*3.1. Call Auction Stage*

Objective function—the social welfare of the call auction stage can be expressed as follows:

$$\pi\_1 = \max \sum\_{i \in I} \sum\_{j \in \mathcal{I}} \left( (P\_{bb,i} - P\_{bd,ij}) \times Q\_{bd,ij} + (P\_{sd,ij} - P\_{sb,j}) \times Q\_{sd,ij} \right) \tag{1}$$

where *i* and *I* represent the index and set of market buyers, respectively; *j* and *J* represent the index and set of market sellers; *Pbb*,*<sup>i</sup>* (yuan/kWh) represents the bid price of market buyer *i*; *Pbd*,*ij* (yuan/kWh) represents the deal price of market buyer *i* with market seller *j*; *Qbd*,*ij* (kWh) represents the energy cleared of market buyer *i* with market seller *j*; *Psb*,*<sup>j</sup>* (yuan/kWh) represents the bid price of market seller *j*; *Psd*,*ij* (yuan/kWh) represents the deal price of market seller *j* with market buyer *i*; *Qsd*,*ij* (kWh) represents the energy cleared of market seller *j* with market buyer *i*; and π1 (yuan) represents the social welfare of the call auction stage.

All the constraints are given by the KMPEX before the bidding. The constraints faced by the call auction stage include bidding constraints and clearing constraints.

Market bidding constraints are defined as follows:

$$\begin{cases} \ 0 \le Q\_{lb;i} \le \overline{Q}\_{lb;i} \\\ 0 \le Q\_{sb;j} \le \overline{Q}\_{sb;j} \end{cases} \tag{2}$$

$$\begin{cases} \ P\_{\alpha} \le P\_{bb,i} \le P\_{\beta} \\\ P\_{\alpha} \le P\_{sb,j} \le P\_{\beta} \end{cases} \tag{3}$$

where *Qbb*,*<sup>i</sup>* (kWh) represents the quantity caps of buyer bid; *Qsb*,*<sup>j</sup>* (kWh) represents the quantity caps of seller bid; and *P*α (yuan/kWh) and *P*β (yuan/kWh) represent the lower and upper bounds of the bid price of buyer *i* and seller *j*, respectively.

Market clearing constraints are defined as follows:

$$\begin{cases} \sum\_{j \in I} Q\_{bd,ij} \le Q\_{bb,i} \\ \sum\_{i \in I} Q\_{sd,ij} \le Q\_{sb,j} \end{cases} \tag{4}$$

$$\sum\_{i \in I} \sum\_{j \in \mathcal{J}} Q\_{hd,ij} = \sum\_{i \in I} \sum\_{j \in \mathcal{J}} Q\_{sd,ij} \tag{5}$$

$$\begin{cases} Q\_{bd,ij} \ge 0\\ Q\_{sd,ij} \ge 0 \end{cases} \tag{6}$$

$$\begin{cases} \begin{aligned} P\_{\Lambda,ij} &= P\_{lb,j} - P\_{sb,j} \\ P\_{bd,ij} &= P\_{bb,i} - P\_{\Lambda,ij}/2 \\ P\_{sd,ij} &= P\_{sb,j} + P\_{\Lambda,ij}/2 \end{aligned} \end{cases} \tag{7}$$

Equation (4) indicates that the accumulated transaction quantity does not surpass the bid quantity. Equation (5) is the power balance constraint, which represents that the total amount of electricity traded between buyers and sellers is balanced. In Equation (6), the lower bound takes nonnegative values for *Qbd*,*ij* and *Qsd*,*ij*, which means that transfers of electricity between buyers or between sellers are not allowed in this transaction.

All sellers (*i* ∈ *I*) and buyers (*j* ∈ *J*) are paired into *I* × *J* pairs, and the DOP of *I* × *J* pairs are calculated by Equation (7), where *<sup>P</sup>*Δ,*ij* represents the DOP between marker buyer *i* and marker seller *j*, which is not negative to meet the transaction conditions. *Pbd*,*ij* and *Psd*,*ij* represents the buying and selling prices for the financial settlement of each pair, respectively, which are determined according to a modified pay-as-bid (PAB) principle that adjusts the effective prices through their DOP [4].
