**4. Optimization Method**

Once an optimization problem is formulated carefully, it is required to solve the problem via an optimization method. To choose an optimization method to solve a problem, it is necessary to consider the number of optimization variables, the type of variables, the number of objective functions, the number of constraints, and the convexity or non-convexity of the problem and the other characteristics [27–29]. In this regard, the optimization methods can be classified into three major groups; (1) exact methods based on mathematical calculations, (2) heuristic methods, and (3) combination of the exact methods and heuristic methods.

### *4.1. Multi-Objective Optimization Using* ε*-Constraint Method*

According to [33,34], the ε-constraint method is considered as one of the classic methods for multi-objective optimization. This method is in line with the exact methods. In addition to its efficiency and simplicity, this method is applicable to both convex and non-convex problems. The main idea of the ε-constraint method is to reformulate the multi-objective problem as a single-objective problem. Then, by iteratively solving the single-objective problem, a Pareto Front is obtained. In the following, the details of the ε-constraint method are explained [34].

Considering a multi-objective problem (Ψ(*X*)), as shown in Equation (39), subjected to different constraints that should be optimized, the following steps should be taken.

$$\Psi(X) = (f\_1(X), f\_2(X), \dots, f\_l(X)), \qquad \mathbf{i} = 1, 2, \dots, n \tag{39}$$

where *fi*(*X*) denotes the *ith* objective function and *n* shows the maximum number of existing objective functions.


$$f\_i^{\text{min}} \le f\_i(X) \le f\_i^{\text{max}} \tag{40}$$

4. To generate different values for *ei*,*ni* , Equations (41) and (42) are used to minimize and maximize the objective function, respectively. By dividing the domain of the *ith* objective function into *qi* equal parts using Equations (41) and (42), *qi* different values are obtained for *ei*,*ni* . It should be noted that *ni* denotes the number of available generated values for *ei*,*ni*.

$$c\_{i,n\_i} = f\_i^{\max} - (\frac{f\_i^{\max} - f\_i^{\min}}{q\_i}) n\_i, \qquad n\_i = 0, 1, \dots, q\_i \tag{41}$$

$$x\_{i,n\_i} = f\_i^{\min} - (\frac{f\_i^{\max} - f\_i^{\min}}{q\_i}) n\_{i\prime} \qquad \qquad n\_i = 0, 1, \ldots, q\_i \tag{42}$$

5. By using the obtained values from Step 4, it can be derived that *fi*(*X*) ≤ *ei*,*ni* or *fi*(*X*) ≥ *ei*,*ni* . For different values of *ei*, a set of solutions is obtained, which forms the Pareto front of the problem.

According to the above-mentioned descriptions, to solve the probabilistic multi-objective RPP, the following equation is formed.

$$\min f\_1 \tag{43}$$

subjected to

$$\begin{cases} \qquad f\_2 \le c\_{2, \nu\_2} \\ \qquad f\_3 \le c\_{3, \nu\_3} \\ \text{Equations (29)-(38)} \end{cases} \tag{44}$$

### *4.2. Fuzzy Decision Maker (FDM)*

As already mentioned, after solving a multi-objective optimization problem, a set of optimal solutions is obtained, called the Pareto Front. While only one solution from the Pareto Front can be chosen as the final optimal solution to the problem, which is known as the Best Compromise Solution (BCS). One way to choose the BCS is to use the Fuzzy Decision Maker (FDM). Having used a fuzzy membership function, each of the optimal solutions is mapped between 0 and 1. For the *kth* objective function, *Fk*, the linear fuzzy membership is defined as Equation (45) [24], and it is supposed that all the objective functions are minimized.

$$\mathcal{F}\_k = \begin{cases} 1, & F\_k \le F\_k^{\min} \\ \frac{F\_k^{\max} - F\_k}{F\_k^{\min} - F\_k^{\max}}, & F\_k^{\min} \le F\_k \le F\_k^{\max} \\ 0, & F\_k \ge F\_k^{\max} \end{cases} \tag{45}$$

where *F* ˆ *k* represents the *kth* normalized objective function. In addition, *Fmin k* and *Fmax k* are used to express the minimum and maximum values of the *kth* objective function, respectively.

After obtaining the fuzzy values of each objective function using Equation (45), there are several ways to find the BCS. In this paper, to obtain the BCS, the min-max method, which is introduced in [35], is used.

$$BCS = \max\left(\min\{\mathbb{P}\_1, \mathbb{P}\_2, \dots, \mathbb{P}\_k\}\right) \tag{46}$$

### **5. Simulation Results and Discussions**

To evaluate the performance of the proposed ε-constraint method in the presence of various objectives, containing expected total VAR cost (*f*1), expected active power losses, expected voltage stability index (*f*2), and expected loadability factor (*f*3), two deterministic and three probabilistic cases are studied, as follows:


All the cases are implemented in GAMS environment Ver. 25.1.2 [36–39], and are solved using the CONOPT 3 Solver [40], in an ASUS laptop, with 8 GB of RAM and 2.4 GHz. The descriptions of the case study are presented in the next subsection.

### *5.1. Case Study Descriptions and Simulation Results*

The test system is the IEEE 30-bus test system, which has 6 generation units, 4 transformers, and 41 branches. The initial settings of the generators' voltage magnitude and transformers tap settings are obtained from [30]. Figure 1 shows the single line diagram of the IEEE-30-bus test system. Also, both the output active and reactive power of generators are set according to [41]. The loads' data and line data are available in [42]. It is assumed that there is not any VAR source in the case study.

**Figure 1.** Single line diagram of the IEEE-30 bus test system [43].

To allocate appropriately the VAR compensation devices, firstly, the *L*-index should be determined for the load buses. Then, the load buses with high values of *L*-index are taken into account as the candidate buses for the VAR compensation devices installation. It is observed that after implementing the proposed methodology, the load at bus 24, bus 25, bus 26, bus 29, and bus 30 obtain the higher values of *L*-index than the other load buses. As a result, the VAR compensator buses are found. After the allocation of VAR compensation devices, it is supposed that the capacities of the VAR compensators can be set to zero. The initial conditions of the system considering the full-load and 1-year planning are stated in Table 2.


**Table 2.** Control variables and objectives under initial conditions.

The control variables limits are expressed in Table 3. The per-unit energy cost is equal to 0.06 (\$/h) [3], the fixed installation cost (*ei*) for all VAR sources is 1000 (\$), and the variable installation cost (*CCi*) for all VAR sources is 3.0 (\$/kVAR) [44].


**Table 3.** Control variables limits.

The voltage magnitude at the load buses, which are considered as the state variables, must be limited between 0.95 (p.u.) and 1.05 (p.u.). To show the effectiveness of the proposed method, two deterministic and three probabilistic cases are considered in the following subsections.

5.1.1. Case A: Deterministic Multi-Objective RPP without Considering the Loadability Factor

In order to validate the efficiency of the proposed method for multi-objective RPP, the ε-constraint method is applied to a deterministic multi-objective RPP problem. The obtained results are also compared with the approach presented in [3]. The deterministic multi-objective RPP aims to minimize the total VAR cost and voltage stability index. Thus, it is expected to achieve a reduction in active power losses and an improvement in the voltage stability index. Table 4 shows the obtained results from deterministic multi-objective RPP without considering the loadability factor in IEEE 30-bus test system with the initial settings. The duration of the load for deterministic multi-objective RPP is assumed to be 8760 h for full-load condition and without changes in the load level. According to Table 4, 15 Pareto optimal solutions are obtained by the ε-constraint method. After that, the min-max approach chooses the fifth solution (highlighted row) as the BCS. The active power losses for the BCS are 4.9813 MW.


**Table 4.** Obtained Pareto optimal solutions for the deterministic multi-objective RPP without considering the loadability factor.

Considering the same operating condition, the ε-constraint method is compared with the Multi-Objective Differential Evolution (MODE) algorithm, which is recommended to solve the deterministic multi-objective RPP [3]. For the BCS, the results of the comparison are presented in Table 5. As it can be observed from Table 5, the ε-constraint method shows better performance compared with the MODE algorithm in minimizing total VAR cost and active power losses. The superiority of the ε-constraint method is confirmed by a 6.2578 % reduction in total VAR cost and a 9.3815 % decrease in the active power losses over the Base Case. However, as it can be observed from Table 5, the voltage stability index of the conventional method is better than the proposed approach.

**Table 5.** Comparison of the obtained results between the ε-constraint method and MODE algorithm for the BCS under the same operating conditions.


The optimal values of the control variables for Case A are represented in Table 6. As it can be observed, only one VAR source has a value of zero. Note that the fixed installation VAR cost is also considered for all VAR sources with the value of zero during the planning studies. It is apparent from Case A that the ε-constraint is an effective method to generate Pareto optimal solutions for the multi-objective RPP.

**Table 6.** The optimal values of the control variables for the deterministic multi-objective RPP without considering the loadability factor.


5.1.2. Case B: Deterministic Multi-Objective RPP Considering the Loadability Factor

In this part, the ε-constraint method is applied to the multi-objective RPP problem in a complex form. However, the uncertainties of the load demand and wind power generation are not considered in this case. In comparison with Case A, another objective, which is called the loadability factor, is added to the problem. Therefore, the main objectives in this part include minimizing the total VAR cost, reducing the active power losses, improving the voltage stability index, and maximizing the loadability factor. In order to solve a deterministic multi-objective RPP considering the loadability factor, it is assumed that the system is under full-load condition. The duration of the load is assumed to be 8760 h. Table 7 provides the simulation results of the deterministic multi-objective RPP problem considering the loadability factor. In addition, this table illustrates that among the 15 generated Pareto optimal solutions, the eighth solution (highlighted row) is the BCS through the min-max approach. The active power losses for the BCS is 9.3494 MW. Also, it can be observed that the total VAR cost and active power losses are dramatically increased for BCS in comparison with Case A. Moreover, the voltage stability index is not improved compared with Case A. Nevertheless, the loadability factor is improved in Case B. The main reason behind the deterioration of active power losses and voltage stability index is due to the enhancement of the loadability factor. It should be noted that the loadability factor can hugely affect the active power losses and voltage stability of power systems.


**Table 7.** Obtained Pareto optimal solutions for the deterministic multi-objective RPP considering the loadability factor.

Table 8 depicts the optimal values of the control variables for Case B. As it can be observed, three VAR sources have a value of zero. It should be noted that the fixed installation VAR cost is also considered for all VAR sources with the value of zero during the planning studies.

**Table 8.** The optimal values of the control variables for the deterministic multi-objective RPP considering the loadability factor.


### 5.1.3. Case C: Probabilistic Multi-Objective RPP Considering the Load Demand Uncertainty

In Cases A and B, the multi-objective RPP in power systems is solved using the deterministic approach. However, with the increasing level of uncertainty, probabilistic multi-objective is required for the RPP problem. In Case C, the probabilistic multi-objective RPP considering three di fferent scenarios for the load level is performed. Each scenario for the load level consists of two main parts: (1) probability of the load level and (2) duration of the load. The overall duration of the load is assumed to be 8760 h, which is the expected time horizon for the RPP. The specifications of the system loading are described in Table 9.

**Table 9.** The specifications of the system loading for probabilistic multi-objective RPP considering the load demand uncertainty.


The simulation results obtained from the probabilistic multi-objective RPP considering the load demand uncertainty are given in Table 10. As it can be observed from this table, 15 Pareto optimal solutions are generated using the ε-constraint method. Using the min-max approach, the eighth solution (highlighted row) is chosen as the BSC. It is worth mentioning that the expected active power losses are 9.5049 MW for the BCS. From Table 10, it is clear that the expected total VAR cost is reduced compared with the Base Case. The expected voltage stability index and the expected loadability factor also show improvement towards the initial conditions. However, with more considerations, it is revealed that the expected active power losses are increased. This fact stems from the evident increase in the loadability factor. As a common incidence in power systems, following the escalation of the loadability factor, the active power losses increase and the system becomes voltage unstable. Generally, from the power systems operators' perspective, monitoring of the voltage magnitude at the load buses as a way of preventing voltage collapse is in high priority. Therefore, the voltage profile of the load buses for each loading scenario is plotted for the BCS, as shown in Figure 2.

**Table 10.** Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering the load demand uncertainty.


**Figure 2.** Voltage profile of the load buses for probabilistic multi-objective RPP considering the load demand uncertainty.

The optimal values of the control variables for the BSC among the different load scenarios are given in Table 11. Having checked this table closely, it is noticed that some VAR sources have a value of zero. Consequently, the variable cost of those VAR sources is equal to zero. However, those VAR sources are allocated, and their fixed VAR installation costs are considered for the planning studies in this paper.


**Table 11.** The optimal values of the control variables among the load scenarios for probabilistic multi-objective RPP considering the load demand uncertainty.

It is clear from Figure 2 that the voltage magnitude of the load buses remains in the range of 0.95 p.u. and 1.05 p.u. for all three load scenarios. Although the loadability factor is improved, the voltage stability index of the system is ensured from the voltage magnitude point of view. Therefore, by making an allowance for the load demand uncertainty, the obtained total VAR cost seems to be more realistic. In the same way, the voltage stability index and the loadability factor are more reliable due to including more scenarios for the planning horizon.

5.1.4. Case D: Probabilistic Multi-Objective RPP Considering the Wind Power Generation Uncertainty

In Case D, it is assumed that a wind farm is located at a PQ node. After generating the wind speed scenarios using the Weibull distribution and power curve, a probabilistic multi-objective RPP is performed. In this case, the IEEE 30-bus test system is modified based on [45]. Hence, a wind farm with a rated power of 40 MW is added to bus 22. The wind farm data is derived from [45] and is presented in Appendix A. To evaluate the impact of the wind farm, six scenarios for the output power of the wind farm are generated. The duration of the load is assumed to be 1460 h and without changes in the load level. The generated wind scenarios and their details are given in Table 12.


**Table 12.** Generated wind scenarios for the probabilistic multi-objective RPP considering the wind power generation uncertainty.

The obtained results from the probabilistic multi-objective RPP using the generated wind scenarios are represented in Table 13. As it is observed from this table, among the 15 generated Pareto optimal solutions using the ε-constraint technique, the eighth solution (highlighted row) is selected as the BSC after applying the min-max approach. The corresponding value of expected active power losses is 8.5777 MW. Compared with the Base Case and Case A, it is clear that the expected total VAR cost has had a remarkable reduction for the best compromise solution. In addition, the enhancement of expected voltage stability index and expected loadability factor is undeniable towards the Base Case and Case A. In addition, the expected active power losses are elevated in contrast with the Base Case. However, the expected active power losses show a reduction of roughly 1 MW, when it is compared with Case A. The main reason for this reduction is the existence of the wind farm in the case study.

**Table 13.** Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering the wind generation uncertainty.


The optimal values of the control variables for the BCS over the wind scenarios are represented in Table 14. Taking a look at Table 14, it is shown that the VAR sources gain the value of zero in almost all scenarios. Therefore, the variable VAR investment cost for those VAR sources equals to zero. However, the VAR sources are allocated and their fixed VAR investment costs are taken into account during the planning horizon.


**Table 14.** The optimal values of the control variables among the wind scenarios for probabilistic multi-objective RPP considering the wind power generation uncertainty.

The voltage profile of the load buses for the BCS is plotted in Figure 3. As it can be observed, the voltage magnitude of the load buses is kept in the interval of 0.95 p.u. and 1.05 p.u. during all wind scenarios. Hence, it can be concluded that based on a proper RPP and having adequate reactive power reserve, the voltage magnitude of the load buses are restricted with specific limits.

**Figure 3.** Voltage profile of the load buses for probabilistic multi-objective RPP considering the wind power generation uncertainty.

5.1.5. Case E: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties

To have a more comprehensive overlook of the probabilistic RPP, in Case E, it is preferred to perform RPP in the presence of two stochastic input variables, including the load demand and wind power generation. Hence, considering the level of the load and wind power generation as the stochastic input variables, combined load-wind scenarios are generated via the proposed technique. After determining the load-wind scenarios, a probabilistic multi-objective RPP is performed to evaluate the existing objectives, while the number of random input variables increases. Taking the IEEE 30 bus-test system with initial settings as the benchmark, 18 combined load-wind scenarios are generated. In general, three load scenarios and six wind scenarios are used to generate 18 combined load-wind scenarios. The descriptions of generated load-wind scenarios are given in Table 15.


**Table 15.** Generated load-wind scenarios for probabilistic multi-objective RPP considering load demand and wind power generation uncertainties.

Table 16 shows the obtained results for the probabilistic multi-objective RPP using the generated load-wind scenarios. As seen from Table 16, by applying the min-max method among the 15 generated Pareto optimal solutions using the ε-constraint approach, the eighth solution (highlighted row) is selected as the BCS. The associated value to expected active power losses is 8.5575 MW. It can be observed that the expected Total VAR cost is considerably reduced, while the expected voltage stability index and the expected loadability factor are not significantly improved towards Case B. In contrast with Case A and the Base Case, both the expected voltage stability index and the expected loadability factor are improved compared with case B. Considering the expected active power losses, no substantial decrease is observed in Case C when it is compared with Case B. Compared with Cases A and C, a reduction of about 1 MW in expected active power losses can be estimated. Due to enhancing the expected loadability factor, the expected active power losses escalate relative to the Base Case.


**Table 16.** Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering load demand and wind power generation uncertainties.

The optimal values of the control variables among 18 generated load-wind scenarios for the BCS are represented in Table 17. As it can be observed from Table 17, the VAR sources gain the value of zero in most of the scenarios for the BCS. Hence, the fixed installation cost is calculated for the VAR sources that gain the value of zero.

In order to investigate the impact of VAR planning in bus voltage magnitude over the different load-wind scenarios, the voltage profile of load buses is plotted for each load-wind scenario in Figure 4. This Figure shows that the voltage magnitude of the load buses is limited to the range of 0.95 p.u. and 1.05 p.u. for all scenarios. As a result, the voltage magnitude of the load buses is regulated within the predefined limits.


**Figure 4.** Voltage profile of the load buses for probabilistic multi-objective RPP considering the wind power generation uncertainty.

5.1.6. Case F: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties Incorporating Reactive Power from Wind Farms

In order to evaluate the impact of generated reactive power by wind farms on RPP, the reactive power of wind farms is taken into account during the planning process. Assuming the constant PF operation for the proposed wind farms, the generated reactive power is calculated for each scenario based on Table 18. The value of PF for the proposed wind farms is taken to be 0.98. It should be noted that the generated reactive power by the proposed wind farms is calculated using Equation (47), as follows:

$$Q\_{W\_{i,S}} = P\_{W\_{i,S}} \tan(\cos^{-1}(PF))\tag{47}$$

where *QWi*,*<sup>S</sup>* and *PWi*,*<sup>S</sup>* indicate the generated reactive and active power by the proposed wind farms, respectively.

**Table 18.** The characteristics of generated reactive power by the proposed wind farm for different scenarios.


Similar to the former case studies, after performing the probabilistic multi-objective RPP considering the reactive power injection by the proposed wind farms, 15 Pareto optimal solutions are obtained, as shown in Table 19.

**Table 19.** Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering load demand and wind power generation uncertainties incorporating the generated reactive power by the proposed wind farms.


From Table 19, it is clear that for the BCS, all the objectives are slightly improved towards Case E. Although this enhancement does not seem to be significant, it shows the penetration of generated reactive power by the proposed wind farms on planning studies. Moreover, the related active power loss reaches 8.4807 MW, which shows a reduction with respect to Case E. The optimal values of the control variables among 18 generated load-wind scenarios incorporating the generated reactive power by the proposed wind farms for the BCS are represented in Table 20.


*Appl. Sci.* **2020**, *10*, 2859

It can be inferred from Table 3 that in almost all scenarios, the VAR sources gain the value of zero, except for the installed VAR compensator at bus 30. In addition, it is revealed that the expected value of the required VAR compensator device at bus 30 reduces while the generated reactive power of the hypothetical wind farms is taken into account. As a result, wind farms have the capability to participate in VAR planning. This leads to a reduction in the size and amount of VAR sources. Therefore, practical power systems show less desire to install new VAR support while numerous large-scale wind farms with su fficient generated reactive power are available.
