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We consider a model of a distribution feeder connecting multiple loads to thesub-station. Voltage is controlled directly at the head of the line(sub-station), however, voltage anywhere further down the line is subject tofluctuations, caused by irregularities of real and reactive distributed powerconsumption/generation. The lack of a direct control of voltage along the linemay result in the voltage instability, also called voltage collapse -phenomenon well known and documented in the power engineering literature.Motivated by emerging photo-voltaic technology, which brings a new source ofrenewable generation but also contributes significant increase in power flowfluctuations, we reexamine the phenomenon of voltage stability and collapse. Inthe limit where the number of consumers is large and spatial variations inpower flows are smooth functions of position along the feeder, we derive a setof the power flow Ordinary Differential Equations (ODE), verify phenomenon ofvoltage collapse, and study the effect of disorder and irregularity ininjection and consumption on the voltage profile by simulating the stochasticODE. We observe that disorder leads to nonlinear amplification of the voltagevariations at the end of the line as the point of voltage collapse isapproached. We also find that the disorder, when correlated on a scalesufficiently small compared to the length of the line, self-averages, i.e. thevoltage profile remains spatially smooth for any individual realization of thedisorder and is correlated only at scales comparable to the length of the line.Finally, we explain why the integrated effect of disorder on the voltage at theend of the line cannot be described within a naive one-generator-one-loadmodel.

Voltage Collapse and ODE Approach to Power Flows: Analysis of a Feeder Line with Static Disorder in Consumption/Production