Figure 3.16 Residual Scatter Diagram against Best Performance Regression Equation
If we now redraw the residual scatter diagram with residues calculated using the new equation, the results are shown in Figure 3.16. The new value of standard error of estimate is only Seen = 18.5 t/m for the chosen 'best' points which represent the target for performance assessment. The variability around the 'best' (or target) performance is larger than previously around the average performance. There are now only three values within the new significance range of +/- 1 x SEEN = 18.5 t/m.
However, we may decide to set the upper significance limit at 3 x SEEN = 55.5. It is a customary procedure when the correlation coefficient of the target line regression equation is strong (near to 1). If the correlation is week, then the significance interval determined by 3 x SEEN will be too wide. In other words, too many instances of substandard performance will be accepted without question. It does not make sense to establish a lower significance limit, because it may look like we are restricting the requirements for performance improvement.
In our case study, applying the 3 x SEEN significance limit will reveal that variations in energy use are too large in all but three months of the observed period, and individual residues are, of course, larger than in the previous case. This is as much as statistical methods can help us in performance evaluation. Finding the reasons for excessive variability and preventing recurrences is clearly beyond the power of statistics.
When performance begins to improve, the points on the residual scatter diagram will be closer to the significance limit line, and eventually all the points should be below the line. When all points are below the significance limit, it means that consistent performance improvements over the targeted best practice are achieved, and this requires an update of the target line, i.e., the corresponding regression equation.
Additional useful information that regression equation can provide is the quantification of a concept of fixed and variable energy consumption (Fig. 3.17). The original regression equation is:
The intercept value of 403.59 will be roughly equivalent to the fixed energy consumption, and '0.8326 x P' will show the rise in energy consumption for additional units of production P. If P = 1, then the additional justified variation in energy will be 0.8326 for an additional unit of product.
For instance, when the production output is 1347 tons, the estimated energy use by regression equation will be 1525.1 tons of steam, and the actual energy use was 1771 tons (from Table 3.1). The energy/production relationship, as described by the regression equation, can provide the following breakdown of this energy value:
• Fixed energy consumption contributes to 403.59 tons of steam use.
• The variable energy portion contributes to 0.8326 x 1347 = 1121.51 tons of steam use.
• The energy residue e = 1771 — 1525.2 = 245.9 tons cannot be explained by the underlying energy/production relationship, as it is caused by the factors that influence energy performance.
Again, we should be careful when coming to any conclusions based on extrapolation and on a mechanical interpretation of statistical parameters. We have shown already that for the data subset based on best performance, quite different statistics emerge and provide a different breakdown of energy variability. Which data sets are representative and over what range can be decided only on a case-by-case basis and by relying on an understanding of the process and of the energy/production relationship.
Another way to present data from Table 3.1 is shown in Figure 3.18. Here, instead of energy, the specific energy consumption (SEC) is plotted against production. It is again an approximate linear relationship, but in this case the slope is negative, i.e. the line is declining. It means that with increased production, energy consumption per unit of product is decreasing and vice versa.
This brings us back to the impact of market demand variations. If production output is falling, it means that we are moving back along the x-axis in Figure 3.18, which in turn means that energy consumption
Figure 3.18 Relationship between Production and Specific Energy Consumption
Production [t /m]
Figure 3.18 Relationship between Production and Specific Energy Consumption per unit of product is increasing. As a result, the production at low capacity utilization is more expensive and should be avoided.
3.8 Summary of Presenting and Analyzing the Energy/Production Relationship
The key to energy and environmental performance improvement is to understand the energy/production relationship and the impact of influencing factors on energy/production variability. In this chapter, we have explored the ways of expressing and analyzing this relationship. Let us review them briefly. Energy/production relationship is determined by:
• process design and technology requirements;
• operational procedures;
• capacity utilization;
• a combination of influencing factors affecting day-to-day operation.
A prerequisite for the analysis of the actual energy/production relationship is the availability of reliable data. Monthly data at the factory level are suitable only as an indication of potential problems. For performance improvement we need daily energy/production data for each designated ECC. A scatter diagram with the resulting data pattern is the best visual representation of the energy/production relationship.
Correlation analysis provides a measure of strength for the association between energy and production along an assumed linear relationship, given by the correlation coefficient r which takes values between 0 and 1. If r is close to 1, it implies a very strong correlation, and indicates good energy management practice and control of energy use in production.
Regression analysis provides the equation which is the best approximation of the underlying energy/production relationship given by available data. The regression equation can be used to estimate the energy value for any given production output. When coefficients of regression equations are calculated by the least square method, the result will be the best-fit line that will go through the center of the data scatter. The measure of variability in energy values is given by the standard error of estimate.
To evaluate the significance of variations, we need to establish a meaningful reference or base line against which individual variations can be judged. A common sense approach suggests selecting several instances of achieved best performances in the observed period, checking beforehand that these have happened under normal circumstances. A new regression line can then be calculated using these data points only and the resulting equation can be used as a baseline for assessing the significance of variability in energy values.
We should be careful not to rely too much on the statistical indicators, and they should not be used mechanically to draw conclusions. They must be used with care to support the analytical study of a particular data pattern that represents an energy/production relationship. In any case, statistics cannot explain causes for particular variability patterns. Only inductive reasoning, based on an understanding of the underlying energy/production relationship and on factors that affect energy performance, can provide explanations for particular deviations and suggest corrective actions.
In practice, data interpretation of energy performance is based on analytical thinking supported by correlation and regression analyses. Such data interpretation is the process of creating knowledge of the causes and effects of variability in the energy/production relationship that will result in operational guidelines, which will ultimately bring the factory to achieve the best practice operation.
Morvay, Z., Gvozdenac, D., Kosir, M. (1999) Systematic Approach to Energy Conservation in the Food Processing Industry, Manual for Training Programme 'Energy Conservation in Industry' (training course performed in Thailand), supported by Thai ENCON Fund, EU Thermie, ENEP program. Olson, C.L. (1987) Statistics: Making Sense of Data, W C B/McGraw-Hill, December.
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