Notebooks · lib/power/ybus

The bus admittance matrix

A network of n buses joined by branches is summarized by the n by n bus admittance matrix Y: the bus current injections are I = Y V. Each branch is a pi section (series admittance ys = 1/(r + jx), total charging susceptance b split between its ends) behind an ideal transformer of complex ratio t = tap e^(j shift) at the from end, so its two-port admittances are

yff = (ys + jb/2) / |t|^2     yft = -ys / conj(t)
ytf = -ys / t                 ytt = ys + jb/2

and it adds yff at (f, f), yft at (f, t), ytf at (t, f) and ytt at (t, t). A bus shunt Gs + jBs (MW and MVAr consumed at 1 per unit voltage) adds (Gs + jBs) / baseMVA on the diagonal. These are the branch model and the assembly of MATPOWER's makeYbus (MATPOWER manual, section 3.2); a tap ratio of 0 means 1, a line rather than a transformer.

Y is stored in compressed sparse column form with complex values. Each branch's four admittances are kept too, for the branch flows.

The fast-decoupled method needs two real matrices made the same way from a simplified network (MATPOWER's makeB, the XB version): B' with no bus shunts, no line charging, unit tap ratios and no resistance, B'' with no phase shifts. make_ybus takes those simplifications as flags.

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[1]lines 29–200

from power.case import BR_B, BR_R, BR_STATUS, BR_X, BS, GS, SHIFT, TAP

record Ybus {
    n: Int
    colptr: [Int]
    rowind: [Int]
    vals: CArr
    yff: CArr
    yft: CArr
    ytf: CArr
    ytt: CArr
}

#: which parts of the network make_ybus leaves out
record YbusOpts {
    no_bus_shunt: Bool
    no_charging: Bool
    unit_tap: Bool
    no_resistance: Bool
    no_shift: Bool
}

DEG: Num = pi / 180.0

def full_network() -> YbusOpts {
    return YbusOpts { no_bus_shunt: false, no_charging: false, unit_tap: false, no_resistance: false,
                      no_shift: false }
}

#| Y for buses 0..nb-1, branch k joining fbus[k] to tbus[k] (internal
#| numbers); out-of-service branches add nothing.
def make_ybus(nb: Int, fbus: [Int], tbus: [Int], bus: Arr, branch: Arr, base_mva: Num, o: YbusOpts) -> Ybus {
    nl: Int = len(fbus)
    yff = czeros(nl)
    yft = czeros(nl)
    ytf = czeros(nl)
    ytt = czeros(nl)
    status = branch[:, BR_STATUS]
    res = zeros(nl) if o.no_resistance else branch[:, BR_R]
    reac = branch[:, BR_X]
    charging = zeros(nl) if o.no_charging else branch[:, BR_B]
    taps = branch[:, TAP]
    shifts = zeros(nl) if o.no_shift else branch[:, SHIFT] * DEG
    for k to nl {
        if status[k] > 0.0 {
            ys = 1.0 / (res[k] + reac[k] * 1.0i)
            bc = charging[k]
            ratio = taps[k]
            if ratio == 0.0 or o.unit_tap {
                ratio = 1.0
            }
            ang = shifts[k]
            tc = ratio * (cos(ang) + sin(ang) * 1.0i)
            yt = ys + bc / 2.0 * 1.0i
            yff[k] = yt / (ratio * ratio)
            yft[k] = -ys / conj(tc)
            ytf[k] = -ys / tc
            ytt[k] = yt
        }
    }

    #: count the entries of each column, a diagonal for every bus
    cnt = int_zeros(nb)
    for j to nb {
        cnt[j] = 1
    }
    for k to nl {
        cnt[fbus[k]] += 1
        cnt[tbus[k]] += 1
    }
    ptr = int_zeros(nb + 1)
    for j to nb {
        ptr[j + 1] = ptr[j] + cnt[j]
    }
    total: Int = ptr[nb]
    ri = int_zeros(total)
    rv = czeros(total)
    fill = int_zeros(nb)
    for j to nb {
        fill[j] = ptr[j]
        ri[fill[j]] = j
        sh = 0.0 + 0.0i
        if not o.no_bus_shunt {
            sh = (bus[j, GS] + bus[j, BS] * 1.0i) / base_mva
        }
        else {
            sh = bus[j, GS] / base_mva + 0.0i
        }
        rv[fill[j]] = sh
        fill[j] += 1
    }
    for k to nl {
        f = fbus[k]
        t = tbus[k]
        rv[ptr[f]] += yff[k]
        rv[ptr[t]] += ytt[k]
        #: column t holds row f (yft); column f holds row t (ytf)
        ri[fill[t]] = f
        rv[fill[t]] = yft[k]
        fill[t] += 1
        ri[fill[f]] = t
        rv[fill[f]] = ytf[k]
        fill[f] += 1
    }

    #: merge repeated rows within each column (parallel branches): count, then fill
    pos = int_zeros(nb)
    for j to nb {
        pos[j] = -1
    }
    colptr = int_zeros(nb + 1)
    for j to nb {
        for p from ptr[j] to ptr[j + 1] {
            if pos[ri[p]] != j {
                pos[ri[p]] = j
                colptr[j + 1] += 1
            }
        }
        colptr[j + 1] += colptr[j]
    }
    m: Int = colptr[nb]
    rowind = int_zeros(m)
    vals = czeros(m)
    for j to nb {
        k: Int = colptr[j]
        for p from ptr[j] to ptr[j + 1] {
            i = ri[p]
            if pos[i] >= colptr[j] and pos[i] < k and rowind[pos[i]] == i {
                vals[pos[i]] += rv[p]
            }
            else {
                pos[i] = k
                rowind[k] = i
                vals[k] = rv[p]
                k += 1
            }
        }
    }
    return Ybus { n: nb, colptr: colptr, rowind: rowind, vals: vals, yff: yff, yft: yft, ytf: ytf,
                  ytt: ytt }
}

#| I = Y V.
def ybus_mul(y: Ybus, v: CArr) -> CArr {
    out = czeros(y.n)
    for j to y.n {
        vj = v[j]
        for p from y.colptr[j] to y.colptr[j + 1] {
            out[y.rowind[p]] += y.vals[p] * vj
        }
    }
    return out
}

test "a two-bus line: Y is ys [[1, -1], [-1, 1]] plus half the charging at each end" {
    bus = zeros(2, 13)
    branch = zeros(1, 13)
    branch[0, BR_R] = 0.01
    branch[0, BR_X] = 0.1
    branch[0, BR_B] = 0.02
    branch[0, BR_STATUS] = 1.0
    fb: [Int] = [0]
    tb: [Int] = [1]
    y = make_ybus(2, fb, tb, bus, branch, 100.0, full_network())
    ys = 1.0 / (0.01 + 0.1i)
    I = ybus_mul(y, [1.0 + 0.0i, 0.0 + 0.0i])
    assert_near(abs(I[0] - (ys + 0.01i)), 0.0, 1e-13)
    assert_near(abs(I[1] + ys), 0.0, 1e-13)
}

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

from power.case import BR_B, BR_R, BR_STATUS, BR_X, BS, GS, SHIFT, TAP

record Ybus {
    n: Int
    colptr: [Int]
    rowind: [Int]
    vals: CArr
    yff: CArr
    yft: CArr
    ytf: CArr
    ytt: CArr
}

#: which parts of the network make_ybus leaves out
record YbusOpts {
    no_bus_shunt: Bool
    no_charging: Bool
    unit_tap: Bool
    no_resistance: Bool
    no_shift: Bool
}

DEG: Num = pi / 180.0

def full_network() -> YbusOpts {
    return YbusOpts { no_bus_shunt: false, no_charging: false, unit_tap: false, no_resistance: false,
                      no_shift: false }
}

#| Y for buses 0..nb-1, branch k joining fbus[k] to tbus[k] (internal
#| numbers); out-of-service branches add nothing.
def make_ybus(nb: Int, fbus: [Int], tbus: [Int], bus: Arr, branch: Arr, base_mva: Num, o: YbusOpts) -> Ybus {
    nl: Int = len(fbus)
    yff = czeros(nl)
    yft = czeros(nl)
    ytf = czeros(nl)
    ytt = czeros(nl)
    status = branch[:, BR_STATUS]
    res = zeros(nl) if o.no_resistance else branch[:, BR_R]
    reac = branch[:, BR_X]
    charging = zeros(nl) if o.no_charging else branch[:, BR_B]
    taps = branch[:, TAP]
    shifts = zeros(nl) if o.no_shift else branch[:, SHIFT] * DEG
    for k to nl {
        if status[k] > 0.0 {
            ys = 1.0 / (res[k] + reac[k] * 1.0i)
            bc = charging[k]
            ratio = taps[k]
            if ratio == 0.0 or o.unit_tap {
                ratio = 1.0
            }
            ang = shifts[k]
            tc = ratio * (cos(ang) + sin(ang) * 1.0i)
            yt = ys + bc / 2.0 * 1.0i
            yff[k] = yt / (ratio * ratio)
            yft[k] = -ys / conj(tc)
            ytf[k] = -ys / tc
            ytt[k] = yt
        }
    }

#: count the entries of each column, a diagonal for every bus
    cnt = int_zeros(nb)
    for j to nb {
        cnt[j] = 1
    }
    for k to nl {
        cnt[fbus[k]] += 1
        cnt[tbus[k]] += 1
    }
    ptr = int_zeros(nb + 1)
    for j to nb {
        ptr[j + 1] = ptr[j] + cnt[j]
    }
    total: Int = ptr[nb]
    ri = int_zeros(total)
    rv = czeros(total)
    fill = int_zeros(nb)
    for j to nb {
        fill[j] = ptr[j]
        ri[fill[j]] = j
        sh = 0.0 + 0.0i
        if not o.no_bus_shunt {
            sh = (bus[j, GS] + bus[j, BS] * 1.0i) / base_mva
        }
        else {
            sh = bus[j, GS] / base_mva + 0.0i
        }
        rv[fill[j]] = sh
        fill[j] += 1
    }
    for k to nl {
        f = fbus[k]
        t = tbus[k]
        rv[ptr[f]] += yff[k]
        rv[ptr[t]] += ytt[k]
        #: column t holds row f (yft); column f holds row t (ytf)
        ri[fill[t]] = f
        rv[fill[t]] = yft[k]
        fill[t] += 1
        ri[fill[f]] = t
        rv[fill[f]] = ytf[k]
        fill[f] += 1
    }

#: merge repeated rows within each column (parallel branches): count, then fill
    pos = int_zeros(nb)
    for j to nb {
        pos[j] = -1
    }
    colptr = int_zeros(nb + 1)
    for j to nb {
        for p from ptr[j] to ptr[j + 1] {
            if pos[ri[p]] != j {
                pos[ri[p]] = j
                colptr[j + 1] += 1
            }
        }
        colptr[j + 1] += colptr[j]
    }
    m: Int = colptr[nb]
    rowind = int_zeros(m)
    vals = czeros(m)
    for j to nb {
        k: Int = colptr[j]
        for p from ptr[j] to ptr[j + 1] {
            i = ri[p]
            if pos[i] >= colptr[j] and pos[i] < k and rowind[pos[i]] == i {
                vals[pos[i]] += rv[p]
            }
            else {
                pos[i] = k
                rowind[k] = i
                vals[k] = rv[p]
                k += 1
            }
        }
    }
    return Ybus { n: nb, colptr: colptr, rowind: rowind, vals: vals, yff: yff, yft: yft, ytf: ytf,
                  ytt: ytt }
}

#| I = Y V.
def ybus_mul(y: Ybus, v: CArr) -> CArr {
    out = czeros(y.n)
    for j to y.n {
        vj = v[j]
        for p from y.colptr[j] to y.colptr[j + 1] {
            out[y.rowind[p]] += y.vals[p] * vj
        }
    }
    return out
}

test "a two-bus line: Y is ys [[1, -1], [-1, 1]] plus half the charging at each end" {
    bus = zeros(2, 13)
    branch = zeros(1, 13)
    branch[0, BR_R] = 0.01
    branch[0, BR_X] = 0.1
    branch[0, BR_B] = 0.02
    branch[0, BR_STATUS] = 1.0
    fb: [Int] = [0]
    tb: [Int] = [1]
    y = make_ybus(2, fb, tb, bus, branch, 100.0, full_network())
    ys = 1.0 / (0.01 + 0.1i)
    I = ybus_mul(y, [1.0 + 0.0i, 0.0 + 0.0i])
    assert_near(abs(I[0] - (ys + 0.01i)), 0.0, 1e-13)
    assert_near(abs(I[1] + ys), 0.0, 1e-13)
}

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