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1. Introduction - Firenze University Press

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e equal at both time intervals and the time-interval period boundary candidate is deselected as a TS<br />

boundary (Fig 3).<br />

Fig 2: Determining the inaccuracy between the input<br />

and approximated supply<br />

2.3. MILP model formulation<br />

300<br />

Fig 3: Acceptance/ rejection of the candidate time<br />

period boundary as a TS boundary<br />

A two-stage MILP model has been developed for minimising the number of TSs at acceptable<br />

inaccuracy. During the first stage, the number of TSs is minimised, depending on the tolerance of<br />

inaccuracy specified by the models’ users. During the second stage, the inaccuracy is minimised at<br />

a fixed minimum number of TSs, determined during the first optimisation stage.<br />

Initially there is NI number of time-intervals and, hence, NI +1 boundaries of time-intervals indexed<br />

by the following index and set: i for the time-boundaries of the time-intervals, iI.<br />

The difference between the real input-supply and approximated-supply is calculated during each<br />

time period separately:<br />

SD RS – AS , iI, (1)<br />

i i i<br />

Because the difference, SDi can have a positive or negative value, it can be represented as the<br />

difference between the positive variables PDi and NDi:<br />

SD PD – ND , iI, (2)<br />

i i i<br />

Note that, when the SDi has a positive value, the NDi is zero, as a result of minimising the<br />

inaccuracy. When SDi has a negative value, the PDi is zero. For minimal inaccuracy the difference<br />

between the real and approximated supply should be the lowest possible.<br />

EDi PDi NDi,<br />

iI, (3)<br />

In (3) the positive value is obtained for the difference between real and approximated supply load.<br />

Further equations relate to the accepting / rejecting of the time-interval boundary as a TS boundary.<br />

The decision is made by the binary variable yi. When there is a positive (4) or negative difference<br />

(5) between the two consecutively-approximated supply loads, there is a TS boundary and the value<br />

of yi is <strong>1.</strong> If there is no difference between these supplies, there is no TS boundary and the value of<br />

yi is 0.<br />

ASi1ASi LV yi<br />

, iI, i NI + 1, (4)<br />

ASi1ASi LV yi<br />

, iI, i NI + 1, (5)<br />

In order to present the selected TS boundaries, the binary variable is multiplied by the observed<br />

time-period boundary:

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