**4. Discussion**

#### *4.1. Additional Possible Phenomena in the Setup*

Figure 6 depicts the TSW of reserve sub-markets in a similar experiment as detailed before and depicted in Figures 2 and 4: The parameter *u* is decreased.

**Figure 6.** Total social welfare (TSW) of the reserve sub-markets as the parameter *u* decreases in example 2. *TSWR*<sup>+</sup> and *TSWR*− denote the total social welfare of the positive and the negative reserve market respectively.

In this figure however, the increase of *TSWR*<sup>+</sup> is not monotone. The explanation for this is that if for example, an energy bid has a positive uncertainty of 27% and a negative uncertainty of 26%, the following happens. As *u* is decreased to 27%, the bid becomes positively uncertain. If the cost of the implied SRDB is acceptable and the MSC holds, the bid will be still accepted in the energy market and its SRDB will be also accepted, increasing the TSW of the reserve market (compared to the *u* = 28% scenario). However, in the case of *u* = 26%, the bid becomes bi-uncertain and instead of one, two SRDBs must be paid for. In this case, it is plausible that the MSC does not hold anymore, resulting in the rejection of all three bids of the order (according to bid acceptance rules). This may be interpreted as a loss of a bid in the positive reserve sub-market, resulting in the decrease of the TSW.

#### *4.2. Total Amount of Allocated Reserve*

At a given value of *u*, the amount of reserve resulting from the SRDBs is explicitly defined by Equation (4). On the other hand, as discussed before, several principles may be applied to determine the total amount of allocated reserves in the power system [6–9]. While the advantage of the proposed methodology is that it allocates reserves and the corresponding cost only in the case of accepted uncertain energy bids (due to MSCs) and it is flexible as the current power mix changes, there are no explicit guarantees for the amount of total allocated reserves for the whole system. The total reserve allocated by SRDBs may not meet the thumb rule that stating that the amount of allocated reserve must be at least equal the capacity of the largest unit in service (consider for example, a not-uncertain nuclear power plant and several smaller uncertain renewable sources). This approach is however is generally accepted in the context of event-driven reserves, while, as mentioned earlier, we are focussing on non event-driven resources. The main aim of the proposed approach is not to handle such large and conservative reserve needs (which most of the time may be handled by long term contracts by the system operator) but to provide a framework in which the (hour-level) actual reserve requirements and costs implied by the uncertainties of imminent power mix are automatically allocated.

On the other hand, Equation (4) may be modified by a normalization factor *c* as described in Equation (20) in order to tune the total amount of allocated reserves.

$$q\_{j}^{RD+} = -q\_{j}^{ESllb} \mathfrak{c} u\_{k}^{-} \qquad q\_{j}^{RD-} = \mathfrak{s}\_{j}^{ESllb} = -q\_{j}^{ESllb} \mathfrak{c} u\_{k}^{+} $$
 
$$q\_{j}^{RD+} = -q\_{j}^{ESl-} \mathfrak{c} u\_{k}^{-} \qquad q\_{j}^{RD-} \stackrel{ESI \rightarrow}{\ } = -q\_{j}^{ESlI \rightarrow} \mathfrak{c} u\_{k}^{+}, \tag{20}$$

If the computational capacity is sufficient, this tuning can be carried out via an outer control loop, calculating the dispatch and the total SRDB amount as a function of *c*. There are no guarantees that any amount of total SRDB-reserves may be allocated with this method (after increasing *c* above a certain level, none of the MSCs will hold, thus all uncertain bids will be rejected and no SRDB-reserve will be allocated) but this additional parameter may be a useful tool to achieve certain regulation aims.
