# Design Optimization in Constrained Applications: Page 3 of 4

Power injection to the grid. A power constraint is most common in utility-scale arrays, which inject the energy produced directly into the grid. In those situations, the grid operator often dictates the maximum instantaneous power the grid can handle at any time. As a result, there is a very clear power limit on what an array can produce, which the designer can achieve by simply matching the inverters’ rated output power to the grid’s requirements.

Power-injection constraints can also arise when regulatory procedures change significantly based on system capacity. A jurisdiction might have an expedited permit or interconnection process for arrays up to 200 kWac, whereas larger systems are subject to a more demanding set of requirements. These policies can artificially limit system capacity. If the permitting process gets significantly more stringent above 200 kWac, you might decide to constrict the ac output power to 200 kW—even if the roof area would otherwise support a 250 kWac system—just to keep the permitting process simple and/or inexpensive.

In this scenario, inverter ac power capacity is considered a fixed variable and dc system capacity becomes the key design variable. Since dc capacity determines system revenue and therefore profit, this variable drives financial performance. The designer incrementally increases the dc nameplate power while looking to maximize financial metrics. The marginal revenue of each group of modules drops successively, since the clipping losses get larger as the dc-to-ac ratio increases. However, the designer can continue increasing array capacity as long as the marginal revenue covers the marginal cost of adding modules, racking and wiring. Inverter power limiting is no longer a loss factor to minimize, but instead is a necessary tradeoff to increase revenue and maximize the financial performance of the array. This optimization exercise tends to push the dc system capacity to 1.4–1.7 times that of the ac inverter capacity, depending on the array location.

## Calculating the Optimal Design

To illustrate the importance of constraints on system design, I will start from a reference design with fixed cost and revenue assumptions and show how different design constraints lead to different optimal system configurations. For the purposes of this exercise, I am using the project’s net present value (NPV) as the optimization objective. The details of the reference design are as follows:

System capacity: 1 MWdc
Array area: 2.85 acres
Array tilt angle: 20°
Array azimuth: 180°
Interrow spacing: 2.4 feet
Location: San Francisco

In the scenarios that follow, I fix one of three major design constraints at a time—area, budget or ac capacity—while leaving the other two variables unconstrained. As shown in Table 1, this exercise results in three distinctly different designs.

Area-constrained scenario. Here I fix the array area based on that of the reference design while adjusting other variables to maximize NPV. The most profitable system design results from packing the modules closely together and reducing the tilt angle. These two variables go hand in hand: by dropping the tilt of the modules, you can reduce the row spacing without incurring a significant amount of interrow shading.

Optimizing the area-constrained array results in a 5° tilt angle and a row spacing of 0.5 feet. Note that the lower tilt in this design reduces the specific yield by 4% compared to the reference design, from 1,532 to 1,474 kWh/kWp. However, the reduced spacing increases the system capacity by 40%. Therefore, the overall energy production grows by 35%. The area-constrained design is even slightly more profitable. While the 4% lower productivity would typically translate to a lower profit margin, the larger array is able to amortize the fixed costs over a larger base. As a result, the ROI of the system is slightly better than that of the reference design.

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