Adrian A. Drăgulescu
January 22, 2024

Introduction

Electricity demand as a timeseries has various interesting characteristics. In general, it depends on external factors like temperature, day of the week, time of the day, on wether the day is a holiday or not, on the customer type, etc.

If we denote by \( L_i \) the electric load for hour \(i\), then the weight of hour \(i\) relative to the other hours of the day is by definition \[ w_i = \frac{L_i}{\langle L\rangle} \] where \( \langle L\rangle \) is the average load for that day. The intraday pattern of electricity consumption as reflected by \( w_i \), is called the hourly load shape.

Over many years, the hourly load shapes have been relatively stable for "similar" days, that is for days falling around the same time of the year and with close enough temperature after adjusting for day of the week patterns. This has started to change in recent years as more and more solar panels have been installed across the US. On most regions of the country on sunny days the electricity demand now shows a pronounced dip in the middle of the day due to solar generation that offsets native customer demand.

This change in hourly long shape has been popularized by the California Independent System Operator in 2012 with the catchy phrase duck curve. While the mid-day dip in electricity demand on sunny days is evident in the data, is it not directly obvious how to quantify or compare qualitatively the strength of this effect between different days and electric regions.

The increase in solar penetration is only expected to increase over the next few years, making the need for such a measure even more important. This document proposes such a measure and shows several examples its use.

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.

Examples

The plot above shows \(D\) the strength of the duck curve between Jan20-Dec23 using ISONE RT demand data between the hours beginning 7 and 19. As expected, we see the points to exhibit a downward sloping trend year over year reflecting the solar build out in the region over this period.

📝 Note The data above is for RT demand not load. Over time, as more and more batteries are added to the system and charge during the mid-day dip in load, it will lead to an increase in \(D\). That's why we recommend calculating \(D\) using the RT load not RT demand data.

As mentioned before, we are now in position to compare the strength of solar penetration between different geographical regions. Of course, local weather conditions and customer mix will be different between regions, but over time a summary static on \(D\) will reveal existing differences. For example, the plot below compares the RT demand in Maine and NH over the Mar23-Apr23 period. As you can see, Maine values tend to be lower on most days. This is mainly due to the fact that Maine has put a very strong community solar program in place which has incentivized solar development in the state.