On the modeling of a solar, wind and fossil fuel energy source by means of the thermostatted kinetic theory

Abstract

This paper is devoted to the modeling of a hybrid energy distribution network with storage, within the thermostatted kinetic theory framework. The network consists of a non-renewable energy source and a renewable energy source. The energy storage is modeled by the introduction of the external force field coupled to the thermostat term. The activation parameters of the energy sources are assumed time-dependent in order to mimic the time-dependent efficiency of different specific energy sources. In particular a solar energy source, a wind energy source and a fossil fuel energy source are modeled. A computational analysis is performed to show the effects of the intermittent activation on the plan of quality improvement of the energy provided to the customers and on the construction of the energy storage. Discussions and future research perspectives are proposed in the last section of the paper.

The hybrid thermostatted kinetic theory framework

This appendix deals with the hybrid thermostatted kinetic theory framework for the modeling of a network ${\mathcal {N}}$ composed of ${\mathbf {N}}=2$ energy sources: a non-renewable energy source NR and a renewable energy source R. The energy storage is also taken into account.

A discrete quality parameter $u\in [0,1]$ identifies the energy sources and specifically the energy source NR is associated with $u_1=1/3$ (low quality), while the energy source R is associated with $u_2=2/3$ (high quality). Each energy source is a functional subsystem and is described by means of the distribution function $f_i(t,w):[0,+\infty [\times D_w\rightarrow {\mathbb {R}}^+$, for $i\in \{1,2 \}$, over the microscopic state $w\in D_w\subset {\mathbb {R}}^+$ which models the continuous energy variable. Accordingly, $f_i(t,w)dw$ represents the number of customers served at time t with an energy value $w\in [w,w+dw]$ which is produced by the energy source $u_i$, for $i\in \{1,2 \}$.

The introduction of an external positive force field ${\mathbf {F}}=(F_1,F_2)\in ({\mathbb {R}}^+)^2$, acting on each functional subsystem, models the construction of the energy storage. The external force takes a certain amount of the energy produced by the sources and allocates it into the energy storage. From the mathematical point of view, the external force field ${\mathbf {F}}$ moves the system far from the equilibrium. Therefore, a thermostat term $\alpha _{(2,2)}[{\mathbf {F}},{\mathbf {f}}](t)$, where ${\mathbf {f}}=\left( f_1(t,w),f_2(t,w) \right)$ is also considered in order to control the global activation energy of the system and allows the existence of a nonequilibrium stationary state.

The macroscopic quantities are represented by the moments of the distribution function. Specifically the global (p, q)-th order moment writes:

$$\begin{aligned} {\mathbb {E}}^{[{\mathbf {F}}]}_{(p,q)}[{\mathbf {f}}](t)=\displaystyle \sum _{i=1}^2 u_i^p\int _{D_w}w^q\,f_i(t,w)\,\mathrm{d}w,\quad \forall \, p,q\in {\mathbb {N}}. \end{aligned}$$

(21)

In particular, it is assumed that ${\mathbb {E}}^{[{\mathbf {F}}]}_{(0,0)}[{\mathbf {f}}](t)=1$ for $t\in [0,+\infty [$ (normalization condition). In the case ${\mathbf {F}}={\mathbf {0}}$, the notation ${\mathbb {E}}^{[{\mathbf {F}}={\mathbf {0}}]}_{(p,q)}[{\mathbf {f}}](t)$ simplifies into ${\mathbb {E}}_{(p,q)}[{\mathbf {f}}](t)$.

The energy sources interact via the network ${\mathcal {N}}$ thus modifying their microscopic state. The time evolution of the distribution function $f_i(t,w)$ is given by balancing the inlet and outlet flows into the elementary volume $[w,w+dw]$:

$$\begin{aligned} \partial _t f_{i}(t,w) +\partial _w\left( \left( F_i - \alpha _{(2,2)}[{\mathbf {F}},{\mathbf {f}}](t)\, w\right) f_{i}(t,w)\right) =J_{i}[{\mathbf {f}}](t,w) ,\quad i\in \{1,2 \}, \end{aligned}$$

(22)

where $J_{i}[{\mathbf {f}}](t,w)=G_{i}[{\mathbf {f}}](t,w)-L_{i}[{\mathbf {f}}](t,w)$, for $i\in \{1,2 \}$, being $G_{i}[{\mathbf {f}}](t,w)$ and $L_{i}[{\mathbf {f}}](t,w)$ the gain term and the loss term, respectively, that write as follows:

$$\begin{aligned} G_{i}[{\mathbf {f}}](t,w)=&\displaystyle \sum _{l=1}^2\, \sum _{j=1}^2 \int _{D_w\times D_w} \eta _{lj}\left( w_*,w^*\right) \, {\mathcal {B}}_{l j}^{i}\, {\mathcal {A}}_{l j}\left( w_*,w^{*};w \right) \nonumber \\&\times f_{l}(t,w_*)\,f_{j}(t,w^*)\,\mathrm{d}w_*\,\mathrm{d}w^*, \end{aligned}$$

(23)

$$\begin{aligned} L_{i}[{\mathbf {f}}](t,w)=&\displaystyle f_{i}(t,w)\, \sum _{j=1}^2 \,\int _{D_w} \, \eta _{ij}\left( w,w^* \right) \, f_{j}(t,w^{*})\, dw^*, \end{aligned}$$

(24)

and the functional parameters have the following meaning:

• $\eta _{lj}\left( w_*,w^*\right) : D_w \times D_w\rightarrow {\mathbb {R}}^+$ denotes the interaction rate between the energy source $u_l$, with the energy value $w_*$, and the energy source $u_j$, with the energy value $w^*$.

• ${\mathcal {A}}_{l j}\left( w_*,w^{*};w \right) :D_w \times D_w\times D_w \rightarrow {\mathbb {R}}^+$ denotes the probability of the evolution of the energy value $w_*$ of the source $u_l$ into the energy value w, after the interaction with the source $u_j$ with the energy value $w^*$. Then one has

  $$\begin{aligned} \displaystyle \int _{D_w}{\mathcal {A}}_{l j}\left( w_*,w^{*};w \right) \,\mathrm{d}w=1,\quad \forall \,w_*,w^*\in D_w. \end{aligned}$$

  (25)

• ${\mathcal {B}}_{l j}^{i}$ denotes the probability of the activation of the energy source $u_i$ after the interaction between the energy source $u_l$ and the energy source $u_j$. Then one has

  $$\begin{aligned} \displaystyle \sum _{i=1}^2{\mathcal {B}}_{l j}^{i}=1,\quad \forall l,j\in \{1,2 \}. \end{aligned}$$

  (26)

The thermostat term $\alpha _{(2,2)}[{\mathbf {F}},{\mathbf {f}}](t)$ reads:

$$\begin{aligned} \alpha _{(2,2)}[{\mathbf {F}},{\mathbf {f}}](t)=\frac{\sum _{j=1}^2\,F_j u_j^2 \int _{D_w} w\,f_{j}(t,w)\,\mathrm{d}w}{\displaystyle {\mathbb {E}}^{[{\mathbf {F}}]}_{(2,2)}[{\mathbf {f}}](t)}, \end{aligned}$$

(27)

and controls the global (2, 2)-order moment ${\mathbb {E}}^{[{\mathbf {F}}]}_{(2,2)}[{\mathbf {f}}](t)$.

The model analyzed in this work is derived from the framework (22) by considering the following assumptions:

• The interaction rate $\eta _{lj}\left( w_*,w^*\right) : D_w \times D_w\rightarrow {\mathbb {R}}^+$ is modeled as follows:

  $$\begin{aligned} \eta _{lj}(w_*,w^*)=\alpha +\beta \delta _{lj},\quad \alpha ,\beta \in {\mathbb {R}}^+, \end{aligned}$$

  (28)

  where $\delta _{lj}$ denotes the delta of Kronecker. In particular it is assumed that the autointeraction rates are greater than the interaction rates between two different energy sources, i.e. $\eta _{ll}>\eta _{lj}$.

• The probability ${\mathcal {A}}_{l j}\left( w_*,w^{*};w \right) :D_w \times D_w\times D_w \rightarrow {\mathbb {R}}^+$ is modeled as follows:

  $$\begin{aligned} {\mathcal {A}}_{l j}(w_*,w^*;w)=\delta (w-d_{lj}(w_*,w^*)), \end{aligned}$$

  (29)

  where $\delta (\cdot )$ denotes the delta of Dirac and

  $$\begin{aligned} d_{lj}(w_*,w^*)={\left\{ \begin{array}{ll}w_*-\lambda _1 &{} \text {if}\quad l=1\quad \text {and}\quad j=2, \\ w_*+\lambda _2 &{} \text {if}\quad l=2\quad \text {and} \quad j=1,\\ w_*&{} \text {if}\quad l=j, \end{array}\right. } \end{aligned}$$

  (30)

  and $\lambda _1,\lambda _2\in [0,1]$. The choice of the definition (30) relies on the following assumptions: the energy source NR should reduce its energy value, the energy source R should increase its energy value, the autointeraction does not modify the energy value of an energy source.

• The probability ${\mathcal {B}}_{l j}^{i}$ is modeled as follows:

  $$\begin{aligned} {\mathcal {B}}_{l j}^{i}={\left\{ \begin{array}{ll}\epsilon _{lj}(t)&{} \text {if}\quad i=l, \\ 1-\epsilon _{lj}(t)&{} \text {if}\quad i\ne l,\end{array}\right. } \end{aligned}$$

  (31)

  where $\epsilon _{lj}(t):[0,+\infty [\rightarrow [0,1]$, for $l,j\in \{1,2 \}$. Specifically, $\epsilon _{1j}(t)$ and $\epsilon _{2j}(t)$, for $j\in \{1,2 \}$, denotes the probability of the activation of the energy source NR and R, respectively.