  Building the conditional probability distribution of wind power forecast
errors benefits both wind farms (WFs) and independent system operators (ISOs).
Establishing the joint probability distribution of wind power and the
corresponding forecast data of spatially correlated WFs is the foundation for
deriving the conditional probability distribution. Traditional parameter
estimation methods for probability distributions require the collection of
historical data of all WFs. However, in the context of multi-regional
interconnected grids, neither regional ISOs nor WFs can collect the raw data of
WFs in other regions due to privacy or competition considerations. Therefore,
based on the Gaussian mixture model, this paper first proposes a
privacy-preserving distributed expectation-maximization algorithm to estimate
the parameters of the joint probability distribution. This algorithm consists
of two original methods: (1) a privacy-preserving distributed summation
algorithm and (2) a privacy-preserving distributed inner product algorithm.
Then, we derive each WF's conditional probability distribution of forecast
error from the joint one. By the proposed algorithms, WFs only need local
calculations and privacy-preserving neighboring communications to achieve the
whole parameter estimation. These algorithms are verified using the wind
integration data set published by the NREL.
