## 2 Modeling the Problem under Study
### Problem Objective Function  
To solve the problem of planning, different objective functions or a combination of several objective functions can be considered multi-objective. There are three objective functions in optimization, which are:  
1. Costs. 2. Reduce system line losses. 3. Reliability indicators. All system development costs are included in two terms, the fixed costs (the construction of new units of distributed generation and reactive power sources and the cost of updating new feeders) and variable costs (the cost of maintenance and operation of the system, which is basically this cost). Existing posts are power supplies that can power at a given variable price ceiling. The cost function is described by [4]:  
\[\text{Cost}_{total}=C_{FC}+C_{WT}+C_{PV}+C_{MS}+\ C_{Q} \tag{25}\]  
Support is common in many parts of the world. The cost of a fuel cell is equal to [31]:  
\[C_{FCi}=0.04\times kWh^{-1}\times\frac{P_{FCi}}{\eta_{i}} \tag{26}\]  
\[PLR_{i}=\frac{P_{g,i}}{P_{\text{max}i}} \tag{27}\]  
\[\left\{ \begin{array}{ll}if&PLR_{i}<0.05\Rightarrow\eta_{i}=0.2716\\ if&PLR_{i}\geq 0.05\Rightarrow\eta_{i}=0.2716 PLR_{i}^{5}-2.9996 PLR_{i}^{4}+3.6503 PLR_{ i}^{3}-2.0704 PLR_{i}^{2}+0.3747\end{array} \right. \tag{28}\]  
The technology of using wind and solar to generate electricity is the new fastest-growing source of electricity supply globally. Wind energy is mainly generated by huge three-bladed wind turbines mounted on top of tall turrets, which work like reverse fans. Instead of using electricity to generate wind, these turbines utilize the wind to generate electricity. The resulting savings primarily preserve fossil energy resources and, more importantly, convert them into large amounts of high value-added petrochemicals. Secondly, the production of electricity from this type of energy is free of any environmental pollution, which contributes significantly to preserving the healthy nature of the human environment and thus provides a path to sustainable economic and social development. The following relations can express the cost function for wind and solar systems:  
\[C_{PV,i}=a+b\times P_{PV,i} \tag{29}\]  
\[C_{WT,i}=a+b\times P_{WT,i} \tag{30}\]  
where \(a=\frac{C_{CP}(kW^{-1})\times Cap(kW)\times G}{Lif_{elim}(Yamr)\times 365 \times 24\times LF}\) and \(b=C_{F}(kW^{-1})+C_{O\&M}(kW^{-1})\) are involved in this relationship. The cost of power purchased from the main network is equal to [4]:  
\[C_{MS}=price.P_{MC} \tag{31}\]  
In recent years, the electricity industry in some countries of the world has become a competitive energy market in various ways, referred to as the restructured electricity industry. In the restructured electricity industry, the transmission system is a meansof access for electricity sellers and buyers, which is unique and remains in the hands of the government. Its operation is left to the independent system operator. In such systems, where the production, transmission, distribution, and sale of electricity are made by different organs and companies, the accurate pricing of power or energy has a special feature and complexity. To transfer power with suitable and safe reliability from the point of the system (vendor), one of the most important services is the reactive power supply service in the system, without which it is not possible to transfer real power in the power system. For various reasons, such as novelty and complexity, the issue of reactive power service pricing in the restructured electricity industry has received less attention. The cost of installing reactive power sources is equal to [32]:  
\[C_{Q}=\sum_{i=1}^{n\epsilon}fc_{i}+C_{i}Q_{i} \tag{32}\]  
The goal is to minimize system line losses after installation and \(DG\) power injection into the distribution network.