(3) Voltage constraint

$$\mathcal{U}\_{i,\min} \le \mathcal{U}\_i \le \mathcal{U}\_{i,\max} \tag{20}$$

$$\mathcal{U}\_{\text{dc},i,\text{min}} \le \mathcal{U}\_i \le \mathcal{U}\_{\text{dc},i,\text{max}} \tag{21}$$

where *Ui*,max and *Ui*,min are the upper and lower limits of the AC node voltage amplitude; *U*dc,*i*,max and *U*dc,*i*,min are the upper and lower limits of the DC node voltage amplitude.

(4) Branch flow constraint

$$S\_j \le S\_{j, \text{max}} \tag{22}$$

$$\mathcal{S}\_{\text{dc},j} \le \mathcal{S}\_{\text{dc},j,\text{max}} \tag{23}$$

$$\mathcal{S}\_{\text{VSC},j} \le \mathcal{S}\_{\text{VSC},j,\text{max}} \tag{24}$$

where *S<sup>j</sup>* is the power on branch *j*; *Sj*,max is the maximum allowable capacity on AC branch *j*; *S*dc,*j*,max is the maximum allowable capacity on DC branch *j*; *S*VSC,*j*,max is the maximum allowable capacity on VSC.

(5) Pollution compensation cost

$$\sum\_{i \in N\_{\rm DG}} E\_i(t) \ge L\_{tol}(t) \text{ t = 1, 2, \dots, 8760 \tag{25}$$

where ∑ *i*∈*NDG Ei*(*t*) is the total capacities of the DGs at the time *t*, and *Ltol*(*t*) is the sum of the loads of all nodes at time *t*.
