Time-Dependent Capital Cost

Time-dependent capital cost describes the decrease in the value of a component over its useful life as a consequence of obsolescence and/or wearing, and it is in general proportional to its initial capital cost. We consider two types of components, depending on how their loss of value is defined. A first category corresponds to components that have a maximum expected duration; examples are, for instance, PV panels and wind turbines, whose datasheets typically define maximum operational life. For this class, we call this cost *depreciation cost*, defined for component *i* as:

$$C\_{\text{depreciation},i}(t) = C\_{\text{capital},i} \cdot \text{CRF} \cdot \frac{t}{T\_s} \tag{2}$$

where *Ccapital*,*<sup>i</sup>* is defined as in Equation (1), and *Ts* serves as a time normalization factor and it denotes the number of simulation samples (Δ*T*) per year, in order to express the depreciation over each sample time (e.g., if the Δ*T* is 1s, *Ts* = 3600 × 24 × 365). *CRF* is the *capital recovery factor*, a ratio used for estimating the present value during the lifespan of a system if invested at a particular interest rate [37]. *CRF* is expressed by:

$$CRF = \frac{R \cdot (1 + R)^n}{(1 + R)^n - 1} \tag{3}$$

where *R* is the interest rate, and *n* denotes the number of years of operations of the system (i.e., component lifetime).

A second class of components has instead a lifetime not defined a priori, but rather determined by their usage characteristics; this is for instance the case of batteries, whose lifetime is defined upon reaching a given value of usable capacity, which depends on several usage-related factors [38]. To distinguish from the former category, we call the time-dependent capital cost for this class as *wear-out cost*, defined for component *i* as:

$$\mathcal{C}\_{\text{zout},i}(t) = \mathcal{C}\_{\text{capital},i} \cdot \frac{L(t)}{L\_{\text{max}}} \tag{4}$$

where *L*(*t*) is the loss of "functionality" over time and *Lmax* is the maximum value of the loss after which the component is considered not functional and it needs to be replaced. Thus, when *L*(*t*) reaches *Lmax*, the entire capital cost has been consumed.

*L*(*t*) clearly depends on the type of component and requires an ad-hoc model. Using a battery as an example, *L*(*t*) will be the capacity loss due to both calendar and cycle aging, which are affected by several parameters [39], most of which can be attributed to how the power is drawn from the battery, and thus from information derived from the power layer. *Lmax* is the maximum capacity loss before the battery is considered as depleted (usually a loss of 20% from the initial capacity). This information must be provided at runtime by power simulation, as will be explained later in this Section.

In summary, the *time-dependent capital cost* depends on the type of component; the term *Ccapital*,*<sup>i</sup>*(*t*) will correspond to either Equation (2) or Equation (4) depending on the characteristic of component *i*.
