The rapidly growing penetration levels of distributed photovoltaic (PV) systems requires more comprehensive studies to understand their impact on distribution feeders. IEEE P.1547 highlights the need for Quasi-Static Time Series (QSTS) simulation in conducting distribution impact studies for distributed resource interconnection. Unlike conventional scenario-based simulation, the time series simulation can realistically assess time-dependent impacts such as the operation of various controllable elements (e.g. voltage regulating tap changers) or impacts of power fluctuations. However, QSTS simulations are still not widely used in the industry because of the computational burden associated with running yearlong simulations at a 1-s granularity, which is needed to capture device controller effects responding to PV variability. This paper presents a novel algorithm that reduces the number of times that the non-linear 3-phase unbalanced AC power flow must be solved by storing and reassigning power flow solutions as it progresses through the simulation. Each unique power flow solution is defined by a set of factors affecting the solution that can easily be queried. We demonstrate a computational time reduction of 98.9% for a yearlong simulation at 1-s resolution with minimal errors for metrics including: number of tap changes, capacitor actions, highest and lowest voltage on the feeder, line losses, and ANSI voltage violations. The key contribution of this work is the formulation of an algorithm capable of: (i) drastically reducing the computational time of QSTS simulations, (ii) accurately modeling distribution system voltage-control elements with hysteresis, and (iii) efficiently compressing result time series data for post-simulation analysis.