# Design Optimization in Constrained Applications: Page 4 of 4

## Inside this Article

**Budget-constrained scenario.** Here I fix the budget while giving the system free rein in terms of array area. In this case, the design goal is to maximize specific yield. As detailed in Table 1, raising the tilt angle to 30° and spreading the interrow spacing out to 8 feet increases the specific yield by 3%, from 1,532 to 1,581 kWh/kWp.

Of course, expanding the array area does not come without costs. Since our budget is fixed, I have accounted for this by decreasing the array capacity to account for the costs associated with the additional land requirements (modeled at $500 per acre per year) as well as the longer wire runs. In spite of the fact that the array area nearly doubles over that of the reference case, optimizing for specific yield in the budget-constrained scenario still results in 2% improvement in NPV.

**Power-constrained scenario.** Lastly, I constrain the system based on capacity, assuming that the maximum allowable power injection to the grid is 810 kWac, but do not constrain area or budget. In this case, dc system capacity has a significant impact on the overall system economics, since a larger dc system has greater revenue potential. Maximizing NPV in this scenario results in a dc system capacity of 1.38 MWp, which is a 1.7:1 dc-to-ac ratio.

This design approach results in a significant amount of inverter power limiting, with clipping losses of approximately 9.1%. Yet a dc loading of 1.7 optimizes the profit of the array by maximizing revenue with an eye toward controlling costs. While locations with higher insolation result in a lower dc-to-ac ratio, the optimal inverter loading will still be considerably higher than that of the reference design.

**Cross-applying the results. **Based on these scenarios, we see that applying three different design constraints to one location, with one set of cost assumptions, leads to three very different optimal designs. It may seem like a counterintuitive or even flawed premise that a single set of cost and revenue assumptions can lead to different optimal designs just based on the primary design constraint, but we can test this premise by cross-applying the optimal designs.

To illustrate, let us swap the area-constrained array with the budget-constrained array and evaluate how the designs perform when applied to a different set of design constraints. On the one hand, the area-constrained design results in tilt of 5° and spacing of 0.5 feet, effectively maximizing power density to increase revenue. On the other hand, the budget-constrained design results in a 30° tilt and an 8-foot-wide row spacing, effectively maximizing the specific yield of the array. Though the increase in profit is greatest in the area-constrained scenario, the budget-constrained design still generates more profit than the reference design.

Figure 2 shows what happens if we swap the design choices between these two applications, which causes the economics of both arrays to fall greatly. Applying the area-constrained design (low tilt and tight spacing) to the budget-constrained scenario results in an array with the same capacity as the reference design but a lower energy yield; as a result, the system NPV drops 40% to $620,000. Meanwhile, applying the budget-constrained design (high tilt and wide spacing) to the area-constrained scenario results in a 50% smaller array capacity; as a result, the NPV drops to under $300,000.

## Optimization Objectives

The selection of the design objective is important to any optimization process. In the previous examples, I maximized profit dollars by optimizing for NPV. Other common optimization objectives include profit margin, levelized cost of energy (LCOE) and initial cost. While these all seem like good objectives, choosing one over another can lead to a different design outcome.

To illustrate, let us consider the difference between optimizing for profit dollars and for profit margin in the power-constrained scenario above. As shown in Figure 3, optimizing for NPV results in a dc-to-ac ratio of 1.7:1, whereas optimizing for profit margin results in a dc-to-ac ratio of 1.4:1. Based on these results, we see that increasing the dc-to-ac ratio above that of the reference design (1.23) initially improves both profit margin and profit dollar. This is because increasing the dc loading initially improves both specific yield and profitability. Above a dc loading value of 1.4, however, profit margin starts to decline and profit dollars accrue more slowly. This is because the specific yield and revenue associated with each new group of modules starts to fall due to inverter power limiting. Because of these losses, it ceases to be profitable to add modules above a dc loading of 1.7.

The fact that different optimization objectives could lead to different designs is not self-evident. After all, when arrays underperform, all performance metrics generally suffer. For example, module or inverter failures or unexpected shading all reduce the array’s profitability and raise the system’s LCOE. This suggests that optimizing PV system designs based on one performance metric will optimize others as well. Figure 3 illustrates that this is not necessarily the case. In this example, the decision about whether to optimize based on profit margin or NPV will swing system capacity by more than 20%, from a dc loading of 1.4 to 1.7, respectively.

Which of these objectives will serve you better? It depends. If money is constrained, then you want to spend each dollar as effectively as possible, in which case profit margin is a better objective to go for. If money is not so tight, you might prefer to spend a bit more and receive more profit dollars.

If we take a step back, we can see that the design objectives for PV power systems are far less complex than those in other industries. Consider the many objectives that engineers must consider when designing a car: fuel efficiency, torque, acceleration, styling, driver visibility, weight, turn radius, length, cost, storage area, range, crash safety rating, reliability and so forth. For better or worse, electricity is a commodity. By definition, therefore, generating the lowest-cost electricity is generally the singular focus of PV system design activities. This is true even when you are making more-nuanced design decisions, such as optimizing for reliability or to streamline O&M activities. The solar industry’s primary optimization objective is to reduce LCOE, regardless of whether designers are optimizing based on initial or future costs.

**CONTACT:**

**Paul Grana** / Folsom Labs / San Francisco, CA / folsomlabs.com

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