## 842 The Cost of Misoptimization

What is the penalty for not optimizing correctly? In general, the following causes could prevent correct optimization:

• Insufficient accuracy of the algorithm or program for calculating the performance

• Incorrect information on economic data (e.g., the factor P in Eq. 8.55)

• Incorrect information on technical data (e.g., the factor K in Eq. 8.55)

• Unanticipated changes in the use of the building and the resulting changes in electrical load

Misoptimization would produce a different design at a value Ccap different from the true optimum Ccapo. For the example of insulation thickness, the effect on the life cycle cost can be seen directly with the solid curve in Figure 8.6. For example, a ±10% in Ccapo would increase Cf by only +1%. Thus, the penalty is not excessive for small errors for this simple example.

This relatively large insensitivity to misoptimization is a feature much more general than the insulation example. As shown by Rabl (1985), the greatest sensitivity likely to be encountered in practice corresponds to the curve

Clife,true(Ccap°,g"ess) = * ( ) ("upperbound") (8.56)

also shown in Figure 8.7 with the label "upper bound." Even here, the minimum is broad; if the true energy price differs by ±10% from the guessed price, the life cycle cost increases only 0.4% to 0.6% over the minimum. Even when the difference in prices is 30%, the life cycle cost penalty is less than 8%.

Cost Penalty

Cost Penalty

upper bound lower bound

true guess

FIGURE 8.7

Life cycle cost penalty versus energy price ratio.

upper bound lower bound

true guess

FIGURE 8.7

### Life cycle cost penalty versus energy price ratio.

Errors in the factor K (due to wrong information about price or conductivity of the insulation material) can be treated the same way, because K and P play an entirely symmetric role in the above equations. Therefore, curves in Figure 8.7 also apply to uncertainties in other input variables.

The basic phenomenon is universal: any smooth function is flat at an extre-mum. The only question is, how flat? For energy investments, that question has been answered with the curves of Figure 8.7. We can conclude that misoptimization penalties are definitely less then 1% (10%) when the uncertainties of the input variables are less than 10% (30%).

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