# Considering the Life-Cycle Cost of Distributed Energy-Storage Planning in Distribution Grids
## 3 ESS Installation Site and Capacity Optimization
### The Second-Stage Optimization  
#### 3.2.1 Life-Cycle Cost Model  
The expensive cost of the ESS is one of the biggest barriers impeding wide-area applications, at present, and this study proposed a comprehensive LCC model which considered various types of costs. The LCCs not only included the initial investment, but the operation and maintenance costs applied in the grid optimal planning, substation equipment maintenance, etc. [19]. [PERSON] et al. considered the cost of the ESS with capital investment, replacement costs, operation and maintenance costs, and disposal cost [17]. Capital investment and replacement costs were the greatest proportion in the whole project cycle. Battery is one of the short-life equipment in the distribution system, frequent charging/discharging, and over charging/discharging can overwhelmingly reduce the lifetime and decrease the whole system economy. To reduce the replacement costs, expansion of the ESS capacity can be adopted, which can prolong the lifetime of the battery and reduce the replacement times. All of the costs considered in this study are presented as follows:  
1. Capital investment  
The main components of the ESS include the storage unit, the PCS, and the necessary supporting facilities, so the capital cost can be expressed as:  
\[C_{sys}=(C_{E}E_{rate}+C_{P}P_{ESS,rate}+C_{B}E_{rate})\frac{\sigma(1+\sigma)^ {Y}}{(1+\sigma)^{Y}-1} \tag{13}\]  
where \(C_{sys}\) is the capital cost of battery. \(E_{rate}\) and \(P_{ESS,rate}\) are the rated capacity and the power rating of the ESS, respectively, and the power rating is the maximum of the ESS charging/discharging power. \(Y\) is the project cycle (year) and \(\sigma\) is the discount rate (%). Apart from this, the other variables were same as shown in Equation (7). The rated power can be evaluated as the minimum value of the maximum power allowed by the whole battery pack (\(E_{r}\cdot E_{rate}\)) and the maximum actual required charging/discharging power of ESS(\(P_{ESS,max}\)) [20]:  
\[P_{ESS,rate}=\min\{Er\cdot E_{rate},P_{ESS,max}\} \tag{14}\]  
where \(Er\) is the energy rate of the ESS (kW/kW-h) and \(P_{ESS,max}\) is the maximum actual required charging/discharging power of the ESS. Generally, the maximum charging/discharging power of Li-ion batteries is strictly related to their rated capacity, so the typical parameter \(Er\) is used to represent the ratio between their maximum power and the rated capacity, the value of \(Er\) is set as 1 [20]. In some cases, the maximum power allowed by the battery could exceed the maximum actual required charging/discharging power, therefore, the minimum value should be chosen to guarantee economy of the ESS. 2. Replacement cost  
The actual project cycle is generally 5 to 20 years. Although the lifetime of the supporting facilities can meet the project cycle need, the lifetime of the ESS and the PCS cannot be used for 20 years, without replacement. Annual replacement cost in the project cycle is formulated as Equation (15). \[C_{rep}=C_{E}E_{rate}\sum_{\varepsilon=1}^{k}\left(\frac{(1-\beta)^{cn}}{(1+ \sigma)^{cn}}\frac{\sigma(1+\sigma)^{Y}}{(1+\sigma)^{Y}-1}\right)+C_{P}P_{ESS, rate}\frac{(1-\beta)^{10}}{(1+\sigma)^{10}}\frac{\sigma(1+\sigma)^{Y}}{(1+\sigma)^{Y}-1} \tag{15}\]  
where \(\beta\) is the average annual decline rate of the ESS capital cost and \(k\) is the replacement times of battery (round up, \(k=Y/n-1\)), n is the average lifetime of the chosen ESS, \(\varepsilon\) is the times of replacement. The lifetime of the PCS was chosen as 10 years, in this study [21]. 3. Operation and maintenance cost  
Operation and maintenance cost of ESS consists of fixed and variable operation and maintenance costs. The fixed operation and maintenance costs(\(C_{FOM}\)) are composed of the labor cost and management cost, related to the types of ESS and the rated power. Then \(C_{FOM}\) can be calculated as:  
\[C_{FOM}=C_{f}P_{ESS,rate} \tag{16}\]  
where \(C_{f}\) is the operation and maintenance cost per kW.