# The Model

The approach assumes that one has a system model that provides a quantitative measure of perfor­mance, say, efficiency h, as a function of some design parameter u. Furthermore, we assume that we know (or can reasonably guess) the relationship between this design parameter and the system cost X. The model shows that the rational economic optimum occurs for that value of the parameter u for which the relative incremental efficiency gains are precisely balanced by the associated relative incremental cost increases: That is, optimum cost-effectiveness occurs at that value of u for which the logarithmic derivatives of h and X are equal.

The fundamental objective of all solar system design is, of course, to maximize the energy delivered per dollar. The most sophisticated approach would be to deal with annual energy delivery and annualized costs. However, for a preliminary analysis, it should be sufficient to work with in­stantaneous efficiency and initial costs since these are directly related to the annualized quantities. We begin by defining the quantity R, which is directly related to the energy per unit cost, as

J?( /Of)

R(«) = ‘8J’

X(u)

where h is the instantaneous solar conversion efficiency under some fixed set of operating condi­tions (e. g., delivery temperature and insolation) and X is the system cost per unit collection area. The quantity u represents some design parameter on which both efficiency and cost depend. In our simple model, the most cost effective system will be that for which R is a maximum.

Formally, we solve for the condition of maximum R(u) by setting the derivative with respect to u equal to zero as follows,

di? 1 /drA / ?7 / dX

d и XduJ A2 / d и )

?7 / 1 d?] 1 dX

X h du X du

i] f d (In?7) d(lnA)

X du du

Thus, optimum cost-effectiveness occurs at that value of u for which the logarithmic derivatives of h and X are equal: That is, when

d (In?7) d (In X)

d и d и (8-3)

Application of this model requires not only that both h(u) and X(u) are known but also that they can be represented by continuous and differentiable functions of u. In practice, these idealized conditions are unlikely to be met, particularly for the cost function. On the other hand, a great deal can be learned simply by representing the behavior by appropriate parametric models. Clearly, the optimum will depend strongly on the shape of both functions. It is important to note that h is fun­damentally bounded (it cannot be greater than unity!), whereas X is not. It is quite possible that in FIGURE 8.1: Schematic representation of a model for the rational optimization of performance and cost trade-offs. The cost and performance both increase with decreasing value of some design para­meter u. the very region in which h(u) is approaching its limit, X(u) will be increasing rapidly. If this is the case, maximizing h(u) alone is clearly a misguided strategy.

The procedure is illustrated graphically in Figures 8.1—8.3 for a case in which both functions are monotonically decreasing with increasing u: That is, reducing the value of u results in higher system efficiency, but this is achieved at a higher cost. It is the difference in the shape of these func­tions that determines the optimum.

In our illustration (Figure 8.1), the efficiency approaches a limiting value (here, 0.85) as u goes to zero and falls off slowly as u increases. In contrast, the cost X is nearly constant at large u but begins to increase relatively rapidly as it approaches zero. We have deliberately chosen these forms to emphasize the effects noted earlier; however, such a qualitative behavior is not at all unreason­able. The behavior of the ratio R is shown in Figure 8.2, and both logarithmic derivatives are shown in Figure 8.3. Note that for such functions, the optimum is rather broad and occurs for values of h that are significantly lower than its maximum value. To the left of the optimum (smaller value of u), marginal increases in cost more than offset the small incremental gains in efficiency, whereas to the right of the optimum (larger value of u), cost savings are too small to make up for the loss in performance.  Design Parameter и (arbitrary units)

FIGURE 8.2: The maximum performance per unit cost [maximum R(u)] is quite broad and occurs for values of u corresponding to significantly less than maximum performance. FIGURE 8.3: The value of u at which the logarithmic derivatives of the cost and performance functions cross (are equal) determines the cost performance optimum.

Updated: August 18, 2015 — 2:52 am