*2.2. Model Linearization of OLTC Transformer*

Nonlinear OLTC transformer constraints formulated in (8) can be linearized by introducing additional binary variables [25]. To linearize the OLTC model, we introduced the binary encoding of a tap position in the following manner:

$$m\_{\mathbb{S}} = \sum\_{a=0}^{\text{lin}} \mathfrak{2}^{a} \cdot \tau\_{a,\mathbb{s}} \tag{15}$$

where we had a new binary variable τ*a*,*s*. Given this, the OLTC transformer model in (8) can be reformulated into Equation (14):

$$\begin{aligned} \text{tr}\_s &= \text{tap}\_{\text{min}} + \Delta \text{tap} \sum\_{a=0}^{\text{bin}} \mathfrak{L}^a \cdot \mathfrak{r}\_{a,s}, \quad \sum\_{a=0}^{\text{bin}} \mathfrak{L}^a \cdot \mathfrak{r}\_{a,s} \le \text{tap}\_{\text{max}}, \\ \pi\_{a,s} &\in \{0, 1\}, \quad \forall \ a \in \{0, \dots, \text{bin}\} \end{aligned} \tag{16}$$

Considering the previous formulation and by introducing a new variable ρ*a*,*<sup>s</sup>* = τ*a*,*s*·*tra*,*s*, ∀ *a* ∈ {0, ... , *bin*} and big number *S*, voltage formulation within OLTC transformer substation is defined as:

$$u\_{1,s} = u\_{0,s} \cdot \text{tap}\_{\text{min}} \cdot \text{trs}\_s + u\_{0,s} \cdot \Delta \text{tap} \sum\_{a=0}^{\text{bin}} 2^a \cdot \rho\_{a,s}, \qquad u\_{0,s} = \text{const.} $$

$$0 \le \text{trs}\_s - \rho\_{a,s} \le (1 - \tau\_{a,s})S \tag{17}$$

$$0 \le \rho\_{a,s} \le \tau\_{a,s} S, \quad \forall \ a \in \{0, \dots, \text{bin}\}, \text{ S} - \text{lig number}$$

By these means, we obtained a linear mathematical model of the OLTC transformer. Where:

