  Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze
transmission system behavior, has been studied for decades. Despite a variety
of available methods, existing PLF approaches rarely take system control into
account. However, system control, as an automatic buffer between the
fluctuations in random variables and the variations in system states, has a
significant impact on the final PLF result. To consider control actions'
influence, this paper proposes the first analytical PLF method for the
transmission grid that takes into account primary and secondary frequency
controls. This method is based on a high-precision linear power flow model,
whose precision is even further improved in this paper by an original
correction approach. This paper also proves that if the joint probability
distribution (JPD) of random variables is expressed by a Gaussian mixture model
(GMM), then the JPD of system states (e.g., nodal voltages) is an infinite GMM.
By leveraging this proposition, the proposed method can generate the joint PLF
of the whole system, is applicable to random variables obeying any
distributions, and is capable of capturing their correlation. The high accuracy
and satisfactory efficiency of this method are verified on test cases scaling
from 14 to 1354 buses.
