Proposal

For an illustration of the issue, below is the hourly shape associated with the electricity demand for two different days in ISO New England.

The dip in the middle of the day is very evident; March 29th was a sunny day whereas March 14th was a cloudy day. Note how for the early morning hours and late evening hours the two curves are very similar; they are simply shifted vertically relative to one another. The only difference appears during the hours when the sun is up, in this case between hours 7 and 19.

In order to quantify the effect of the solar generation for an hourly shape \( w(t) \), between two hours \(t_i\) and \(t_f\) we introduce the measure

\[ D = \frac{1}{t_f - t_i} \int_{t_i}^{t_f} w(t) dt - \frac{w_i + w_f}{2},\] where \(w_i = w(t_i)\) and \(w_f = w(t_f)\).

The value of \(D\) is the ratio of the difference of the area under the weights \(w(t)\) and the area of the right-angle trapezoid defined by the points \([t_i, 0]\), \([t_i, w_i]\), \([t_f, w_f]\), \([t_f, 0]\) to the time interval \(t_f - t_i\).

This definition has several nice properties. It is a dimensionless quantity. Its value is signed. A sunny day, with a dip in the hourly shape will have a negative value for \(D\), whereas a cloudy day will exhibit a positive value. Indeed, for the two days mentioned in the figure above, \(D=-0.16\) on March 29th and \(D=0.02\) on March 14th. A day with a more pronounced dip in its hourly shape than another day will have a lower \(D\) value. Therefore, as solar penetration increases in a region, this should be evident by simply plotting the values of \(D\) over time. The calculation of \(D\) is very fast once the integral is written in discrete form.