  This paper presents a tensor-recovery method to solve probabilistic power
flow problems. Our approach generates a high-dimensional and sparse generalized
polynomial-chaos expansion that provides useful statistical information. The
result can also speed up other essential routines in power systems (e.g.,
stochastic planning, operations and controls).
  Instead of simulating a power flow equation at all quadrature points, our
approach only simulates an extremely small subset of samples. We suggest a
model to exploit the underlying low-rank and sparse structure of
high-dimensional simulation data arrays, making our technique applicable to
power systems with many random parameters. We also present a numerical method
to solve the resulting nonlinear optimization problem.
  Our algorithm is implemented in MATLAB and is verified by several benchmarks
in MATPOWER $5.1$. Accurate results are obtained for power systems with up to
$50$ independent random parameters, with a speedup factor up to $9\times
10^{20}$.
