*3.2. Objective Functions*

For probabilistic multi-objective RPP, the aim is to satisfy three main objectives. These objectives include the minimization of total VAR investment cost, minimization of voltage stability index (*L*-index), and maximization of loadability factor, which lead to a reduction in total active power losses and improvement of voltage stability.

### 3.2.1. Minimization of Total VAR Investment Cost

One of the important objectives in the RPP is the total cost of VAR planning. In spite of allocating the optimal location and capacity for VAR sources, optimal VAR planning can handle the RPP problem from economic aspects. For this reason, the first objective function is a cost-based objective function comprising two main parts, as follows:

(1). The first part evaluates the expected cost of energy loss (*Wc*) during the generated scenarios and is expressed as follows [16–26]:

$$\mathcal{W}\_{\mathfrak{c}} = \pi\_{\mathfrak{s}} (\ln \sum\_{s \in \Omega\_{\mathfrak{s}}} t\_{\mathfrak{s}} P\_{\text{loss},\mathfrak{s}}) \tag{10}$$

where *Ploss*,*<sup>s</sup>* shows the active power losses during the *sth* scenario, *ts* represents the duration of the *sth* scenario, *h* is a constant parameter that is related to the first part cost-based objective function and identifies the per-unit energy cost, and π*s* denotes the probability of the *sth* scenario. To calculate the total active power losses, Equation (11) can be used as follows [27–30]:

$$P\_{\text{loss},s} = \sum\_{\substack{l \in \Omega\_{Line} \\ l \neq \dots \neq l \text{-}line}} G\_{(l,s)} \left( V\_{i,s}^2 + V\_{j,s}^2 - 2V\_{i,s}V\_{j,s} \cos \left( \theta\_{i,s} - \theta\_{j,s} \right) \right) \tag{11}$$

where *Vi*,*<sup>s</sup>* and *Vj*,*<sup>s</sup>* are the sending and receiving ends voltage magnitude of the *lth* transmission line for the *sth* scenario, respectively, θ*i*,*<sup>s</sup>* and <sup>θ</sup>*j*,*<sup>s</sup>* are the sending and receiving ends voltage

angles of the *lth* transmission line for the *sth* scenario, respectively, and *<sup>G</sup>*(*l*,*<sup>s</sup>*) is used to designate the conductance of the *lth* transmission line for the *sth* scenario.

(2). The second part measures the expected cost of VAR investment (*Ic*) during the generated scenarios and is derived as follows [14–17]:

$$I\_{\mathcal{E}} = \sum\_{s \in \Omega\_{\theta}} \pi\_s(\sum\_{i \in \Omega\_{\text{Comp}}} (c\_i + \mathcal{C}\_{\text{Ci}} Q\_{\text{Ci},s})) \tag{12}$$

where *ei* and *CCi* are the fixed and variable installation costs of VAR sources, respectively.

Accordingly, the first objective function (*f*1) can be derived as follows:

$$f\_1 = F\_\varepsilon = \mathcal{W}\_\varepsilon + I\_\varepsilon \tag{13}$$

where *Fc* shows the expected Total VAR Cost (TVC).

### 3.2.2. Minimization of Voltage Stability Index

In this paper, the *L*-index is proposed as the voltage stability index that is a well-known static voltage index [31]. In order to estimate the static voltage stability of the power system, the *L*-index should be calculated for all load buses (*PQ* buses). All the load buses that have higher values of the *L*-index than others are considered as the weak buses. Weak buses mostly suffer from a lack of reactive power and are prone to the voltage collapse. Equation (14) can be used to calculate the *L*-index (*Lj*) for the *jth* load bus, as follows [31]:

$$L\_j = \left| 1 - \sum\_{i=1}^{\Omega\_{\mathcal{E}}} F\_{ji} \frac{\overline{V\_i}}{\overline{V\_j}} \right|, \qquad \forall j \in \Omega\_{PQ} \tag{14}$$

where *Vi* shows the voltage of the *ith* generator, and *Vj* represents the voltage of the *jth* load bus. *Fji* can be derived from *Ybus* matrix of the system. Thus, by rearranging the current and voltage equations in power systems, as shown in Equation (15), the consecutive *Ybus* matrix is achieved. Thereafter, using the arrays of the consecutive *Ybus* matrix, the *Fji* matrix can be calculated as Equation (16).

$$
\begin{bmatrix} I\_{\mathcal{S}} \\ I\_{I} \end{bmatrix} = \begin{bmatrix} \mathcal{Y}\_{\mathcal{S}\mathcal{S}} & \mathcal{Y}\_{\mathcal{S}^{I}} \\ \mathcal{Y}\_{I\mathcal{S}} & \mathcal{Y}\_{\mathcal{U}} \end{bmatrix} \begin{bmatrix} V\_{\mathcal{S}} \\ V\_{I} \end{bmatrix} \tag{15}
$$

$$F\_{ji} = -\left[\boldsymbol{Y}\_{ll}\right]^{-1} \boldsymbol{Y}\_{l\underline{\mathcal{G}}\prime} \qquad \forall j \in \Omega\_{P\underline{\mathcal{Q}}\prime} \,\forall i \in \Omega\_{\mathcal{S}} \tag{16}$$

where *Ig* and *Il* show the current of generators and loads, respectively, and *Vg* and *Vl* are the voltage of generators and loads, respectively. In addition, *Ygg*, *Ygl*, *Ylg*, and *Yll* are the submatrices of the consecutive *Ybus* matrix. It should be noted that only *Yll* arrays of the consecutive *Ybus* matrix are related to the PQ nodes. Also, the consecutive *Ybus* matrix is a symmetric matrix. Therefore, *Ylggl* = *Ylg*.

By minimizing the values of the *L*-index at the weak buses, there is a possibility to increase the level of static voltage stability in power systems. The voltage stability of power systems can be determined by the *L*-index when the maximum value of the *L*-index (*Lmax*) is assigned to the static voltage stability level in power systems, as follows:

$$L\_{\max} = \max\limits(L\_j), \qquad \forall j \in \Omega\_{P\mathbb{Q}} \tag{17}$$

To improve the static voltage stability of power systems, it is necessary to minimize *Lmax*. It should be noted that the equations proposed for the *L*-index are related to the deterministic problem. In the case of a probabilistic problem, considering all necessary modifications on the *Ybus* matrix in each scenario, after re-formulating Equations (14)–(17), new equations can be rewritten as follows:

$$L\_{j,s} = \left| 1 - \sum\_{i=1}^{\Omega\_{\tilde{\mathcal{I}}}} F\_{\bar{j},s} \frac{\overline{V\_{i,s}}}{\overline{V\_{j,s}}} \right|, \qquad \forall j \in \Omega\_{\mathcal{PQ}\_{\mathcal{I}}} \; \forall s \in \Omega\_{\mathcal{s}} \tag{18}$$

where *Lj*,*<sup>s</sup>* indicates L-index value for the *jth* load bus and *sth* scenario, *Vi*,*<sup>s</sup>* and *Vj*,*<sup>s</sup>* are the voltage of the *ith* generator and *jth* load bus for the *sth* scenario, respectively. For each scenario, the *Fji*,*<sup>s</sup>* matrix can be calculated as Equation (20).

$$
\begin{bmatrix} I\_{\mathbb{g},s} \\ I\_{l,s} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \boldsymbol{Y}\_{\mathbb{g}\mathbb{g},s} & \boldsymbol{Y}\_{\mathbb{g}l,s} \\ \boldsymbol{Y}\_{l\mathbb{g},s} & \boldsymbol{Y}\_{l\mathbb{I},s} \end{bmatrix} \begin{bmatrix} \boldsymbol{V}\_{\mathbb{g},s} \\ \boldsymbol{V}\_{l,s} \end{bmatrix} \end{bmatrix} \qquad \forall \mathbf{s} \in \Omega\_{\mathbb{g}} \tag{19}
$$

$$F\_{j\bar{\imath},s} = -\left[Y\_{\Pi,s}\right]^{-1}Y\_{l\underline{\jmath},s\prime} \qquad \forall j \in \Omega\_{PQ\_{\prime}} \,\forall i \in \Omega\_{\underline{\wp}\prime} \,\forall s \in \Omega\_{s} \tag{20}$$

where *Ig*,*<sup>s</sup>* and *Il*,*s* denote the current of generators and loads for the *sth* scenario, respectively, *Vg*,*<sup>s</sup>* and *Vl*,*<sup>s</sup>* are the voltage of generators and loads for the *sth* scenario, respectively. Also, *Ygg*,*s*, *Ygl*,*s*, *Ylg*,*s*, and *Yll*,*<sup>s</sup>* are the submatrices of the consecutive *Ybus* matrix for the *sth* scenario.

According to the aforementioned descriptions, Equations (18)–(20) can be obtained for each scenario. The maximum value of the *L*-index for each scenario can be derived as follows:

$$L\_{\text{max},s} = \max\{L\_{j,s}\}, \qquad \forall j \in \Omega\_{PQ\_{\prime}} \,\forall s \in \Omega\_{s} \tag{21}$$

Consequently, the second objective function (*f*2), which is the expected value of the static voltage stability index during the generated scenarios, can be derived as follows:

$$f\_2 = \sum\_{s \in \Omega\_s} \pi\_s L\_{\text{max},s} \tag{22}$$

### 3.2.3. Maximization of the Loadability Factor

The injected active power (*Pi*) and reactive power (*Qi*) at the *ith* bus can be expressed in terms of the voltage (*V*), the elements of the *Ybus* matrix of the system, and the loadability factor (Γ) as follows [32]:

$$P\_i = P\_{Gi} - (1 + \Gamma)P\_{Di} - \text{Re}\{V\_i \sum\_{j=1}^{N\_R} \left(V\_j \mathcal{Y}\_{i,j}\right)^\*\} \tag{23}$$

$$Q\_i = Q\_{Gi} - (1 + \Gamma) Q\_{Di} - \operatorname{Im} \{ V\_i \sum\_{j=1}^{N\_B} \left( V\_j \mathcal{Y}\_{i,j} \right)^\* \} \tag{24}$$

where *PGi* and *QGi* are the active and reactive power generation at the *ith* bus, respectively, *PDi* and *QDi* represent the base-case active and reactive power consumption at the *ith* bus, respectively, and *NB* denotes the total number of buses.

Maximizing the loadability factor is defined as the third objective function, in this paper. However, considering the random nature of the problem, a probabilistic formulation is required. Therefore, by re-formulating Equations (23) and (24), a stochastic formula is derived to obtain the expected loadability factor, as shown in Equations (25) and (26).

$$P\_{i,s} = P\_{Gi,s} - (1 + \Gamma(s))P\_{Di,s} - \text{Re}\{V\_{i,s} \sum\_{j=1}^{N\_B} \left(V\_{j,s} Y\_{i,j,s}\right)^\*\}, \qquad \forall s \in \Omega\_s \tag{25}$$

$$Q\_{i,s} = Q\_{\text{Gi},s} - (1 + \Gamma(s))Q\_{\text{Di},s} - \text{Im}\langle V\_{i,s} \sum\_{j=1}^{N\_B} \left( V\_{j,s} Y\_{i,j,s} \right)^\* \rangle\_{\prime} \qquad \forall s \in \Omega\_s \tag{26}$$

where *Pi*,*<sup>s</sup>* and *Qi*,*<sup>s</sup>* denote the injected active and reactive power at the *ith* bus for the *sth* scenario, respectively, *PGi*,*<sup>s</sup>* and *QGi*,*<sup>s</sup>* represent the active and reactive power generation at the *ith* bus for the *sth* scenario, respectively, *PDi*,*<sup>s</sup>* and *QDi*,*<sup>s</sup>* are the base-case active and reactive power consumption at the *ith* bus for the *sth* scenario, respectively, <sup>Γ</sup>(*s*) denotes the loadability factor for the *sth* scenario; *Vi*,*<sup>s</sup>* and *Vj*,*<sup>s</sup>* indicate the voltage of the *ith* bus *jth* bus for the *sth* scenario, respectively, and lastly, the elements of the *Ybus* matrix for the *sth* scenario are shown by *Yi*,*j*,*s*.

According to the above-mentioned descriptions, the third objective function (*f*3), which is the expected value of the loadability factor, can be derived as follows:

$$f\_{\mathfrak{J}} = \sum\_{s \in \Pi\_s} \pi\_s \Gamma(s) \tag{27}$$

Finally, the optimization criteria subjected to equality and inequality constraints are as follows:

$$\text{Optimization Criteria} = \begin{cases} \min(f\_1) \\ \min(f\_2) \\ \max(f\_3) \end{cases} \tag{28}$$
