# I Dream of Gini for Power Equality

One way that equality is measured is with the Gini coefficient, or Gini for short, which is in the range between zero and one. For values of zero there is maximum equality, every value is the same. For Gini values of one there is maximum inequality. The Gini coefficient is usually applied to income distributions so that if everyone has the same wealth (or income) then Gini = 0 and if one person has all the wealth then Gini = 1. We can use this same notion of equality and apply it to power equality. Using the PENS resilience indexes we can calculate the Gini coefficient over a period of time for different regions. As we have done in earlier blogs we will apply this to different metro areas. Gini will be calculated for all the sensors on a daily basis with respect to an index and the values averaged over a year. The rankings of metro areas by Gini values for each index can be used as a means for determining which areas are in need of remediation.

As the US power grid is very resilient overall, most of the index values will be very close to one. Consequently, most of the Gini values will be close to zero. To work around this feature of the grid, rather than looking at the resilience index itself, we look at one minus the index. This will give much more relative range and a better dispersion of the values and therefore rankings of metro areas. The interpretation of 1 – index is that rather than looking at the equality of resilience we are looking at the complement of it, namely, the equality of non-resilience or resilience inequality. Thus, in this case we are focusing on the inequality (unbalanced) rather than the quality (excellence). For example, considering PQI, the PENS Quality Index, which is a non-linear measure of the voltage from its reference value, 1 – PQI would be a measure of power “non-excellence” or power quality deficiency and by measuring the Gini value we find the inequality of power non-excellence.

Given this new representation we can look at the Lorenz curve which can be used to calculate the Gini value and provides a pictorial and intuitive presentation. Here the Gini value is calculated from the average of the daily 1-PQI value for each sensor. Several examples of metro area Lorenz curves for 2022 are shown in Fig. 1. The diagonal red line denotes the “line of equality” where every sensor has the same average daily value of 1-PQI. The ratio of the gray area divided by the gray area plus the blue area is the Gini value. When the Lorenz curve is all blue with no gray, that is perfect equality where all the values are the same.

We can also calculate the daily Gini by using the daily 1-PQI for all the sensors in a given metro and plot the daily variation over the course of a year. This is shown for some selected metro areas in Fig 2. Note the spikes on some days indicating severe inequality across the metro area, meaning some outliers were driving the daily Gini value.

The interesting contrast between the full-year Gini from the average 1-PQI (Fig. 1) and the full-year Gini from daily 1-PQI (Fig. 2) indicates that on a daily basis there is significant inequality of power non-quality, but that over the course of a year the inequality is much less (values closer to 0). In other words, the inequality on a daily basis is large, but the average over the course of a year is much more equitable.

In some cases there is correlation between the daily and yearly Gini values for a metro area. For example, Fig. 2 for Indianapolis the span of daily Gini values is more than 0.2 which gives a yearly Gini of about 0.5, compared with Baltimore or Miami which has a daily Gini range of about 0.1 and a much smaller yearly Gini of about 0.2. Detroit was fairly stable but had a rough time in the summer months which led to an intermediate value (in this sample of metro areas) of Gini of 0.3.

Another way of looking at the daily/yearly discrepancy is to recall that large Gini values indicate inequality, which is in this case means that the large daily Gini values are being driven by outliers. The affect of these outliers is greatly diminished when taking the daily average which is why the yearly Gini values are much smaller than the daily values.

When considering remedies it means that if one is considering the daily values the remedy should be directed to the area of the outliers, while when considering yearly values the remedy should be directed to the overall metro area with the greatest inequality.

To further examine the ranks by day and year, the Gini ranks by metro area are given in the table below. The Gini values are from smallest (more equality) to largest (less equality). According to a Kendall Tau analysis the two rankings have a Kendall rank correlation of -0.25 indicating a slight anti-correlation between the two ranking methods.

Table 1. Gini rankings from most equitable to least equitable

Note that an area may have a good degree of equality measured by Gini but still be poor overall by other resilience measures. Houston is an example of this, which from previous blog articles was among the worst regions, meaning that while Houston scores poorly on a number of metrics related to power resilience, at least everyone is more equally sharing that poor performance. On the other hand, Detroit scores poorly both on power resilience and power equality.

The point here is that we can use the power quality complement, or non-quality of power, values to construct meaningful Lorenz curves and Gini values which can then be used for ranking different areas, which can in turn be used to inform remediation or policy decisions to bring equality laggards more into the fold, either within a metro area or for a metro area as a whole. Overall, as we’ve seen in these blogs it is a non-trivial question about which area is in need of which kind of remediation and a question that policy makers should be well aware of.