Optimal and energy efficient operation of conveyor belt systems with downhill conveyors

Abstract

Downhill conveyors are important potential energy sources within conveyor belt systems (CBSs). Their energy can be captured using regenerative drives. This paper presents a generic optimisation model for the energy management of CBSs that have downhill conveyors. The optimisation model is able to optimally schedule three configurations of a case-study CBS that is connected to the grid and operated under a time-of-use tariff. The three suggested drive configurations showcase potential energy savings/incomes that can be obtained from implementing: (a) variable speed control, (b) internal use of downhill conveyor energy and (c) the export of energy to the grid. The results show that a CBS with a daily energy consumption of 924 kWh can be reconfigured and controlled to reduce consumption by 53 or 100 % or be made to generate 1984 kWh, depending on the configuration. Analysis of the investment in each of the three configurations is assessed using a life-cycle cost and payback period (PBP). The daily operation simulation results show that the use of regenerative drives and variable speed control is able to provide energy savings in CBSs. The cost analysis shows that the configuration that enables sale of energy to the grid is the most profitable arrangement, for the case study plant under consideration. The sensitivity analysis indicates that the PBPs are more sensitive to the annual electricity price increases than changes in the discount rate. Combining regenerative drives and optimal operation of CBS generates energy savings that give attractive PBPs of less than 5 years.

Appendix:: Appendix: Discrete version of the transport equation

The flow model Eq. 1 is discretised into N _x samples in space and N _t samples in time over a given total time period using a finite difference method . The implicit backward finite difference method is chosen due to its stability (Trefethen 1996). Therefore, Eq. 1 becomes,

$$\begin{array}{@{}rcl@{}} &&\frac{q(i,n~+~1)~-~q(i,n)}{\Delta t}\\&&+v_{n}\frac{q(i~+~1,n+1)-q(i-1,n+1)}{2{\Delta} x}~=~0. \end{array} $$

(14)

A backward space derivative difference is used for the tail end of the conveyor because q(N _x + 1,n) is invalid. Let, $\gamma _{n}~=~v_{n}\frac {\triangle t}{\triangle x}$, then the algebraic manipulation of the Eq. 14 results into,

$$ q(i,n+1)=\left\{\begin{array}{ll} \begin{array}{l} 2\{q(i-1,n)-q(i-1,n+1)\}/\gamma_{i}+q(i-2,n+1)\\ \{q(N_{x},n)-q(N_{x}-1,n+1)\}/(\gamma_{i}+1) \end{array} & \begin{array}{l} i=2,3,{\ldots} N_{x}-1\\ i=N_{x} \end{array}\end{array}\right.. $$

(15)

Equation 15 can be also be written in matrix and vector form as,

$$ \widehat{A}_{n}\mathbf{q}_{n~+~1}~=~\{\mathbf{q}{}_{n}~+~\bar{\mathbf{b}}_{n}~\cdot~ I_{n}~/~v_{n}\}\text{ where } \widehat{A}_{n}\in\mathcal{R}^{N_{x}~\times~ N_{x}}\text{ ,} \bar{\mathbf{b}}_{n}\in\mathcal{R}^{N_{x}~\times~1}, $$

(16)

where q _n is a vector q(i,n) ∀ i ∈ [1,N _x ] and, the matrices $\widehat {A}_{n}$ and vectors $\bar {\mathbf {b}}_{n}$ are defined by,

$$ \widehat{A}_{n}~=~\left[\begin{array}{lllllll} 1 & \gamma_{n}/2 & 0 & {\cdots} & 0 & 0 & 0\\ -\gamma_{n}/2 & 1 & -\gamma_{n}/2 & & 0 & 0 & 0\\ 0 & -\gamma_{n}/2 & 1 & {\ddots} & 0 & 0 & 0\\ {\vdots} & & {\ddots} & {\ddots} & {\vdots} & {\vdots} & \vdots\\ 0 & 0 & 0 & & 0 & 0 & 0\\ 0 & 0 & 0 & & -\gamma_{n}/2 & 1 & \gamma_{n}/2\\ 0 & 0 & 0 & & 0 & -\gamma_{n} & (1~+~\gamma_{n}) \end{array}\right]\text{and }\bar{\mathbf{b}}_{n}~=~\left[\begin{array}{ll} \gamma_{n}~/~2\\ 0\\ 0\\ 0\\ \vdots\\ 0\\ 0 \end{array}\right]. $$

(17)