In the coming decade, large space structures with high-power sources will be orbiting the Earth at ionospheric altitudes. Exposed voltages on any part of the structure will cause a current flow between the exposed element and the ambient plasma, which in turn may cause a power loss or arcing. In an effort to better understand high-voltage current collection, a series of rocket and laboratory experiments will be conducted whereby, initially, positive high-voltage spheres will be exposed to an ambient plasma. In support of this program, the authors have solved the time-dependent nonlinear fluid equations for ions and electrons and the Poisson equation in order to study the response of a plasma to positive high-voltage spheres. Simulations were conducted for a range of voltages and for both low and high density surrounding plasmas. In all of the simulations, the sheath and current collection were found to be highly dynamical and the plasma response exhibited four distinct phases. The phases include a rapid initial current surge, the formation and subsequent propagation of both ion and electron density shells away from the sphere, a burst of localized large-amplitude wave activity, and eventually a quasi-steady state. The steady state results are consistent with the classical Langmuir-Blodgett theory.