A novel intervals-based algorithm for the distribution short-circuit calculation

Abstract

In this paper, a novel intervals-based distribution short-circuit calculation algorithm is proposed. In distribution networks, there are various renewable energy-based generators (solar panels, wind generators, small hydro turbines, etc.), as well as loads with uncertain generation and consumption. Therefore, short-circuit calculation has to consider all these uncertainties. The algorithm proposed in this paper deals with above-mentioned uncertainties, as well as correlations among them. Algorithm testing is performed on two test examples of distribution networks, 6-bus and 1003-bus, for the verification of its robustness and efficiency on real-life, large-scale systems. The results demonstrate that the proposed algorithm provides highly accurate results and that it is able to solve real-life short-circuit problems with a higher precision than the traditional deterministic short-circuit calculation algorithms.

Appendixes

1.1 Appendix A

For DG (photovoltaic or wind) and load, active power is analyzed when weather parameters (from historical database and/or forecasted ones) are changed. The following time-dependent inputs are used for simulations: solar radiation, solar elevation, air temperature, wind speed, wind direction, and atmospheric pressure.

Weather data are classified by using the Self-Organization Map Artificial Neural Network (SOM ANN), where the SOM ANN represents a clustering concept with self-organizing characteristics that can efficiently group different input patterns. The correlation coefficients between dependent inputs are calculated from clustered weather data and corresponding powers from DG units or loads, where the feedforward artificial neural networks (FF ANNs) with back propagation are used for approximating the output active power of unmonitored elements. For details about the applied methodology, see [33].

In this paper, correlation coefficients are calculated from weather data (from weather historical database and/or forecasted values) classified into the winning neuron of the SOM ANN and subsequently from the corresponding internal active powers of DG units and output active powers of loads obtained by FF ANNs. The correlation coefficient between ath and bth inputs classified into the winning neuron is as follows:

$$ k_{ab} = \frac{{{\text{Cov}}(a,b)}}{{\sigma_{a} \sigma_{b} }}, $$

(7.1.1)

where the covariance between ath and bth inputs (z_a and z_b, respectively) in (7.1.1) can be calculated from the set of input samples as [37]:

$$ {\text{Cov}}(a,b) = \frac{1}{N}\sum\nolimits_{n = 1}^{N} {(z_{an} - \bar{z}_{a} )(z_{bn} - \bar{z}_{b} )} , $$

(7.1.2)

where N is the number of samples for a (b)th input, while the mean value for the set of a (b)th input samples is:

$$ \bar{z}_{a(b)} = \frac{1}{N}\sum\nolimits_{n = 1}^{N} {z_{a(b)n} } . $$

(7.1.3)

Standard deviation for a (b)th input in (7.1.1) is as follows:

$$ \sigma_{{a\text{(}b\text{)}}} = \sqrt {\frac{1}{N}\sum\nolimits_{n = 1}^{N} {(z_{a(b)n} - \bar{z}_{a(b)} )^{2} } } . $$

(7.1.4)

1.2 Appendix B

The following equations, in the complex numbers form, present the short-circuit calculation in the ∆-circuit in the bus k:

Single phase

$$ \hat{I}_{xk}^{\Delta } = \frac{{\hat{U}_{xk}^{{}} }}{{\hat{Z}_{xxk} }}, $$

(7.2.1a)

$$ \hat{I}_{yk}^{\Delta } = 0, $$

(7.2.1b)

$$ \hat{I}_{zk}^{\Delta } = 0, $$

(7.2.1c)

$$ x,y,z \in \{ {\text{a,b,c}}\} . $$

(7.2.1d)

Two phases

$$ \hat{I}_{xk}^{\Delta } = \frac{{\hat{U}_{xk} - \hat{U}_{yk} }}{{\hat{Z}_{xxk} - \hat{Z}_{{{\text{y}}xk}} - \hat{Z}_{xyk} + \hat{Z}_{{{\text{yy}}k}} }}, $$

(7.2.2a)

$$ \hat{I}_{yk}^{\Delta } = \frac{{\hat{U}_{yk} - \hat{U}_{xk} }}{{\hat{Z}_{xxk} - \hat{Z}_{yxk} - \hat{Z}_{xyk} + \hat{Z}_{yyk} }}, $$

(7.2.2b)

$$ \hat{I}_{zk}^{\Delta } = 0, $$

(7.2.2c)

$$ x,y,z \in \{ {\text{a,b,c}}\} . $$

(7.2.2d)

Two phases with ground

$$ \hat{I}_{xk}^{\Delta } = \frac{{\hat{Z}_{yyk} \hat{U}_{{{\text{a}}k}} - \hat{Z}_{yxk} \hat{U}_{{{\text{b}}k}} }}{{\hat{Z}_{xxk} \hat{Z}_{yyk} - \hat{Z}_{xyk} \hat{Z}_{yxk} }}, $$

(7.2.3a)

$$ \hat{I}_{xk}^{\Delta } = \frac{{\hat{Z}_{xxk} \hat{U}_{{{\text{b}}k}} - \hat{Z}_{xyk} \hat{U}_{{{\text{a}}k}} }}{{\hat{Z}_{xxk} \hat{Z}_{yyk} - \hat{Z}_{xyk} \hat{Z}_{yxk} }}, $$

(7.2.3b)

$$ \hat{I}_{zk}^{\Delta } = 0, $$

(7.2.3c)

$$ x,y,z \in \{ {\text{a,b,c}}\} . $$

(7.2.3d)

Three phases

$$ \hat{I}_{{{\text{a}}k}}^{\Delta } = \frac{{(\hat{Z}_{3} - \hat{Z}_{4} )(\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{b}}k}} ) + (\hat{Z}_{2} - \hat{Z}_{1} )(\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{c}}k}} )}}{{\hat{Z}_{1} \hat{Z}_{4} - \hat{Z}_{2} \hat{Z}_{3} }}, $$

(7.2.4a)

$$ \hat{I}_{{{\text{b}}k}}^{\Delta } = \frac{{\hat{Z}_{4} (\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{b}}k}} ) - \hat{Z}_{2} (\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{c}}k}} )}}{{\hat{Z}_{1} \hat{Z}_{4} - \hat{Z}_{2} \hat{Z}_{3} }}, $$

(7.2.4b)

$$ \hat{I}_{{{\text{c}}k}}^{\Delta } = \frac{{ - \hat{Z}_{3} (\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{b}}k}} ) + \hat{Z}_{1} (\hat{U}_{{{\text{a}}k}} - \hat{U}_{{{\text{c}}k}} )}}{{\hat{Z}_{1} \hat{Z}_{4} - \hat{Z}_{2} \hat{Z}_{3} }}, $$

(7.2.4c)

$$ \begin{aligned} \hat{Z}_{1} = - \hat{Z}_{{{\text{aa}}k}} + \hat{Z}_{{{\text{ab}}k}} + \hat{Z}_{{{\text{ba}}k}} - \hat{Z}_{{{\text{bb}}k}} ,\quad \hat{Z}_{2} = - \hat{Z}_{{{\text{aa}}k}} + \hat{Z}_{{{\text{ab}}k}} + \hat{Z}_{{{\text{ca}}k}} - \hat{Z}_{{{\text{cb}}k}} , \hfill \\ \hat{Z}_{3} = - \hat{Z}_{{{\text{aa}}k}} + \hat{Z}_{{{\text{ac}}k}} + \hat{Z}_{{{\text{ba}}k}} - \hat{Z}_{{{\text{bc}}k}} ,\quad \hat{Z}_{4} = - \hat{Z}_{{{\text{aa}}k}} + \hat{Z}_{{{\text{ac}}k}} + \hat{Z}_{{{\text{ca}}k}} - \hat{Z}_{{{\text{cc}}k}} . \hfill \\ \end{aligned} $$

(7.2.4d)