**1. Introduction**

Reactive Power Planning (RPP) in power systems can be considered as one of the most di fficult and complicated problems due to its complex variables, constraints, and optimization algorithms [1]. It is related to optimal sizing and allocation of VAR sources in power systems to satisfy prescheduled objectives, such as determining the optimal allocation and minimizing the operation costs [2,3]. The main objective of RPP is to achieve feasible operation with a satisfactory voltage profile with a lack of VAR support conditions. According to the concept of VAR planning in power systems, various objectives functions can be defined for the RPP problem. These objectives may consist of cost-based objective functions or objective functions that maximize or minimize indices, such as voltage stability margin or system loadability [4,5]. Moreover, it is possible to express the RPP as a multi-objective optimization problem, which optimizes several goals simultaneously [1].

Moreover, there is an increasing interest in using Renewable Energy Resources (RESs), such as wind farms and solar power plants, in power systems due to their technical, environmental, and economic advantages [4–6]. However, with the high penetration of RESs in power systems, the challenges associated with RPP are dramatically increased. One of the main challenges that can a ffect

the RPP is the uncertainty in the generation availability of RESs. Uncertainty in the sources' parameters leads to di fficulties in proper decision-making in the planning of power systems. Furthermore, owing to the stochastic nature of the load demands in electric power systems, additional uncertainties should be considered in RPP.

In RPP research studies, the probabilistic decision-making process based on either source uncertainty or load demand uncertainty is a well-developed research topic. Nevertheless, the probabilistic multi-objective RPP in power systems considering the uncertainty of loads and wind farms at the same time has not been fully investigated. In [7], a novel approach for dynamic VAR planning to improve the short-term voltage stability and transient stability is proposed. The impact of FACTS devices in RPP is analyzed in [8,9]. However, in both studies, an attempt has been made to explain the problem in a deterministic context. A multi-objective RPP that mainly focuses on voltage stability is introduced in [3]. Nonetheless, it is modeled based on a deterministic approach. In [10], a multi-objective approach for RPP with wind generations is presented. In this study, various objectives, such as system loadability, power losses, and cost of reactive power investment are considered. In [11], the RPP is solved using the Genetic Algorithm (GA) to reach coordination in controlling the reactive power in the presence of wind farms and FACTS devices. The loadability factor of the system is optimized by the optimal allocation of wind farms and FACTS devices. This procedure is implemented when loads with constant Power Factor (PF) and wind farms without uncertainty are assumed. Using the Benders decomposition method and considering the high penetration of wind generation, the RPP problem is tackled as two-stage stochastic programming in [12]. Using the Di fferential Evolutionary Algorithm (DEA), the RPP is solved in a wind integrated system in [13]. A major problem with the suggested model is that it only includes the uncertainty in wind power generation. In [14], a multi-stage stochastic model for RPP is extended, which involves the uncertainty of loads. Nonetheless, the proposed model describes the probabilistic behavior of the system in the absence of wind farms. In [15], a mixed-integer quadratic model for long term VAR planning is proposed. An attempt was made to minimize the operation and investment cost of new VAR sources and the load shedding risk through multi-objective optimization. Though the uncertainty in demand is completely taken into account, the proposed model does not consider uncertainty in wind power generation. In [16], a stochastic model based on chanced constrained programming for RPP is defined. The proposed model is solved using GA. Although the uncertainty is modeled in the power generation, it optimizes only one objective, including operational and investment costs. A chanced constrained model is proposed for probabilistic RPP in [17]. The proposed model is solved through two-stage stochastic programming. The main disadvantage of the proposed model is that it only considers the load as a random parameter. Besides, only the investment cost of new VAR sources is taken into account as the main objective function.

The main drawback of all the mentioned research studies is that the optimal RPP considering load demand and wind power generation uncertainties at the same time are not fully investigated. This paper aims to address RPP as a probabilistic multi-objective problem in order to reduce the total cost of reactive power investment, minimize the active power losses, maximize the voltage stability index, and improve the loadability factor. The generators' voltage magnitude, the transformers tap settings, and the output reactive power of the VAR sources are considered as the main control variables. To cope with the probabilistic multi-objective RPP problem, the ε-constraint technique is employed. To validate the e fficiency of the proposed method, the IEEE 30-bus test system is implemented in the GAMS environment under five various conditions. The obtained results show the e ffectiveness and high accuracy of the proposed method. Table 1 shows a comparison between the proposed probabilistic multi-objective RPP and the previously published research papers.

The rest of this paper is organized as follows. Section 2 deals with the uncertainty modeling. Problem formulation is presented in Section 3. Section 4 describes the optimization method. Simulation results are given in Section 5. Finally, some brief conclusions are summarized in Section 6.


