**Abbreviations**

The following abbreviations are used in this manuscript:


 The variable nomenclature is

*H* Bid/schedule realization history


The superscripts used in the variables stand for


#### **Appendix A. Structure of the Variable Vector and Formulation of Logical Constraints**

Let us assume that R+ denotes the nonnegative reals, <sup>R</sup>[0,1] denotes the set of real numbers in [0, 1], while B denotes the set of binary numbers (B = {0, 1}).

We denote the numbers of the various bid types submitted to the market as summarized in Table A1.

> **Table A1.** Number of various bid types submitted to the market.


The variable vector of the proposed formulation may be partitioned as

$$\begin{pmatrix} M\mathbf{C}P\\\ Y^{ES}\\\ Y^{ED}\\\ Y^{RS+}\\\ Y^{RD+}\\\ Y^{RS-}\\\ Y^{RD-}\\\ I^{ES}\\\ I^{ED}\\\ I^{RD+}\\\ I^{RD-}\\\ Z\end{pmatrix},\tag{A1}$$

*IESUb* ∈ R*nESUb* + , *IESU*<sup>+</sup> ∈ R*nESU*<sup>+</sup> + , *IESU*− ∈ R*nESU*− + , *IEDUb* ∈ R*nEDUb* + , *IEDU*<sup>+</sup> ∈ R*nEDU*<sup>+</sup> + , *IEDU*− ∈ R*nEDU*− + *IRD*<sup>+</sup> *ESUb* ∈ R*nESUb* + , *IRD*<sup>+</sup> *EDUb* ∈ R*nEDUb* + , *IRD*<sup>+</sup> *ESU*− ∈ R*nESU*− + , *IRD*<sup>+</sup> *EDU*− ∈ R*nEDU*− + , *IRD*− *ESUb* ∈ R*nESUb* + , *IRD*− *EDUb* ∈ R*nEDUb* + , *IRD*− *ESU*+ ∈ R*nESU*<sup>+</sup> + , *IRD*− *EDU*+ ∈ R*nEDU*<sup>+</sup> + .

$$\begin{aligned} I^{ES} &= \begin{pmatrix} I^{ESllb} \\ I^{ESll+} \\ I^{ESll-} \\ I^{ESll-} \end{pmatrix} \quad I^{ED} = \begin{pmatrix} I^{EDllb} \\ I^{EDll+} \\ I^{EDll-} \\ I^{RD+} \\ I^{RD+} \\ I^{RD+} \end{pmatrix} \quad I^{RD-} = \begin{pmatrix} I^{RD-} & Esilb \\ I^{RD-} & I^{RD-} \\ I^{RD-} & I^{RD-} \\ I^{RD-} & I^{RD-} \end{pmatrix} \end{aligned} \tag{A4}$$

,

The vectors *<sup>I</sup>ES*, *<sup>I</sup>ED*, *IRD*<sup>+</sup> and *IRD*− holding the incomes are composed as

*YES* ∈ <sup>R</sup>*nES*+*nESUb*+*nESU*++*nESU*− [0,1] , *YED* ∈ <sup>R</sup>*nED*+*nEDUb*+*nEDU*++*nEDU*− [0,1] , *YRS*<sup>+</sup> ∈ R*nRS*<sup>+</sup> [0,1] , *YRD*<sup>+</sup> ∈ <sup>R</sup>*nRD*++*nESUb*+*nEDUb*+*nESU*−+*nEDU*<sup>−</sup> + , *YRS*− ∈ R*nRS*− [0,1] , *YRD*− ∈ <sup>R</sup>*nRD*−+*nESUb*+*nEDUb*+*nESU*++*nEDU*<sup>+</sup> + .

$$\begin{aligned} \mathbf{y}^{ES} &= \begin{pmatrix} y^{ES} \\ y^{ES \text{LU}} \\ y^{ES \text{LU}} \\ y^{ES \text{LU}} \end{pmatrix} \quad \mathbf{y}^{ED} = \begin{pmatrix} y^{ED} \\ y^{ED} \\ y^{ED} \\ y^{ED} \text{LU} \end{pmatrix} \\ \mathbf{y}^{RS+} &= \begin{pmatrix} \mathbf{y}^{RS+} \\ \mathbf{y}^{RD+} \end{pmatrix} \quad \mathbf{y}^{RD+} = \begin{pmatrix} y^{RD+} \\ y^{RD+} + \text{EUR} \\ y^{RD+} + \text{EUR} \\ y^{RD+} + \text{EUR} \\ y^{RD+} + \text{EUR} \end{pmatrix} \\ \mathbf{y}^{RS-} &= \begin{pmatrix} \mathbf{y}^{RS-} \\ \mathbf{y}^{RD-} \end{pmatrix} \quad \mathbf{y}^{RD-} = \begin{pmatrix} \mathbf{y}^{RD-} \\ \mathbf{y}^{RD-} - \text{EUR} \\ \mathbf{y}^{RD-} - \text{EUR} \\ \mathbf{y}^{RD-} - \text{EUR} \\ \mathbf{y}^{RD-} - \text{EOL} \end{pmatrix} \tag{A3} \end{aligned} \tag{A3}$$

*<sup>Y</sup>ES*, *<sup>Y</sup>ED*, *<sup>Y</sup>RS*+, *<sup>Y</sup>RD*+, *YRS*− and *YRD*− hold the acceptance indicators, 

*MCP* = ⎛⎜⎝ *MCP<sup>E</sup> MCPR*<sup>+</sup> *MCPR*− ⎞⎟⎠ (A2)

where *MCP* holds the market clearing prices, *Energies* **2019**, *12*, 2957

The sub-vector *z* is a binary vector holding the auxiliary variables for logical implications, and as these binary variables are bound to acceptance variables, it is partitioned similarly to *Y*.

$$Z = \begin{pmatrix} Z^{ES} \\ Z^{ED} \\ Z^{RS+} \\ Z^{RD+} \\ Z^{RS-} \\ Z^{RD-} \end{pmatrix} \\ \text{ (A5)}$$

$$\begin{aligned} Z^{ES} &= \begin{pmatrix} z^{ES} \\ z^{ESllb} \\ z^{ESlI+} \\ z^{ESlI-} \end{pmatrix} \quad z^{ED} = \begin{pmatrix} z^{ED} \\ z^{EDlI+} \\ z^{EDlI+} \\ z^{EDlI-} \\ z^{EDlI-} \end{pmatrix} \\\ Z^{RS+} &= \left( \begin{array}{c} z^{RD+} \\ z^{RD+} \\ z^{RD+} \\ z^{RD+} \\ z^{RD+} \end{array} \right) \begin{pmatrix} z^{RD+} \\ z^{RD+} \\ z^{RD+} \\ z^{RD+} \\ z^{RD+} \end{pmatrix} \\\ Z^{RS-} &= \left( \begin{array}{c} z^{RD-} \\ z^{RD-} \\ z^{RD-} \\ z^{RD-} \end{array} \right) \end{aligned} \tag{A6}$$

To give an example how the logical implications corresponding to income formulations and bid acceptance constraints are implemented in the computational framework, let us consider the variable block *zESUb* ∈ B3*nESUb* corresponds to the implications (8) and (9) describing the income of of bi-uncertain energy supply bids, and to the bid acceptance constraints of bi-uncertain energy supply bids. The logical implications using the *zESUb* variables are implemented as follows.

The constraints corresponding to income formulation described in Equations (8) and (9) may be written in the shorter form

$$\begin{array}{rcl} y\_{j}^{ESIIb} > 0 & \rightarrow & f\_{1}^{I} \leq 0 & \& \ f\_{1}^{I} \geq 0\\ y\_{j}^{ESIIb} < 1 & \rightarrow & f\_{2}^{I} \leq 0 & \& \ f\_{2}^{I} \geq 0 \end{array} \tag{A7}$$

where

$$f\_1^I = y\_j^{ESILb} q\_j^{ESILb} p\_j^{ESILb} + q\_j^{ESILb} MCP^E - q\_j^{ESILb} p\_j^{ESILb} - I\_j^{ESILb} \tag{A8}$$

and

$$f\_2^I = y\_j^{ESIIb} q\_j^{ESIIb} p\_j^{ESIIb} - I\_j^{ESIIb} \tag{A9}$$

Bid acceptance constraints of ESUb bids may be written as

$$\begin{aligned} y\_j^{\text{ESl}lb} > 0 &\rightarrow f^{\text{MSC}} \le b^{\text{MSC}}\\ y\_j^{\text{ESl}lb} < 1 &\rightarrow f\_1^{\text{BA}} \le b^{\text{BA}} \text{ or } f\_2^{\text{BA}} \le 0 \end{aligned} \tag{A10}$$

where, according to Equation (10) and the bid acceptance rules of ESUb bids

$$\begin{aligned} f^{\text{MSC}} &= p\_j^{\text{ESILb}} q\_j^{\text{ESILb}} y\_j^{\text{ESILb}} - I\_j^{\text{ESILb}} - I\_j^{\text{RD} + \text{ESILb}} - I\_j^{\text{RD} - \text{ESILb}} \\\ b^{\text{MSC}} &= -S\_j^{\text{ESILb}} \\\ f\_1^{BA} &= \text{MCP}^E \\\ b^{BA} &= p\_j^{\text{ESILb}} \\\ f\_2^{BA} &= \mathcal{y}\_j^{\text{ESILb}} + \mathcal{y}\_j^{\text{RD} + \text{ESILb}} + \mathcal{y}\_j^{\text{RD} - \text{ESILb}} \end{aligned} \tag{A1}$$

Let us note that (as every *y* ∈ [0, 1]) *f BA*2 ≤ 0 ⇔ *yESUb j* = 0 *yRD*<sup>+</sup> *ESUb l* = 0 *yRD*<sup>−</sup> *ESUb j* = 0. All together, the implications may be summarized and reformulated as

$$\|y\_j^{\text{ESLb}} \le 0 \text{ and/or } \left(f\_1^l \le 0 \text{ \&\ } -f\_1^l \le 0 \text{ \&\ } f^{\text{MSC}} \le b^{\text{MSC}}\right) \tag{A12}$$

$$-y\_{\uparrow}^{ESllb} \le -1 \text{ and/or } \left(f\_2^I \le 0 \text{ \&\ } -f\_2^I \le 0 \text{ \&\ } \left(f\_1^{BA} \le b^{BA} \text{ or } f\_2^{BA} \le 0\right)\right) \tag{A13}$$

Formula (A12) may be implemented in the optimization framework as

$$\begin{aligned} y\_j^{ESlb} - z\_{j1}^{ESlb} &\le 0 \\ f\_1^I - B\_1^I (1 - z\_{j1}^{ESlb}) &\le 0 \\ -f\_1^I - B\_1^I (1 - z\_{j1}^{ESlb}) &\le 0 \\ f^{MSC} - B^{MSC} (1 - z\_{j1}^{ESlb}) &\le b^{MSC} \end{aligned} \tag{A14}$$

where the *B*-s are the so called 'big M'-s: *BI*1 = *max*(*f I*1 ), *BMSC* = *max*(*f MSC*).

While, Formula (A13) is implemented as

$$\begin{aligned} &y\_j^{ESlbb} - z\_{j2}^{ESlbb} \le -1\\ &f\_2^I - B\_2^I (1 - z\_{j2}^{ESlbb}) \le 0\\ &(1 - f\_2^I - B\_2^I (1 - z\_{j2}^{ESlb})) \le 0\\ &f\_1^{BA} - z\_{j3}^{ESlbb} B\_1^{BA} \le b^{BA}\\ &f\_2^{BA} - (1 - z\_{j2}^{ESlbb}) B\_2^{BA} - (1 - z\_{j3}^{ESlbb}) B\_2^{BA} \le 0 \end{aligned} \tag{A15}$$

We can see that since we have an implication of the type *A* → *B* or *C* a bi-uncertain energy supply bid requires 3 auxiliary binary variables. Bids, to which only simple acceptance constraints are connected like *ES*, *ED*, *RS*+ and so forth, require only 2 binary variables to formulate the two simple implications. Based on these considerations, the size of the *z* blocks may be easily determined. The other implications may be formulated analogously, using the appropriate variables.

In the case of SRDB-s, for example, *yRD*<sup>+</sup> *ESUb j* , (while the income-related constraints are totally analogous) we formulate the bid acceptance-related constraints as

$$y\_j^{RD+} \stackrel{ESI lb}{>} 0 \quad \rightarrow \quad f^{MSC} \le b^{MSC} \text{ & \& \ f\_1^{BA} \le b\_1^{BA}$$

$$y\_j^{RD+} \stackrel{ESI lb}{<} 1 \quad \rightarrow \quad f\_2^{BA} \le b\_2^{BA} \text{ or } f\_3^{BA} \le 0 \tag{A16}$$

where *f MSC* and *bMSC* is the same as before (MSC condition for the order), and here *f BA*1 ≤ *<sup>b</sup>BA*1 corresponds to the appropriateness of *MCPR*+: *f BA*1 = *MCPR*+, *<sup>b</sup>BA*1 = *pRD*<sup>+</sup> *ESUb j*.

On the other hand, *f BA*2 = <sup>−</sup>*MCPR*+, *<sup>b</sup>BA*2 = −*pRD*<sup>+</sup> *ESUb j* , and *f BA*3 hold the acceptance indicators corresponding to the bids of the order.

#### **Appendix B. Reference Bid Set**

In this appendix, the reference bid set of the example detailed in Section 3 is described.

**Table A2.** Reference ES bid set: The columns correspond to the index of the bid (ID), quantity (*q*),bid price (*p*), positive and negative uncertainty (*u*+, *u*<sup>−</sup>), and *S* respectively.



**Table A3.** Reference ED bid set: The columns correspond to the index of the bid (ID), quantity (*q*), bid price (*p*), positive and negative uncertainty (*u*+, *u*<sup>−</sup>), and *S* respectively.


**Table A4.** Reference (non-SRDB) RS+ bid set: The columns correspond to the index of the bid (ID), quantity (*q*) and bid price (*p*) respectively. The SRDB bids are generated according to *u*, and the implied actual set of uncertain energy bids.

**Table A5.** Reference (non-SRDB) RS- bid set: The columns correspond to the index of the bid (ID), quantity (*q*) and bid price (*p*) respectively. The SRDB bids are generated according to *u*, and the implied actual set of uncertain energy bids.


**Table A6.** Reference (non-SRDB) RD+ bid set: The columns correspond to the index of the bid (ID), quantity (*q*) and bid price (*p*) respectively. The SRDB bids are generated according to *u*, and the implied actual set of uncertain energy bids.


**Table A7.** Reference (non-SRDB) RD- bid set: The columns correspond to the index of the bid (ID), quantity (*q*) and bid price (*p*) respectively. The SRDB bids are generated according to *u*, and the implied actual set of uncertain energy bids.

