3.2.3. FCR Service

The FCR term is assessed by calculating the upward and downward power that the BESS can provide in each interval, depending on the scheduled daily profile of the energy stored. As assumed in the paper, in planning studies, the minimum time-step used to represent the customer variability within the typical day is one hour. Consequently, the offered FCR service is assumed constant within the same hour.

The power exchanged with the grid in the generic *f* th hour is assessed as the difference between the starting and the ending state of charge (SoC):

$$P\_f^{RES} = \frac{So\mathbb{C}\_f - SoC\_{f+1}}{\Delta t}.\tag{8}$$

If the BESS is charging (*SoCf*<sup>+</sup><sup>1</sup> > *SoCf* being *SoCf* the SoC at the beginning of the *f* th hour), it absorbs power from the distribution system, operating as a load (*PBESS <sup>f</sup>* < 0). If, on the contrary, the BESS is discharging (*SoCf*<sup>+</sup><sup>1</sup> < *SoCf*), it delivers power to the distribution system, operating as a generator (*PBESS <sup>f</sup>* > 0). Thus, since ancillary frequency services have to be provided with a symmetric band, the width of the available semi-band in the *f* th hour (Δ*Pf*) is given by the minimum power that the storage device can provide, taking into account the BESS nominal power (*Pn*), the power injected to or absorbed from the grid in the *f* th hour (*PBESS <sup>f</sup>* ), and the limits imposed by the DSO reserved capacity (Cmin\_DSO, Cmax\_DSO):

$$\Delta P\_f = \min\left\{ \left( \mathbf{P}\_n - \left| P\_f^{BESS} \right| \right), \frac{1}{\Delta T} \cdot \left( \mathbf{C}\_{\max, \text{DSO}} - \max\left\{ \text{SoC}\_f, \text{SoC}\_{f+1} \right\} \right), \frac{1}{\Delta T} \cdot \left( \min\left\{ \text{SoC}\_f, \text{SoC}\_{f+1} \right\} - \mathbf{C}\_{\min, \text{DSO}} \right) \right\}. \tag{9}$$

In addition to the operating limits of the storage device, the availability of FCR can be limited hour by hour by the minimum amount of reserve (Δ*PFCR min* ) that has to be provided in order to be eligible as the FCR service provider (it depends on the national regulation). It is predictable that this limit, lowered in the recent years due to the broad diffusion of renewables, will be further reduced (or even deleted) to allow the participation of many small DERs directly connected to the distribution system. If the semi-band calculated is lower than this limit, it is assumed that BESS cannot provide the FCR service for that specific hour.

It is worth noticing that the calculated semi-band Δ*Pf* is the whole amount of available power that the storage can offer in the *f* th hour for the considered TSO services. Thus, it can be split in accordance with the ratio *rFCR* between the two frequency regulation services provided, or it can be dedicated exclusively to FCR (*rFCR* = 1) or aFRR (*rFCR* = 0).

The income from the FCR service is usually remunerated in capacity and energy. The first term is simply obtained by multiplying the cumulative amount of FCR available in the whole year by the average

annual price of remuneration (*pCFCR*). The second term is estimated by assuming reasonable hypotheses on the average request of primary frequency regulation, based on historical measurements. Thus, the actual amount of energy used for this ancillary service in the *f* th hour is calculated by multiplying the available energy (*rFCR*·Δ*Pf*·Δ*t*) by a heuristic factor *fd FCR* (assumed constant). Then, the remuneration of this energy is obtained by multiplying it for the corresponding hourly energy market price (*pEf*). By so doing, the total income from the FCR service in a generic year within the planning horizon is:

$$B\_{yearly}^{FCR} = 365 \cdot \sum\_{f=1}^{N\_f} \left[ \left( r^{FCR} \cdot \Delta P\_f \cdot \Delta t \right) \cdot \left( p\_C^{FCR} + f\_d^{FCR} \cdot p\_{Ef} \right) \right]. \tag{10}$$

The NPV of this frequency service in the whole planning period is finally derived as:

$$B\_{FCR} = \sum\_{i=1}^{N\_{yurs}} a^i \cdot B\_{yearly}^{FCR} = B\_{yerly}^{FCR} \cdot a \cdot \frac{1 + a^{N\_{yurs}}}{1 + a} \,. \tag{11}$$

## 3.2.4. aFRR Service

Operating reserves of this category are typically activated centrally with an activation time between 30 s up to 15 min. Differently from FCR, the aFRR may last more than one hour, but its maximum requested duration (Δ*t aFRR*) differs country by country (e.g., 2 h are requested in Italy).

The available semi-band offered for the aFRR service is calculated as (1—*rFCR*) Δ*Pf*. However, an additional constraint has to be considered, because the power offered must be provided constantly and continuously at least for Δ*t aFRR*. Consequently, the available semi-band for the aFRR service is obtained as the minimum of the following values:

$$\Delta P\_f^{\text{sFRR}} = \min \left\{ \left( 1 - r^{FCR} \right) \cdot \Delta P\_{f^\*} \, , \, \frac{\text{C}\_{\text{max\\_DSO}} - \max \left\{ \text{SoC}\_f, \text{SoC}\_{f+1} \right\}}{\Delta t^{FRR}} , \, \frac{\min \left\{ \text{SoC}\_f, \text{SoC}\_{f+1} \right\} - \text{C}\_{\text{min\\_DSO}}}{\Delta t^{FRR}} \right\} . \tag{12}$$

The equations used to monetize this service are formally the same used for the FCR service, but with a different amount of maximum semi-band available, different capacity price, *pCaFRR*, and different heuristic factor, *fd aFRR*, used to estimate the average amount of energy provided.
