Size optimization refers to any DER-VET case where there is at least one DER being sized by DER-VET instead of the user providing the size a priori. Size optimization converts the maximum charge power, maximum discharge power, and maximum generating power parameters into optimization variables. Doing so requires that binary variables are turned off (no minimum generating power and complications with negative energy prices and ancillary services), that the optimization window size is a whole year, and that there is only one optimization year. DER-VET will formulate and solve a simultaneous dispatch and size optimization problem that considers all costs and benefits, including capital costs, operational costs, and service participation, for a system. Constraint services (e.g. reliability) are respected in size optimization, allowing for size minimization problems that are not sensitive to the costs of the technology.
DER-VET's optimization problem will seek the sizes for all equipment being sized that minimizes the objective function simply represented by the expression (NPV(costs) - NPV(benefits)). Both costs and benefits should increase with size, but there should be a point at some size where the benefits are largest with respect to costs, which is considered the optimal size. Because the present value of costs and benefits is calculated in the objective, the useful life of the equipment needs to be known beforehand, disqualifying dynamic degradation calculations from size optimization problems.
Continuous Size Optimization
All technologies in DER-VET are continuously-sized. This means that DER-VET is allowed to select any size between a minimum (e.g. 0 kW) and a maximum (e.g. 100 kW) value. DER-VET cannot optimally select the number of fixed-size generators, for example, and will always return an optimal power capacity instead. (This can be done by manually adjusting the number of fixed-size generators in a single case until an optimal value is reached)
Not all technology options in DER-VET are compatible with size optimization.
|Internal Combustion Engine||Compatible|
|Single Electric Vehicle||Not Compatible|
|Fleet of Electric Vehicles||Not Compatible|
|Controllable Load||Not Compatible|
|Thermal Technologies||Not Compatible|
Most technologies in DER-VET have a single size parameter - their rated capacity in kW. Energy storage systems, on the other hand, have two size parameters - power capacity and energy capacity. In reality, they have more, which include separate charge and discharge power capacities, and minimum power capacities for both charging and discharging. Minimum power is not compatible with size optimization, so these are set to 0 kW and when doing size optimization, the charging power capacity is constrained to be equal to the discharging power capacity of the storage system. This reduces the number of optimization variables for energy storage size optimization to two - power capacity and energy capacity.
All technologies in DER-VET that can be optimally sized require a cost function. This cost function determines how the capital costs of the technology change with changes in its size variables. Cost functions in DER-VET are linear - no complicated relationships are allowed. This linearity preserves the mixed integer linear programming approach in DER-VET. So, no unit cost reductions in size (as you might expect in the real world) are not allowed.
DER-VET will add all terms in a technology's cost function to arrive at the total capital cost of that technology. For most technologies, this capital cost function follows the form: fixed costs ($) + costs per kW ($/kW) * power capacity (kW) = total capital cost ($). For energy storage systems, the extra size variable is represented as well: fixed costs ($) + costs per kW ($/kW) * power capacity (kW) + costs per kWh ($/kWh) * energy capacity (kWh) = total capital cost ($).
It is important that all costs per kW and costs per kWh are filled in with a positive nonzero number when optimally sizing that size parameter. Otherwise, the optimization will not penalize larger sizes and the size results will be infinite. For energy storage systems, optimally sizing energy capacity and power capacity are independent. If only sizing one, only the cost for that parameter ($/kW or $/kWh) is required.
Just as increasing size needs to be penalized in the optimization, increasing size should also increase the benefit the system can provide. Unlike the costs, though, the increase in benefits should not be linear with size - some diminishing returns need to be included. If both the costs and the benefits were linear with size, then the optimal size would be either 0 or infinity, which does not provide much useful information. Instead, the incremental value of additional size should decrease with increasing size. This ensures there is some value greater than or equal to 0 but less than infinity that minimizes the objective function.
This is most noticeable in DER-VET when modeling systems providing wholesale energy or ancillary services. DER-VET adopts a price taker model formulation for these services. Energy and ancillary service prices are an input to the model and are not modified by the operation of the DERs. So, every kWh of energy or kW of ancillary service provided by a DER is worth the same amount of money, no matter how many kW or kWh are offered. This linear relationship between size and value is not compatible with size optimization. Instead, a fixed power capacity is required when providing these services. The energy capacity of a storage system can be optimally sized when providing wholesale services because the value curve of these services with respect to energy capacity shows the requisite diminishing returns.
Size optimization is only compatible with a single optimization year. Multiple optimizations can be executed separately on different years of data to arrive at several size results, each of which can be evaluated using a fixed-size multiple opt-year DER-VET case. Complicated scenarios where multiple optimization years are included by the user or triggered by DER-VET should not include size optimization.
Degradation is not compatible with size optimization. Degradation couples a system's operation with its life, which is used to calculate the benefits of the system. This complexity is not handled in DER-VET (size optimization requires a known life beforehand), so degradation can not be included when performing size optimization. An iterative approach can work in this case, calculating life using the degradation feature, then executing a size optimization to determine the optimal size with the life from the previous run, and iterating until convergence.
The deferral service is also not compatible with size optimization. The deferral service calculates the minimum size requirements for the DERs at each year of the analysis window and compares them to the know sizes of DERs present. Increasing DER size could potentially push a deferral failure farther into the future, increasing the benefit of the DER, but the number of years in a deferral is not an optimization variable, so cannot be solved for directly. An iterative approach is recommended here.
Size optimization in DER-VET should be used as a first-pass approximation only. Because the linear cost functions in DER-VET cannot fully capture the costs of a a real system, a single optimization year does not incorporate inter-annual variability, and additional size may be desirable in reliability-focused applications for some safety margin. After performing a size optimization, best practices include to run several fixed-size cases to understand what inputs the results are most sensitive to, to include costs that follow a more realistic non-linear pattern, and to understand if hedging or other more complicated strategies make sense.