## 2 Modeling of Distributed Energy Resources (DERs)
### Modeling and Characteristic Analysis of Energy Storage Devices and Back-Up Source  
ES devices and back-up source are used to compensate energy shortage caused by insufficient distributed generator (DG) generation, hence relative analysis is significant for further study on source-load coordination mechanism. Table 1 shows the characteristics of both energy-type and power-type ES devices, including energy density, power density, investment cost, approximate cycle times and response speed. Figure 1: Power output curve of wind turbine (WT). Through observation, it is obvious that two types of ES devices are complementary to each other, and a hybrid storage system can provide enhanced power supply capability. Besides, although flywheel storage and SMES have outstanding performance, they are relatively expensive and are still in the experimental stages, which made them unsuitable for practical application. In this paper, the HESS consists of a lead-acid battery, Li-ion battery and super capacitor. 1. Energy state model of ES device  
The operation status of all ES devices can be generally described via the following formula [38; 39]:  
\[E_{\text{ES}}^{(t)}=E_{\text{ES}}^{(t-1)}(1-\sigma)+P_{\text{ES}}^{(t)}\cdot \eta\cdot\Delta t \tag{3}\]  
where \(E_{\text{ES}}^{(t)}\) is the remaining energy of the ES device at moment \(t\), \(\sigma\) is the self-discharge rate, \(P_{\text{ES}}^{(t)}\) is the charge/discharge power during time interval \(t\), \(\eta\) is the charge/discharge efficiency, \(\Delta t\) is the duration of time interval. The energy state of the ES device is determined by \(E_{\text{ES}}\) at the start of a time interval and charged/discharged power during that period. For different ES devices, the self-discharge rate can vary from 0.1%-0.3%/d (batteries) to 20%-40%/d (super capacitor). 2. Lead-acid battery lifetime model  
The lifetime of a lead-acid battery is relatively short and is significantly influenced by operation conditions compared to super capacitors and a Li-ion battery. While the lifetime of super capacitor (SC) can extend as long as 20 years and a Li-ion battery can be charged/discharged for over 5000 cycles, substitution of a lead-acid battery may be required, which makes the lifetime calculation model of batteries essential for optimization of MG source planning. An existing lifetime calculation model [37] is applied to predict the lifetime of lead-acid batteries. The cycle time of lead-acid batteries is related to depth of discharge (DOD), as illustrated in Figure 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **ES Type** & \begin{tabular}{c} **Energy Density** \\ **(Wh/kg)** \\ \end{tabular} & \begin{tabular}{c} **Power Density** \\ **(W/kg)** \\ \end{tabular} & \begin{tabular}{c} **Investment** \\ **Cost (S/kWh)** \\ \end{tabular} & \begin{tabular}{c} **Cycle** \\ **Times** \\ \end{tabular} & \begin{tabular}{c} **Response** \\ **Speed** \\ \end{tabular} \\ \hline Energy type & \
\begin{tabular}{c} Lead-acid battery \\ Li-ion battery \\ \end{tabular} & 30–50 & 75–300 & 70–420 & \(10^{3}\) & medium \\ \hline Power type &
\begin{tabular}{c} Super capacitor \\ Flywheel \\ SMES \\ \end{tabular} & 5–130 & 400–1600 & 1400–4900 & \(10^{6}\) & fast \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of various types of energy storage (ES) devices. Figure 2: Cycle time of lead-acid batteries. DOD, depth of discharge. The calculation formula can be expressed as:  
\[N=a_{1}+a_{2}e^{a_{3}D_{N}}+a_{4}e^{a_{5}D_{N}} \tag{4}\]  
where \(D_{N}\) is the depth of discharge, \(N\) is the equivalent cycle times under \(D_{N}\), \(a_{1}\)-\(a_{5}\) are coefficients. The cost of a single charge/discharge cycle is represented as:  
\[C_{\text{cycle}}=\frac{C_{\text{cap-B}}}{N} \tag{5}\]  
where \(C_{\text{cap-B}}\) is the Capital investment of batteries, \(C_{\text{cycle}}\) is the cost of a single charge/discharge cycle. Related studies [40; 41] indicate that the cumulative lifetime of a battery is influenced by the state of charge (SOC) under which the battery is operating.