*Formulation*

The consumer interest function and the producer cost function are in accordance with relationships (5) to (8), respectively, in dollars per hour [41].

$$B\_{d\mathbf{j}}\left(P\_{d\mathbf{j}}\right) = -\frac{1}{2}c\_{d\mathbf{j}}P\_{d\mathbf{j}}{}^2 + d\_{d\mathbf{j}}P\_{d\mathbf{j}} + m\_{\mathbf{j}}\tag{5}$$

$$\mathbf{C}\_{\mathfrak{J}^{\bar{i}}}(\boldsymbol{p}\_{\mathfrak{J}^{\bar{i}}}) = \frac{1}{2}\mathbf{a}\_{\mathfrak{J}^{\bar{i}}}\boldsymbol{P}\_{\mathfrak{J}^{\bar{i}}}{}^2 + b\_{\mathfrak{J}^{\bar{i}}}\boldsymbol{P}\_{\mathfrak{J}^{\bar{i}}} + m\_{\bar{i}} \tag{6}$$

$$\mathbb{C}\_k(P\_{\mathrm{DG}\_k}) = \frac{1}{2} \mathbf{a}\_{\mathrm{DG}k} P\_{\mathrm{DG}k}^2 + b\_{\mathrm{DG}k} P\_{\mathrm{DG}k}^2 + c\_{\mathrm{DG}k} \tag{7}$$

$$\mathbb{C}\_{w} = \gamma\_{w} \times d\_{\text{wind}} \times P\_{w} \tag{8}$$

In these relations, *cdj* and *ddj* are slope and width from the origin of the uniform curve of *j* − *th* consumer demand, *agi* and *bgi* are slope and width from the origin of the uniform curve suggested by the generator, *mj* and *mi* are constant coefficients of profit and consumption functions of the *j* − *th* generator and *i* − *th* generator, *Pdj* and *Pgi* are the real power of the *j* − *th* consumption and *i* − *th* generator, *Ck PDGk* provides the cost function of the *k* − *th* DG number, *Nm* is the number of DGs connected to the network and *PDGk* represents the active power generation of the *k* − *th* DG number, *Cw* represents the cost of wind turbine production in each scenario, *dwind* is the recommended price of wind turbine per MW/h of power generation, *Pw* gives the production power of the wind turbine unit in each scenario (MW), *w* is considered as low and high scenarios for the power production of wind turbines in the set of Ω, and γ<sup>w</sup> represents the probabilities for the two scenarios of wind power production, 0.4 and 0.6, that are chosen for the low and high scenarios, respectively.

The formulation of the problem of density management in reconstructed power systems is the maximization of social welfare by considering the power balance constraints and the density of transmission lines. Mathematically, the objective function of the problem (maximizing social welfare) is a nonlinear relation (9):

$$\text{Max } SW = \sum\_{i=1}^{N\_d} B\_{di}(P\_{di}) - \sum\_{i=1}^{N\_{\mathcal{S}}} C\_{\mathcal{S}^{\mathcal{I}}}(P\_{\mathcal{S}^{\mathcal{I}}}) - \sum\_{k=1}^{N\_{\mathcal{W}}} C\_k(P\_{D\mathcal{G}k}) - \sum\_{w \in \Omega} \gamma\_w \times d\_{wind} \times P\_w \tag{9}$$

According to the formulation in relation to (9), the objective function, which is social welfare, is equal to the total profit of consumers minus the total cost of producers, which should be maximized according to the ISO view of this objective function. The equal and unequal constraints are given below. The profit of each manufacturer is obtained according to the following relationship:

$$\text{Profit}\_{\eta} = \left(LMP\_{\eta\\_\text{M}} \times P\_{\eta\\_\text{M}}\right) - \mathbb{C}\_{\eta} \tag{10}$$

where: profit*<sup>q</sup>* is the profit of the producer *q*, *LMPq*\_*<sup>n</sup>* is the local limit pricing in the *n* − th bus where the *q* − th producer is located, *Pq*\_*<sup>n</sup>* is the production capacity of the q − th producer in the *n* − th bus, and *Cq* is the cost of the q − th producer. Hence, we have:

Power balance constraints as:

$$P\_{\mathcal{G}i} + P\_{\text{D}Gi} + P\_W - P\_{di} = \sum\_{j=1}^{N} \frac{1}{\varkappa\_{i-j}} (\delta\_i - \delta\_j)\_\* \text{ for } i \in \mu\_{\text{DG}} \tag{11}$$

Maximum power limitation as:

$$\left|Pl\_{i-j}\right| \le PI\_{i-j}{}^{\max\_{r,l}} \tag{12}$$

Range of variables as:

$$0 \le P\_{\mathbb{S}^l} \le P\_{\mathbb{S}^l}^{\max, \text{ for } \mathbb{s}} \tag{13}$$

$$0 \le P\_{DGk} \le P\_{DG}^{\max} \tag{14}$$

$$0 \le P\_{dj} \le P\_{dj}^{\max, \text{ for } d} \tag{15}$$

$$\mathcal{S}\_i^{min\_{i\_l} max \ for} \tag{16}$$

In these relations, *N* and *NL* are the number of system busbars and the number of lines, respectively. *δ<sup>i</sup>* is voltage angle in the *i* − *th* busbar; *xi*−*<sup>j</sup>* gives the inductive reactance of the connecting line series amongst *i* and *j* buses; uDG provides the network DG set and *Pmax DGk* is the operating rate of the *k* − th DG. *Pli*−*<sup>j</sup>* and *Pli*−*<sup>j</sup> max* are the active power and maximum active power at the connection line between the buses *i* and *j*, respectively. *Pgimax* and *Pdjmax* represent maximum values of *Pgi* and *Pdj*. *δmin <sup>i</sup>* and *<sup>δ</sup>max <sup>i</sup>* are the minimum and maximum values of *δi*. The local limit value is also obtained from the power balance equilibrium in each bus.

Information of line parameters, generator cost coefficients, and consumption rates in different busbars is given in Tables 1–3, respectively.


**Table 1.** Grid line details.

**Table 2.** Grid generator details.


**Table 3.** Consumption details in busbars.


In this work, it is assumed that four distributed generation units without uncertainty and one wind turbine with two high and low production scenarios, according to Table 4, are connected to the 14-bus IEEE test network. The specifications of the cost function of

these units and their capacity and installation location are given in Table 4. Additionally, the network of 14 buses that the IEEE studied is shown in Figure 3.


**Table 4.** Distributed generation (DG) unit details.

**Figure 3.** The studied network of 14 buses by the IEEE.
