Probabilistic Approaches for the Steady-State Analysis of Distribution Systems with Wind Farms

Abstract

This chapter deals with probabilistic approaches for the steady-state analysis (probabilistic load flow) of distribution systems with wind farms. The probabilistic analysis is performed taking into account the randomness of both the distribution system loads and the wind energy production. Several approaches are presented to obtain the probability functions of state and dependent variables (e.g., voltage amplitudes and line flows). These approaches are mainly concentrated on wind farm probabilistic models, using one of the classical probabilistic techniques (e.g., Monte Carlo simulation, convolution process, and special distribution functions) to perform the probabilistic load flow. Numerical applications on a 17-bus balanced test distribution system and on an IEEE 34-bus unbalanced test distribution system are presented and discussed, considering the various wind farm models.

Appendices

Appendix A.1

1.1 Simulation of Gaussian Correlated Random Variables

In order to generate an approximate n-vector R of Gaussian random variables characterized by the mean value μ(R) and by the covariance matrix cov(R), the following procedure was applied [39]:

• 1. Evaluate the eigenvalues l ₁ , …, l _n and the corresponding eigenvectors Ψ ₁, …, Ψ _n of the matrix cov(R);

• 2. Generate an n-vector Γ of uncorrelated Gaussian variables characterized by zero mean values and variances equal to l ₁ , …, l _n ;

• 3. Generate the vector R as:

$$ {\varvec{R}} = \mu \left( {\varvec{R}}\right) + [\varvec{\varPsi}_1, \ldots,\varvec{\varPsi}_n]\varvec{\varGamma}.$$

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1.2 Simulation of Rayleigh Correlated Random Variables

In order to generate random numbers of N _V random variables $\omega_{1} , \ldots ,\omega_{{N_{V} }}$, which are Rayleigh distributed and correlated, the following approximate procedure can be applied [13]:

• 1. With $\mu_{{_{1} }}, \ldots, \mu_{{N_{V} }}^{{}}$ as the mean values, $\sigma_{1} , \ldots ,\sigma_{{N_{V} }}$ as the standard deviations, and $\rho_{ij}$ ($i,\,j = 1, \ldots ,N_{V}$, $i \ne j$) as the correlation factors, the covariance matrix can be built as:

$$ {\varvec{\varOmega }}_{\omega } = \left[ {\begin{array}{*{20}c} {\sigma_{{_{1} }}^{2} } & \cdots & {\rho_{{1N_{V} }}^{{}} } \\ {} & \ddots & {} \\ {\rho_{{N_{V} 1}}^{{}} } & \cdots & {\sigma_{{N_{V} }}^{2} } \\ \end{array} } \right], $$

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and a lower triangular matrix L can be determined so that :

$$ \varvec{\varOmega}_{\omega } = {\mathbf{L}}\;{\mathbf{L}}^{T} $$

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• 2. For each random variable $\omega_{i}$, a vector of random numbers ${\varvec{\upomega}}{}_{i\,}^{*}$, distributed according to a Rayleigh pdf with a mean value of $\mu_{i}$, is generated;

• 3. Starting from $\upomega{}_{i\,}^{*}$, a new variable is determined according to the following relationship:

$$ {\mathbf{z}}_{i}^{*} = \frac{{{\varvec{\upomega}}_{i}^{*} - E\left( {{\varvec{\upomega}}_{i}^{*} } \right)}}{{\sigma \left( {{\varvec{\upomega}}_{i}^{*} } \right)}} $$

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• 4. The random numbers ${\mathbf{r}}{}_{{\text{i}\,}}^{*}$ of correlated Rayleigh variables can be determined as:

$$ {\mathbf{r}}{}_{i\,}^{*} = \left| {{\mathbf{L}}\left( {{\mathbf{z}}{}_{i\,}^{*} } \right)^{T} + \mu_{i} } \right|. $$

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Appendix A.2

The 17-bus balanced test distribution system and the IEEE 34-bus unbalanced test distribution system are shown in Figs. 29 and 30.

Fig. 29

17-bus balanced, 3-phase, test-distribution system

Fig. 30

IEEE 34-bus test-distribution system