16.06.2013 Views

1. Introduction - Firenze University Press

1. Introduction - Firenze University Press

1. Introduction - Firenze University Press

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

TSi yi ti1, iI, i NI + 1, (6)<br />

The number of TSs is obtained from (7). One is added to the sum of the selected TS boundaries, as<br />

the TS boundaries at the beginning and end of the observed time-horizon were excluded within the<br />

model:<br />

NTS yi<br />

1,<br />

(7)<br />

iI, iNI 1<br />

The inaccuracy during each time-interval is determined by multiplying the positive difference<br />

between the real and approximated supplies with the time-horizon of the time-interval.<br />

INi EDi( ti ti1) , iI, i NI + 1, (8)<br />

The overall inaccuracy is a result of summating the inaccuracies over the time-intervals:<br />

INA IN , (9)<br />

iI, iNI 1<br />

i<br />

and this overall inaccuracy is constrained and should be less than or equal to the fraction of the<br />

initial amount of solar irradiation presented as an area (A0) below the measured profile of Fig 1:<br />

INA A , (10)<br />

<br />

0<br />

A (( t t) RS<br />

0 i1 i i<br />

iI, iNI 1<br />

2.4. Optimisation procedure<br />

)<br />

,<br />

(11)<br />

Optimisation is performed over two stages. During the first stage of optimisation, Equations (1-11)<br />

are used with the objective of minimising the number of TSs as follows:<br />

min zI NTS , (12)<br />

This step requires specifying the acceptable error-level (tolerance) . The procedure applies multi-<br />

objective optimisation by the –constraint method, where one objective is considered in the<br />

objective function and the other is inserted in the model as an –constraint. The result from<br />

optimisation is the minimal number of TSs, min NTSI required to meet any constraint about the<br />

inaccuracy limit (10).<br />

However, after the first stage, the inaccuracy is not optimal. In order to obtain a further reduction in<br />

inaccuracy, in the second stage of optimisation the same model using equations (1–11) is used<br />

together with an additional equation (13), which fixes the number of TSs, and the objective as<br />

expressed in (14) ,<br />

NTS min NTSI<br />

, (13)<br />

min zII INA<br />

, (14)<br />

Multi-objective optimisation could also be performed over one stage, with the so called weighted<br />

sum method as sometimes this is faster. In this case, the objective function would be a weighted<br />

sum of NTS and INA with a high enough weight w (e.g. 10,000) for NTS, in order for the minimised<br />

NTS to have priority over the minimum of INA.<br />

z wNTS INA , (15)<br />

301

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!