Estimating the Statistics of Non-Uniform and Time-Varying Distributed Parameter Fluctuations in a Transmission Line

Abstract

The problem of estimating the statistics of the distributed parameters of a transmission line which are random functions of time and spatial location is addressed in this paper. The main idea is to transform the spatial–temporal PDE (partial differential equation) problem into a purely temporal problem using spatial test functions and then apply first-order perturbation theory to express the random components of the line voltage and current as linear functional of the random distributed parameters. We then evaluate the line voltage and current statistical correlations in terms of the distributed parameters correlations. Simulation studies are carried out for variance estimation of the resistance per unit length, and appropriate results are obtained.

Appendix-1

We could, in principle, carry out a simulation along the same line when all the distributed parameters have just a single mode fluctuation, i.e.,

$$\begin{aligned} \delta R\left( t,z\right)&=\delta R\left( t\right) \varphi _1 \left( z\right) \end{aligned}$$

(56)

$$\begin{aligned} \delta L\left( t,z\right)&=\delta L\left( t\right) \varphi _1 \left( z\right) \end{aligned}$$

(57)

$$\begin{aligned} \delta G\left( t,z\right)&=\delta G\left( t\right) \varphi _1 \left( z\right) \end{aligned}$$

(58)

$$\begin{aligned} \delta C\left( t,z\right)&=\delta C\left( t\right) \varphi _1 \left( z\right) \end{aligned}$$

(59)

The line equations become

$$\begin{aligned} \frac{-\partial i\left( t,z\right) }{\partial z}&= \left( G_0+\delta G\left( t,z\right) \right) \upsilon \left( t,z\right) \nonumber \\&\quad +C_0\frac{\partial \upsilon \left( t,z\right) }{\partial t}+\frac{\partial \left( \delta C\left( t,z\right) \upsilon \left( t,z\right) \right) }{\partial t} \end{aligned}$$

(60)

$$\begin{aligned} \frac{-\partial \upsilon \left( t,z\right) }{\partial z}&= \left( R_0+\delta R\left( t,z\right) \right) i \left( t,z\right) \nonumber \\&\quad +L_0\frac{\partial i\left( t,z\right) }{\partial t}+\frac{\partial \left( \delta L\left( t,z\right) i\left( t,z\right) \right) }{\partial t} \end{aligned}$$

(61)

and the first-order perturbed equation is

$$\begin{aligned} \frac{-\partial \delta i\left( t,z\right) }{\partial z}= G_0\delta \upsilon \left( t,z\right) \nonumber \\&\quad +\delta G\left( t\right) \varphi _1\left( z\right) \upsilon ^{\left( 0\right) }\left( t,z\right) \nonumber \\&\quad +C_0\frac{\partial \delta \upsilon \left( t,z\right) }{\partial t}+\frac{\partial \left( \delta C\left( t\right) \upsilon ^{\left( 0\right) }\left( t,z\right) \right) }{\partial t}\varphi _1\left( z\right) \end{aligned}$$

(62)

$$\begin{aligned} \frac{-\partial \delta \upsilon \left( t,z\right) }{\partial z}&= R_0\delta i\left( t,z\right) +L_0 \frac{ \partial \delta i\left( t,z\right) }{ \partial t} \nonumber \\&\quad +\delta R\left( t\right) i^{\left( 0\right) }\left( t,z\right) \varphi _1\left( z\right) \nonumber \\&\quad +\frac{\partial \left( \delta L\left( t\right) i^{\left( 0\right) }\left( t,z\right) \right) }{\partial t}\varphi _1\left( z\right) \end{aligned}$$

(63)

or in terms of test function expansion, we have

$$\begin{aligned} -\sum _n \delta i_n \left( t\right) \varphi _n^{'}\left( z\right)&=G_0\sum _n \delta \upsilon _n \left( t\right) \varphi _n\left( z\right) \nonumber \\&\quad +\delta G\left( t\right) \sum _n \varphi _1\left( z\right) \varphi _n\left( z\right) \upsilon _n^ {\left( 0\right) }\left( t\right) \nonumber \\&\quad +C_0\sum _n \delta \upsilon _n^{'} \left( t\right) \varphi _n\left( z\right) \nonumber \\&\quad +\sum _n \left( \delta C\left( t\right) \upsilon _n^{\left( 0\right) }\left( t\right) \right) ^{'}\varphi _1\left( z\right) \varphi _n\left( z\right) \end{aligned}$$

(64)

$$\begin{aligned} -\sum _n \delta \upsilon _n \left( t\right) \varphi _n^{'}\left( z\right)&=R_0\sum _n \delta i_n \left( t\right) \varphi _n\left( z\right) \nonumber \\&\quad +L_0\sum _n \delta i_n^{'}\left( t\right) \varphi _n\left( z\right) \nonumber \\&\quad +\delta R\left( t\right) \sum _n \varphi _1\left( z\right) \varphi _n\left( z\right) i_n^ {\left( 0\right) }\left( t\right) \nonumber \\&\quad +\sum _n \left( \delta L\left( t\right) i_n^{\left( 0\right) }\left( t\right) \right) ^{'}\varphi _1\left( z\right) \varphi _n\left( z\right) \end{aligned}$$

(65)

Taking the inner product with $\varphi _m\left( z\right)$ and using the definitions $a\left( m,n\right) = \langle \varphi _n^{'}\varphi _m\rangle ,\langle \varphi _n\varphi _m\rangle =\delta \left[ n-m\right]$ and $b\left( m,n\right) = \langle \varphi _1\varphi _n\varphi _m\rangle$, we get

$$\begin{aligned}&\sum _n a\left( m,n\right) \delta i_n \left( t\right) +G_0\delta \upsilon _m \left( t\right) \nonumber \\&\quad + \delta G\left( t\right) \sum _n b\left( m,n\right) \upsilon _n^ {\left( 0\right) }\left( t\right) \nonumber \\&\quad +C_0\delta \upsilon _m^{'}\left( t\right) +\sum _n b\left( m,n\right) \left( \delta C\left( t\right) \upsilon _n^{\left( 0\right) }\left( t\right) \right) ^{'}=0 \end{aligned}$$

(66)

Similarly,

$$\begin{aligned}&\sum _n a\left( m,n\right) \delta \upsilon _n \left( t\right) +R_0\delta i_m \left( t\right) \nonumber \\&\quad + \delta R\left( t\right) \sum _n b\left( m,n\right) i_n^ {\left( 0\right) }\left( t\right) \nonumber \\&\quad +L_0\delta i_m^{'}\left( t\right) +\sum _n b\left( m,n\right) \left( \delta L\left( t\right) i_n^{\left( 0\right) }\left( t\right) \right) ^{'}=0 \end{aligned}$$

(67)

We assume that $\delta R \left( t\right) ,\delta L\left( t\right) ,\delta G\left( t\right)$ and $\delta C\left( t\right)$ have just two frequencies $\omega _1$ and $\omega _2$ of variation. Such a situation can, for example, occur when there are periodic environmental disturbances acting on the line. Thus, we can write

$$\begin{aligned} \delta R\left( t\right)&=\delta R_1\sin \left( \omega _1t\right) +\delta R_2\sin \left( \omega _2t\right) \end{aligned}$$

(68)

$$\begin{aligned} \delta L\left( t\right)&=\delta L_1\sin \left( \omega _1t\right) +\delta L_2\sin \left( \omega _2t\right) \end{aligned}$$

(69)

$$\begin{aligned} \delta G\left( t\right)&=\delta G_1\sin \left( \omega _1t\right) +\delta G_2\sin \left( \omega _2t\right) \end{aligned}$$

(70)

$$\begin{aligned} \delta C\left( t\right)&=\delta C_1\sin \left( \omega _1t\right) +\delta C_2\sin \left( \omega _2t\right) \end{aligned}$$

(71)

Thus, Eqs. $\left( 66\right)$ and $\left( 67\right)$ assume the form

$$\begin{aligned}&\sum _n a\left( m,n\right) \delta i_n \left( t\right) \nonumber \\&\quad +G_0\delta \upsilon _m \left( t\right) \nonumber \\&\quad + \delta G_1 \psi _{G_1}\left( m,t\right) \nonumber \\&\quad +\delta G_2 \psi _{G_2}\left( m,t\right) \nonumber \\&\quad +C_0\delta \upsilon _m^{'}\left( t\right) \nonumber \\&\quad +\delta C_1 \psi _{C_1}\left( m,t\right) \nonumber \\&\quad +\delta C_2 \psi _{C_2}\left( m,t\right) =0 \end{aligned}$$

(72)

$$\begin{aligned}&\sum _n a\left( m,n\right) \delta \upsilon _n \left( t\right) \nonumber \\&\quad +R_0\delta i_m \left( t\right) \nonumber \\&\quad + \delta R_1 \psi _{R_1}\left( m,t\right) \nonumber \\&\quad +\delta R_2 \psi _{R_2}\left( m,t\right) \nonumber \\&\quad +L_0\delta i_m^{'}\left( t\right) \nonumber \\&\quad +\delta L_1 \psi _{L_1}\left( m,t\right) \nonumber \\&\quad +\delta L_2 \psi _{L_2}\left( m,t\right) =0 \end{aligned}$$

(73)

with,

$$\begin{aligned} \psi _{G_k}\left( m,t\right)&=\sum _n b\left( m,n\right) \upsilon _n^{\left( 0\right) }\left( t\right) \sin \left( \omega _k t\right) ; {k=1,2} \end{aligned}$$

(74)

$$\begin{aligned} \psi _{C_k}\left( m,t\right)&=\sum _n b\left( m,n\right) \left( \upsilon _n^{\left( 0\right) }\left( t\right) \sin \left( \omega _k t\right) \right) ^{'}; {k=1,2} \end{aligned}$$

(75)

$$\begin{aligned} \psi _{R_k}\left( m,t\right)&=\sum _n b\left( m,n\right) i_n^{\left( 0\right) }\left( t\right) \sin \left( \omega _k t\right) ; {k=1,2} \end{aligned}$$

(76)

$$\begin{aligned} \psi _{L_k}\left( m,t\right)&=\sum _n b\left( m,n\right) \left( i_n^{\left( 0\right) }\left( t\right) \sin \left( \omega _k t\right) \right) ^{'}; {k=1,2} \end{aligned}$$

(77)

These equations are of the form

$$\begin{aligned} \frac{d}{{dt}}\left[ \begin{array}{l} \delta \underline{i} \left( t \right) \\ \delta \underline{\upsilon }\left( t \right) \end{array} \right] = - \underline{\underline{K}} \left[ \begin{array}{l} \delta \underline{i} \left( t \right) \\ \delta \underline{\upsilon }\left( t \right) \end{array} \right] + \underline{\underline{\psi }} \left( t \right) \left[ \begin{array}{l} \delta {R_1}\\ \delta {R_2}\\ \delta {L_1}\\ \delta {L_2}\\ \delta {C_1}\\ \delta {C_2}\\ \delta {G_1}\\ \delta {G_2} \end{array} \right] \end{aligned}$$

(78)

writing, $\underline{\theta }= \left[ \begin{array}{l} \delta {R_1}\\ \delta {R_2}\\ \delta {L_1}\\ \delta {L_2}\\ \delta {C_1}\\ \delta {C_2}\\ \delta {G_1}\\ \delta {G_2} \end{array} \right] \in \mathbb {R}^8$ and $\underline{\xi }\left( t \right) = \left[ \begin{array}{l} \delta \underline{i} \left( t \right) \\ \delta \underline{\upsilon }\left( t \right) \end{array} \right]$

We can express eq.$\left( 78 \right)$ as:

$$\begin{aligned} \frac{d}{{dt}}\underline{\xi }\left( t \right)&= - \underline{\underline{K}}~ \underline{\xi }\left( t \right) \nonumber \\&\quad + \underline{\underline{\psi }} \left( t \right) \underline{\theta }\end{aligned}$$

(79)

so,

$$\begin{aligned} \underline{\xi }\left( t \right)&= \left( {\int \limits _0^t {\underline{\underline{\Phi }} \left( {t - \tau } \right) \underline{\underline{\psi }} \left( \tau \right) d\tau } } \right) \nonumber \\&\quad \underline{\theta }\buildrel \Delta \over = \underline{\underline{Q}} \left( t \right) \underline{\theta }\end{aligned}$$

(80)

where $\underline{\underline{\Phi }} \left( {t} \right) =\exp \left( -t\underline{\underline{K} }\right)$. So

$$\begin{aligned} {\underline{\underline{R}} _{\xi \xi }}\left( t \right)&= \mathbb {E}\left( {\underline{\xi }\left( t \right) . \underline{\xi }{{\left( t \right) }^T}} \right) \nonumber \\&\quad = \underline{\underline{Q} } \left( t \right) \mathbb {E}\left( {\underline{\theta }\,\,{{\underline{\theta }}^T}} \right) \underline{\underline{Q} } {\left( t \right) ^T} \nonumber \\&\quad = \underline{\underline{Q} } \left( t \right) {\underline{\underline{R}} _{\theta \theta }}\underline{\underline{Q} } {\left( t \right) ^T} \end{aligned}$$

(81)

or

$$\begin{aligned} {\left[ {{R_{\xi \xi }}\left( t \right) } \right] _{\alpha \beta }} = \sum \limits _{\gamma \rho = 1}^\infty {{{\left[ {{{\underline{\underline{R}} }_{\theta \theta }}} \right] }_{\gamma \rho }}{Q_{\alpha \gamma }}\left( t \right) } \,\,{Q_{\beta \rho }}\left( t \right) \end{aligned}$$

(82)

writing, eq.$\left( 82 \right)$ as:

$$\begin{aligned} {\underline{\underline{R}} _{\xi \xi }}\left( t \right) = \underline{\underline{Q}} \left( t \right) \,{\underline{\underline{R}} _{\theta \theta }}\underline{\underline{Q}} {\left( t \right) ^T} \end{aligned}$$

(83)

or equivalently as

$$\begin{aligned} Vec\left( {{{\underline{\underline{R}} }_{\xi \xi }}\left( t \right) } \right) = \left( {\underline{\underline{Q}} \left( t \right) \otimes \underline{\underline{Q}} \left( t \right) . Vec\left( {{{\underline{\underline{R}} }_{\theta \theta }}} \right) } \right) \end{aligned}$$

(84)

We can estimate ${{{\underline{\underline{R}} }_{\theta \theta }}}$ as:

$$\begin{aligned}&Vec\left( {{{\widehat{R}}_{\theta \theta }}} \right) \nonumber \\&\quad = {{\mathrm{argmin}} _{\underline{\eta }}}\,{\sum \limits _t {\left\| {Vec\left( {{{\widehat{R }}_{\xi \xi }}\left( t \right) } \right) - \left( {\underline{\underline{Q}} \left( t \right) \otimes \underline{\underline{Q}} \left( t \right) } \right) \underline{\eta }} \right\| } ^2} \nonumber \\&\quad = {\left[ {\sum \limits _t {\left( {\underline{\underline{Q}} {{\left( t \right) }^T} \otimes \underline{\underline{Q}} {{\left( t \right) }^T}} \right) \left( {\underline{\underline{Q}} \left( t \right) \otimes \underline{\underline{Q}} \left( t \right) } \right) } } \right] ^{ - 1}} \nonumber \\&\qquad \left[ {\sum \limits _t {\left( {\underline{\underline{Q}} {{\left( t \right) }^T} \otimes \underline{\underline{Q}} {{\left( t \right) }^T}} \right) } \,. Vec\left( {{{\underline{\underline{R}} }_{\xi \xi }}\left( t \right) } \right) } \right] \end{aligned}$$

(85)