Assumptions

  1. The distribution power network lies in a 2 dimensional Euclidean space $\mathbf{R}$.
  2. At any spatial point $x$ in $\mathbf{R}^2$, the number density per unit area of power lines in the distribution power network with 1 end located at $x$ is $\rho_{N}$.
  3. At any spatial point $x$ in $\mathbf{R}^2$, the power lines with 1 end located at $x$ are isotropic.
  4. At any spatial point $x$ in $\mathbf{R}^2$, the probability density function of the length of the power lines with 1 end located at $x$ obeys a Pareto distribution with cumulative distribution function: $F(l) = 1 - \left( \frac{l}{l_{\mathrm{m}}} \right)^{1-\alpha}$, where $l$ is the length of the power lines and $l_{\mathrm{m}}$ the lower bound of the power lines. Differentiate $F(l)$ yields the probability density function for $l$: $f(l) = (\alpha - 1) l_{\mathrm{m}}^{\alpha - 1} l^{-\alpha}$.
  5. The impedance per unit length of power lines in the distribution power network is $z$.

Resulting Model

Under the above assumptions, we can derive the power flow equation of the distribution power network:

$$ \Delta I(x) = \frac{\rho_{N} (\alpha - 1) l_{\mathrm{m}}^{\alpha - 1}}{2\pi z} \int_{\mathbf{R}^2 \setminus \mathbf{B}{l{\mathrm{m}}}(x)} \frac{V(x) - V(y)}{|x-y|^{2+\alpha}} dy $$

Here $\Delta I(x)$ is the current source at $x$ (if $\Delta I(x) < 0$, then there is a current sink at $x$), $V(x)$ and $V(y)$ the voltage values at $x$ and $y$ respectively, and $\mathbf{B}{l{\mathrm{m}}}(x)$ is a ball with radius $l{\mathrm{m}}$ centered at $x$.

Note that in the limit where $l_{\mathrm{m}} \rightarrow 0$, the right hand side of equation above can be interpreted as a fractional Laplacian operator acting on the voltage field in $\mathbf{R}^2$, up to some constant (assuming $\rho_{N} l_{\mathrm{m}}^{\alpha - 1}$ is held fixed).

Parameters Setting