    bus[0, BUS_I] = 1.0
    bus[1, BUS_I] = 2.0
    bus[0, BASE_KV] = 138.0
    bus[1, BASE_KV] = 138.0
    gen = zeros(1, GEN_COLS)
    gen[0, GEN_BUS] = 1.0
    gen[0, GEN_STATUS] = 1.0
    gen[0, MBASE] = 200.0
    branch = zeros(1, BUS_COLS)
    branch[0, F_BUS] = 1.0
    branch[0, T_BUS] = 2.0
    branch[0, BR_R] = 0.01
    branch[0, BR_X] = 0.1
    branch[0, BR_STATUS] = 1.0
    machine = zeros(1, M_ZX + 1)
    machine[0, M_MBASE] = 200.0
    machine[0, M_ZX] = 0.2
    c = Case { base_mva: 100.0, bus: bus, gen: gen, branch: branch }
    f = ikss(c, machine, 0.85, true, [0, 1])
    kg = 1.1 / (1.0 + 0.2 * sqrt(1.0 - 0.85 * 0.85))
    zk = kg * 0.2 * 0.5 * 1.0i + (0.01 + 0.1i)
    ib = 100.0 / (sqrt(3.0) * 138.0)
    assert_near(f.ikss_ka[1], 1.1 / abs(zk) * ib, 1e-12)
    assert_near(f.xk[0], kg * 0.1, 1e-12)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

from power.case import BR_R, BR_STATUS, BR_X, BUS_I, Case, F_BUS, GEN_BUS, GEN_STATUS, SHIFT, T_BUS, TAP
from power.contingency import csc_from_triplets
from power.splu import lu_solve, mindeg_order, splu

#: IEC 60909-0 Table 1, maximum currents above 1 kV
CMAX: Num = 1.1
KT_NUM: Num = 0.95
KT_X: Num = 0.6
PIVOT: Num = 0.001
#: the generator machine table's columns (power.raw's Raw.machine)
M_MBASE: Int = 0
M_ZR: Int = 1
M_ZX: Int = 2
BASE_KV: Int = 9
MBASE: Int = 6
#: MATPOWER's bus and branch width, and a gen matrix through PMIN
BUS_COLS: Int = 13
GEN_COLS: Int = 10

record Fault {
    #: per faulted bus: Ik'' (kA), Sk'' = sqrt(3) Un Ik'' (MVA), Zk (per unit, system base)
    ikss_ka: Arr
    skss_mva: Arr
    rk: Arr
    xk: Arr
}

#| Ik'' at each bus row in `buses`, for generators with source impedance
#| machine[g] = (MBASE, R, X''d) on their own base and rated power factor
#| cos_phi; with_kt multiplies transformer impedances by K_T.
def ikss(c: Case, machine: Arr, cos_phi: Num, with_kt: Bool, buses: [Int]) -> Fault {
    nb: Int = int(shape(c.bus)[0])
    nl: Int = int(shape(c.branch)[0])
    ng: Int = int(shape(c.gen)[0])
    row = map()
    for i to nb {
        row[c.bus[i, BUS_I]] = i
    }
    #: the real form of Y: G at (i, j) and (n + i, n + j), -B at (i, n + j), B at (n + i, j)
    ri: [Int] = []
    ci: [Int] = []
    vals = []
    for k to nl {
        if c.branch[k, BR_STATUS] > 0.0 {
            f: Int = int(row[c.branch[k, F_BUS]])
            t: Int = int(row[c.branch[k, T_BUS]])
            z = c.branch[k, BR_R] + c.branch[k, BR_X] * 1.0i
            if c.branch[k, TAP] != 0.0 or c.branch[k, SHIFT] != 0.0 {
                z = z * (KT_NUM * CMAX / (1.0 + KT_X * c.branch[k, BR_X]))
            }
            ys = 1.0 / z
            ri, ci, vals = _put(ri, ci, vals, nb, f, f, ys)
            ri, ci, vals = _put(ri, ci, vals, nb, t, t, ys)
            ri, ci, vals = _put(ri, ci, vals, nb, f, t, -ys)
            ri, ci, vals = _put(ri, ci, vals, nb, t, f, -ys)
        }
    }
    sin_phi = sqrt(max(1.0 - cos_phi * cos_phi, 0.0))
    for g to ng {
        if c.gen[g, GEN_STATUS] > 0.0 {
            b: Int = int(row[c.gen[g, GEN_BUS]])
            xd = machine[g, M_ZX]
            kg = CMAX / (1.0 + xd * sin_phi)
            zg = kg * (machine[g, M_ZR] + xd * 1.0i) * c.base_mva / machine[g, M_MBASE]
            ri, ci, vals = _put(ri, ci, vals, nb, b, b, 1.0 / zg)
        }
    }
    m: Int = 2 * nb
    cp, rr, vv = csc_from_triplets(m, ri, ci, vals)
    lu = splu(m, cp, rr, vv, mindeg_order(m, cp, rr), PIVOT)
    nf: Int = len(buses)
    ik = zeros(nf)
    sk = zeros(nf)
    rk = zeros(nf)
    xk = zeros(nf)
    for q to nf {
        k = buses[q]
        e = zeros(m)
        e[k] = 1.0
        col = lu_solve(lu, e)
        rk[q] = col[k]
        xk[q] = col[nb + k]
        un = c.bus[k, BASE_KV]
        #: per unit current c / |Zk| times the base current baseMVA / (sqrt(3) Un), kA
        ik[q] = CMAX / hypot(rk[q], xk[q]) * c.base_mva / (sqrt(3.0) * un)
        sk[q] = sqrt(3.0) * un * ik[q]
    }
    return Fault { ikss_ka: ik, skss_mva: sk, rk: rk, xk: xk }
}

def _put(ri: [Int], ci: [Int], vals: Arr, n: Int, i: Int, j: Int, y: Complex) {
    ri = append(ri, i)
    ci = append(ci, j)
    vals = append(vals, re(y))
    ri = append(ri, n + i)
    ci = append(ci, n + j)
    vals = append(vals, re(y))
    ri = append(ri, i)
    ci = append(ci, n + j)
    vals = append(vals, -im(y))
    ri = append(ri, n + i)
    ci = append(ci, j)
    vals = append(vals, im(y))
    return ri, ci, vals
}

test "a generator behind a line: Ik'' is c / |K_G Z_G + Z_line| in base current" {
    bus = zeros(2, BUS_COLS)
    bus[0, BUS_I] = 1.0
    bus[1, BUS_I] = 2.0
    bus[0, BASE_KV] = 138.0
    bus[1, BASE_KV] = 138.0
    gen = zeros(1, GEN_COLS)
    gen[0, GEN_BUS] = 1.0
    gen[0, GEN_STATUS] = 1.0
    gen[0, MBASE] = 200.0
    branch = zeros(1, BUS_COLS)
    branch[0, F_BUS] = 1.0
    branch[0, T_BUS] = 2.0
    branch[0, BR_R] = 0.01
    branch[0, BR_X] = 0.1
    branch[0, BR_STATUS] = 1.0
    machine = zeros(1, M_ZX + 1)
    machine[0, M_MBASE] = 200.0
    machine[0, M_ZX] = 0.2
    c = Case { base_mva: 100.0, bus: bus, gen: gen, branch: branch }
    f = ikss(c, machine, 0.85, true, [0, 1])
    kg = 1.1 / (1.0 + 0.2 * sqrt(1.0 - 0.85 * 0.85))
    zk = kg * 0.2 * 0.5 * 1.0i + (0.01 + 0.1i)
    ib = 100.0 / (sqrt(3.0) * 138.0)
    assert_near(f.ikss_ka[1], 1.1 / abs(zk) * ib, 1e-12)
    assert_near(f.xk[0], kg * 0.1, 1e-12)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors