The PV curve of a transfer into bus 38 of the IEEE 118-bus system

How much power can be pushed into bus 38 of MATPOWER's case118, and taken out at the reference bus, before the power flow has no solution? vp.power.cpf traces the voltage at bus 38 against the transfer by continuation, through the nose of the curve where Newton's method on the plain power flow stops converging, and down the low-voltage branch.

The curve

Pseudo-arclength continuation: a tangent step along the curve, then Newton's method on the power flow plus the step equation. Near the nose the trace steps back and approaches with smaller steps.

import numpy as np
import vipyr as vp
from vipyr.figure import figure

r = vp.power.cpf("case118", 38, step=0.2, max_step=1.0, singular_values=True)
print(
    "points:",
    len(r.mw),
    " nose at %.2f MW, Vm(38) = %.4f per unit" % (r.nose_mw, r.v_poi[r.nose]),
)
print("lowest Vm in the network at the nose: %.4f" % r.v_min[r.nose])
upper = np.arange(len(r.mw)) <= r.nose
fig = figure(
    "PV curve, transfer into bus 38 (case118)",
    "transfer (MW)",
    "Vm (per unit)",
)
fig.plot(r.mw[upper], r.v_poi[upper], "bus 38, upper branch")
fig.plot(r.mw[~upper], r.v_poi[~upper], "bus 38, lower branch")
fig.plot(r.mw, r.v_min, "lowest Vm in the network")
fig.vline("nose", r.nose_mw)

points: 71  nose at 1990.69 MW, Vm(38) = 0.6914 per unit
lowest Vm in the network at the nose: 0.6914

Why Newton fails there

The smallest singular value of the power-flow Jacobian falls to nearly zero at the nose: the Jacobian is singular where the two branches meet.

print(
    "smallest singular value: %.3g at the base case, %.3g at the nose"
    % (r.smin[0], r.smin[r.nose])
)
fig = figure(
    "Smallest singular value of the Jacobian along the curve",
    "transfer (MW)",
    "$\\sigma_{\\min}$",
)
fig.plot(r.mw[upper], r.smin[upper], "upper branch")
fig.plot(r.mw[~upper], r.smin[~upper], "lower branch")
fig.yscale("log")

smallest singular value: 0.185 at the base case, 1.58e-05 at the nose

Credits

The methods follow MATPOWER's (BSD-3-Clause). The test cases are MATPOWER's case files, each with its source named in its header.

Source

gallery/power/plot_pv_curve.py, run as written when these pages were built. The output and the figures above are from that run.

The Viv program it calls is lib/power/cpf.viv: its atlas, the defs and the calls between them, or open it in the workbench.