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Ambient Forcing is a novel method to sample random states from manifolds of differential-algebraic equations (DAE). These states can represent local perturbations of nodes in power systems with loads, which introduces constraints into the system. These states must be valid initial conditions to the DAE, meaning that they fulfill the algebraic equations. Additionally, these states should represent perturbations of individual variables in the power grid, such as a perturbation of the voltage at a load. These initial states enable the calculation of probabilistic stability measures of power systems with loads, which was not yet possible, but is important as these measures have become a crucial tool in studying power systems. To verify that these perturbations are network local, i.e. that the initial perturbation only targets a single node in the power grid, a new measure, the spreadability, related to the closeness centrality, is presented. The spreadability is evaluated for an ensemble of typical power grids. The ensemble depicts a set of future power grids where consumers, as well as producers, are connected to the grid via inverters. For this power grid ensemble, we additionally calculate the basin stability as well as the survivability, two probabilistic measures which provide statements about asymptotic and transient stability. We also revisit topological classes, that have been shown to predict the basin stability of grids and explore if they still hold for grids with constraints and voltage dynamics. We find that the degree of the nodes is a better predictor than the topological classes for our ensemble. Finally, ambient forcing is applied to calculate probabilistic stability measures of the IEEE 96 test case.