*3.2. OCECF Model*

The OCECF model aims to reduce the network losses and carbon flow losses as much as possible according to satisfying the constraints of the power grid and maintaining the stability of the power system voltage. Therefore, the OCECF model is able to describe as follows [23,36]:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

$$\begin{aligned} \min & \mu\_1 f\_1(\mathbf{x}) + \mu\_2 f\_2(\mathbf{x}) + (1 - \mu\_1 - \mu\_2) V\_{\mathbf{d}} \\ \text{s.t.} & P\_{\text{Gi}} - P\_{\text{Di}} - V\_i \sum\_{j \in \mathcal{N}\_i} V\_j \Big( g\_{ij} \cos \theta\_{ij} + b\_{ij} \sin \theta\_{ij} \Big) = 0 \\ & Q\_{\text{Gi}} - Q\_{\text{Di}} - V\_i \sum\_{j \in \mathcal{N}\_i} V\_j \Big( g\_{ij} \sin \theta\_{ij} + b\_{ij} \cos \theta\_{ij} \Big) = 0 \\ & P\_{\text{Gi}}^{\text{min}} \le P\_{\text{Gi}} \le P\_{\text{Gi}}^{\text{max}} \ i \in \mathcal{N}\_{\text{G}} \\ & Q\_{\text{Gi}}^{\text{min}} \le Q\_{\text{Gi}} \le Q\_{\text{Gi}}^{\text{max}} \ i \in N\_{\text{G}} \\ & V\_i^{\text{min}} \le V\_i \le V\_i^{\text{max}} \ i \in N\_{\text{B}} \\ & Q\_{\text{Gi}}^{\text{min}} \le Q\_{\text{Gi}} \le Q\_{\text{Gi}}^{\text{max}} \ i \in N\_{\text{C}} \\ & k\_{\text{ti}}^{\text{min}} \le k\_{\text{ti}} \le k\_{\text{fa}}^{\text{max}} \ i \in N\_{\text{k}} \\ & |S\_i| \le S\_{\text{max}}^{\text{max}} \ i \in N\_{\text{L}} \end{aligned} (5)$$

where nonlinear functions *f* 1(*x*) and *f* 2(*x*) are the components of carbon flow loss and active power loss; *V*d is the voltage stability component; μ1 and μ2 are the weight coefficients, μ1 ∈ [0, 1], μ2 ∈ [0, 1], μ1 + μ2 ≤ 1; *x* = [*<sup>V</sup>*, θ, *k*t, *Q*C]<sup>T</sup> corresponds to the voltage value of each node of the power grid *V*, the phase angle of each node θ and the on-load tap changer (OTLC) ratio *k*t, reactive power compensation *Q*C. The remaining variables can be referenced in the nomenclature and *V*d can be described as [23]

$$V\_{\rm d} = \sum\_{j=1}^{n} \left| \frac{2V\_j - V\_{j\rm max} - V\_{j\rm min}}{V\_{j\rm max} - V\_{j\rm min}} \right| \tag{6}$$

where *n* represents the number of load nodes; *Vj* is the node voltage of load node *j*; and *Vj*max and *Vj*min denote the maximal and minimal voltage ranges of load node *j*, respectively.
