*4.3. Market Implementation*

As proposed in Section 3, the values *u*− and *u*+ may be decreased step-by-step from high values. During the simulation we analyzed how this decrease affects the various sub-markets assuming the same original bid set.

Regarding the possible market implementation of the method, this decrease of parameters may be carried out on for example, a monthly scale, thus progressively coupling the initially independent energy and reserve sub-markets via the SRDBs and meanwhile giving time for market participants to adapt to the changing regulation.

A further important question arises in connection with widely used special orders in electricity markets. Block orders [28] submitted to the energy sub-market and exhibiting the fill-or-kill property have not been discussed in the current paper for the aim of simplicity. There is no theoretical obstacle however to apply the fundamental principle of the approach for such orders. Considering a multi-period uncertain block order, the respective SRDBs may be defined for each affected period—in this case, the computations for each period must be carried out simultaneously and the income of the block order has to be formulated as the sum of incomes for the different periods. The MSC in this case can be included in the acceptance rules of the block order.

Minimum income condition orders [29], which are basically hourly step orders bound together by the minimum income condition, also define interdependencies between different periods. The novel income formulation of these orders proposed in Reference [21] and also used in this paper can be easily generalized to account also for the expenses of SRDBs, thus the acceptance rules of such bids may be also generalized for the proposed setup.

The proposed formulation is also compatible with general complex orders [22], orders including load gradient conditions (LGC orders) [30] and markets with so called 'PUN' orders, where the buyers pay uniform price despite multiple price zones [25].

One must keep in mind however that generalizing the proposed framework for interdependent multi-period clearing mechanisms may be computationally demanding because of the high number of integer variables originating from logical expressions. On the other hand, the MILP framework may be efficiently implemented using novel computational paradigms (e.g., Benders decomposition—see Reference [26]).

#### **5. Conclusions and Future Work**

In this article we proposed a method to implement the *uncertain bidder pays principle* in integrated portfolio-bidding electricity markets, in which the bidders of uncertain energy bids compulsorily submit predefined reserve demand bids (supplementary reserve demand bids or SRDBs) to the reserve markets to account for production/demand uncertainty. The profitability of orders corresponding to uncertain energy bids upon acceptance is ensured in the proposed framework by the minimum surplus conditions (MSC), which bounds the total surplus of the order composed by the uncertain energy bid and the connected SRDBs. As the set of uncertain bids is defined by an uncertainty threshold parameter *u*, the proposed framework may be progressively applied, inducing increasing coupling between the initially independent energy and reserve sub-markets. The computational formulation results in a mixed integer linear programming problem (MILP).

Regarding the generalization perspectives of the framework, when we are discussing the implementation of the uncertain bidder pays principle in the context of portfolio-bidding markets, we think that the general question may be formulated as '*how the uncertainty of a unit/consumer becomes a reserve demand bid, which covers the cost of reserve allocation*'. In this paper we proposed an approach which works on the level of single bids. This approach has the benefit that the resulting SRDB (and thus the cost of the reserve) is straightforwardly assigned to a market participant—to

the submitter of the uncertain energy bid. A different approach, where the uncertainty of multiple energy bids bids is represented by one reserve demand bid, would allow the more complex handling of uncertainties (e.g., with the possible application of risk measures [31]) but the problem of cost allocation would become more challenging as well. Nevertheless, the general problem of bid/order formulation from uncertain energy bids means a potential research direction for the future, which we plan to pursue based on the current results. A straightforward generalization of the concept is to formulate the approach for multi-period models with the inclusion of block orders and/or minimum income condition (MIC) orders.

**Author Contributions:** D.C., Á.S. and P.M.S. formulated the research problem and the proposed solution approach, D.C. and Á.S. developed the computational approach, D.C. implemented the codes, D.C. wrote the first draft of the paper, D.C., Á.S. and P.M.S. reviewed and wrote the paper.

**Funding:** This work has been supported by the Fund PD 123900 of the Hungarian National Research, Development and Innovation Office, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the Fund KAP19-1.1-ITK of the Pázmány Péter Catholic University.

**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
