## Meaning and Use of the Regression Line in Energy Performance Evaluation

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Once the regression equation has been obtained, estimates of past or future energy values for a given production output can be made, and variability in energy use determined. Variability can be expressed as the difference between standard or reference and actual energy use. Variability analysis is a systematic process of identifying and evaluating variances in order to provide useful information for measuring efficiency and improving the performance of energy use.

A part of variance in energy use will be caused by variances in production output. This is normal or expected variability. It will be determined by the underlying relationship between energy and production, as expressed by the regression equation. For instance, let us have a closer look at month 5 in Table 3.1 which displays an apparent excessive variability in energy use. This is shown in the scatter diagram in Figure 3.6. If we want to know what should have been the energy consumption for that month according to the obtained regression equation, we should substitute the actual production output of this month (13281 from Table 3.1) into the equation:

The underlying relationship between energy and production warrants the energy use of 1509.28 tons of steam. When we compare the estimated energy use value E' with the measured value of E, we will get the variability portion of energy that is not explained by the variability in production output:

One interpretation of this difference is that energy consumption in the respective month was 314.72 tons more than required for the given production output, based on the underlying average energy/production relationship. Therefore, it comes as no surprise that this month accounted for the highest energy consumption per ton of product (see column 4 of Table 3.1).

On the other hand, we can repeat the same procedure for month 1 when the lowest specific energy consumption was recorded:

If we calculate the difference between the actual and estimated values we will get what in statistical terminology is called a residual e:

There is a minus sign in front of our residual. What does it mean? A quick interpretation will tell us that we have used too little energy for the given production output. But the values of energy and production have been checked and confirmed to be correct, so this production output has really been achieved with the stated energy amount.

How and why this has happened, are the questions that regression analyses cannot answer. In this case, it can only tell us that for reasons unknown we have saved about 148.48 ton of steam in this particular month compared to the expected consumption based on the average underlying relationship.

If we calculate and plot all the residues, a graph will appear that can be used for residual analysis (Fig. 3.12). This is yet another scatter diagram, but this one shows a variability of energy values at the given production level, as calculated from the regression equation. Statistics say that individual

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Production [t /m] Figure 3.12 Residual Scatter Diagram residues should fall within a narrow horizontal band, otherwise our analysis may be based on false assumptions.

In our case, we can be satisfied that the 'narrow band' is defined over ± 200 range, with three outliers dropping outside the range as indicated in the graph. If an identifiable cause exists for the outliers, as it should indeed exist for all the other variations, we can be satisfied with the validity of our analysis. This brings us to the task of quantifying variability in energy values. There are four issues at stake:

• How can variations be quantified?

• What is the reference for energy variation assessment?

• Whether a variation is significant or not? and

• If a variation is significant, what was the cause of it?

Statistics can provide the answers for the first three questions but not fthe fourth.

3.7.1 Quantifying and Understanding Energy/Production Variability

We can see that the points in scatter diagrams (Fig. 3.11) do not fall precisely at, or around, the best-fit line. They vary in a seemingly random way with smaller or greater distances from the best-fit line, similar to energy residues. This is shown by the residual diagram. The fundamental indication of variability, provided by any given energy/production data set, is the amount of scatter around the estimated regression line. The measure for this variability is called the standard error of estimate, which is the deviation between the actual or measured energy E; and the estimated one E;' as computed by using the correlation equation for the same value of production P. The standard error of estimation is given as follows:

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