  Stochastic fluctuations in power injections from distributed energy resources
(DERs) combined with load variability can cause constraint violations (e.g.,
exceeded voltage limits) in electric distribution systems. To monitor grid
operations, sensors are placed to measure important quantities such as the
voltage magnitudes. In this paper, we consider a sensor placement problem which
seeks to identify locations for installing sensors that can capture all
possible violations of voltage magnitude limits. We formulate a bilevel
optimization problem that minimizes the number of sensors and avoids false
sensor alarms in the upper level while ensuring detection of any voltage
violations in the lower level. This problem is challenging due to the
nonlinearity of the power flow equations and the presence of binary variables.
Accordingly, we employ recently developed conservative linear approximations of
the power flow equations that overestimate or underestimate the voltage
magnitudes. By replacing the nonlinear power flow equations with conservative
linear approximations, we can ensure that the resulting sensor locations and
thresholds are sufficient to identify any constraint violations. Additionally,
we apply various problem reformulations to significantly improve computational
tractability while simultaneously ensuring an appropriate placement of sensors.
Lastly, we improve the quality of the results via an approximate gradient
descent method that adjusts the sensor thresholds. We demonstrate the
effectiveness of our proposed method for several test cases, including a system
with multiple switching configurations.
