Intra- and All-Day PV Power Forecasting Using Expansion PDE Models Composed of the L-Transform Components in Nodes of Step-by-Step Evolved Polynomial Binary-Nets

Abstract

Photovoltaic (PV) power is one of the most important energy sources available in backcountry regions or developing southern countries with missing infrastructure. Intra- or all-day statistical models, using the latest environmental and power data records, can predict PV power for a plant-specific location and condition on time. Numerical Weather Prediction (NWP) systems are run every 6 h to produce free prognoses of local cloudiness with a considerable delay and usually not in operational quality. Differential binomial neural networks (D-BNN) are a novel neurocomputing technique that can model the characteristics of the weather. D-BNN decomposes the n-variable Partial Differential Equation (PDE), allowing a complex representation of the near-ground atmospheric dynamics, into a set of 2-input node sub-PDEs. These are converted and substituted using the Laplace transform formulations of Operation Calculus. D-BNN produces applicable PDE components, one by one using the selected binary nodes to extend its sum models. Historical spatial data are examined to pre-assess daily training samples for a specific inputs->output time shift used in forecasting the Clear Sky Index. Iterative 1–9 h and 24-h sequence PV power prediction models using machine learning (ML) and statistics are compared and evaluated. Daily modelling allows for sequence predictions of full PV power (PVP) cycles in operational quality. Reliable PV forecasting is required in load management in plant power supply and consumption.

APPENDIX - Node PDE Conversion Using the L-Transform

D-BNN develops a BNN structure to decompose the n-variable PDE into sub-PDEs in its two input nodes. These are converted using OC into pure rational terms corresponding to the L-transformed node functions. The sum of selected inverse L-transformed rational terms gives the PDE model of an n-variable output function [8].

$$ a + bu + \sum_{i = 1}^n {c_i \frac{\partial u}{{\partial x_i }}} + \sum_{i = 1}^n {\sum_{j = 1}^n {d_{ij} } } \frac{\partial^2 u}{{\partial x_i \partial x_j }} + \ldots = 0\quad\quad u = \sum_{k = 1}^\infty {u_k } $$

(2)

$$\begin{gathered} u\left( {x_1 , \, x_2 ,, \, \ldots ,x_n } \right) - unknown \, \,separable\, \, function\, \, of\, \, n - input\, \, variables \hfill \\ a, \, b, \, c_i , \, d_{ij} ,... - weights\, \, of\, \, terms\quad\quad u_i - partial\, \, sum\, \, functions \hfill \\ \end{gathered}$$

The general linear PDE (2) can describe an unknown separable u function of n inputs, which can be expressed in convergent sum series (2) of partial u_k function solutions of 2-variable simple sub-PDEs defined by the equality of 8 variables.

$$ L\left\{ {\left. {f^{(n)} (t)} \right\}} \right. = p^n F(p) - \sum_{k = 1}^n {p^{n - i} f_{0 + }^{(i - 1)} } \begin{array}{*{20}c} {\,} & {\,} \\ \end{array} L\left\{ {\left. {f(t)} \right\}} \right. = F(p) $$

(3 )

$$ f\left( t \right), \, f^{\prime} \left( t \right), \, \ldots , \, f^{(n)} \left( t \right)-originals \, continuous \, in\, {<}0\, + ,\infty{>} \quad p,t \, - \, complex \, and \, real \, variables $$

The OC binomial conversion of the f(t) function n^th derivatives in an Ordinary differential equation (ODE) is based on the proposition of their Laplace transformation (L-transform) taking into account the initial conditions (3).

$$ F(p) = \frac{P(p)}{{Q(p)}} = \frac{Bp + C}{{p^2 + ap + b}} = \sum_{k = 1}^n {\frac{A_k }{{p - \alpha_k }}} $$

(4)

$$ B, \, C, \, A_k - \, coefficients \,of\, \, elementary\, \, fractions\quad\quad a,b \, - binomial\, parameters $$

The conversion of ODE results in algebraic equations from which the L-transform F(p) can be expressed in the complex form of a pure rational term (4). The pure rational function represents the original function f(t). It can be expressed in the form of elementary sum fractions (4), which are transformed into inverse L images using the OC definitions (5) to obtain the original f(t) of a real variable t in the ODE solution.

$$ F(p) = \frac{P(p)}{{Q(p)}} = \sum_{k = 1}^n {\frac{P(\alpha_k )}{{Q_k (\alpha_k )}}} \frac{1}{p - \alpha_k }\begin{array}{*{20}c} {\,} & {\,} & {\,} \\ \end{array}\quad f(t) = \sum_{k = 1}^n {\frac{P(\alpha_k )}{{Q_k (\alpha_k )}}} e^{\alpha_k \cdot t} $$

(5)

$$ a_k - \, simple \, real \, roots \, of \, the \, multinomial \, Q\left( p \right)\,\,\,F\left( p \right) - L - \, transform \, image $$

D-BNN composes pure rational terms, for example (7), from GMDH binomials (6) [4]  in its block nodes (Fig. 9) to convert specific 2-variable sub-PDEs (2) into the L-transforms of unknown summation u_k functions (2) (Fig. 12).

$$ y\, = \,a_0 \, + \,a_1 x_i \, + \,a_2 x_j \, + \,a_3 x_i x_j \, + \,a_4 x_i^2 \, + \,a_5 x_j^2 $$

(6)

$$x_i , \, x_j -2\, \, input\, \, variables\, \, of \, \,neuron\, \, nodes$$

Fig. 12.

2-input node blocks produce simple (/) and composite neurons to solve PDEs

The inverse L-transform is applied to rational terms in the selected nodes (7), according to OC (5). The sum of the two-variable u_k original, produced in the nodes (Fig. 9), gives a model of the separable output u function (2). Each node block forms simple neurones, for example, (7), to convert and solve specific 2-variable PDEs.

$$ y_i = w_i \frac{b_0 + b_1 x_1 + b_2 sig(x_1^2 ) + b_3 x_2 + b_4 sig(x_2^2 )}{{a_0 + a_1 x_1 + a_2 x_2 + a_3 x_1 x_2 + a_4 sig(x_1^2 ) + a_5 sig(x_2^2 )}} \cdot e^\varphi $$

(7)

$$ \begin{gathered} \varphi = \, arctg\left( {x_1 /x_2 } \right) - phase\, \, representation\, \, of \, \,2 \, \,input\, \, variables\,x_1 , \, x_2 \hfill \\ a_i , \, b_i - binomial\,parameters\quad w_i - \, weights\quad sig-sigmoidal\, \, transform \hfill \\ \end{gathered} $$

The Euler notation of complex numbers (8) corresponds to the expression OC f(t) (5). The radius amplitude r represents a rational term, while the angle phase = arctg(x₂/x₁) of 2 variables of real value can give an inverse L transformation of F(p).

$$ p = \underbrace {x_1 }_{\text{Re}} +\, i \cdot \underbrace {x_2 }_{\text{Im}} = \sqrt {x_1^2 + x_2^2 } \cdot e^{i \cdot \arctan \left( {\frac{x_2 }{{x_1 }}} \right)} = r \cdot e^{i \cdot \varphi } = r \cdot (\cos \varphi + i \cdot \sin \varphi ) $$

(8)