mathematical equation for Power Line communication channel
Agreed to others. You need to narrow down your interest to a frequency band as PLC can be achieved as narrow-band or wide-band. Your channel and noise model would depend on the frequency band you are interested in. For instance, if you are interested in broadband communication (e.g. OFDM), then you need to take transmission/reflection coefficients of the network of interest / topology, attenuation factor (which is a function of distance and frequency) and delay of all the paths in the receive path into account. Of course, the channel is also time-varying and the noise is always assumed to be non-Gaussian. 

A pretty common channel model assumed for broadband communication would be as the following:

\begin{equation} \label{eqn_PLN_channel}
H(f)=\sum^L_{i=1}\left\{\prod_{k=1}^O \Gamma_{ik} \prod_{m=1}^P
T_{im} A(f, d_i) e^{-j2\pi f\tau_i}\right\}
\end{equation}
where $\Gamma_{ik}$ and $T_{im}$ denote the reflection coefficient
and transmission coefficient, respectively, $A(f, d_i)$ is the
distance and frequency dependent attenuation and $\tau_i$ represents
the delay of $i$th path out of all the receive paths ($L$) at the point of signal reception.

Derivation of this equation is simple and stems from what's physically happening in the network. Though, note that the channel depends both on frequency and time. 

On the other hand, usually there are no closed-form estimates of the noise and the noise is numerically estimated based on a model such as Nakagami-m or Generalized Gaussian Distribution.