# Sizing Hybrid Energy Storage Systems for Distributed Power Systems under Multi-Time Scales
## 2 Modeling of Distributed Power System
### Modelling of Distributed Energy Carriers  
#### 2.2.1 Distributed Renewable Power Generation System  
As shown in Figure 1, as one of the most important part of the distributed power system, the wind turbine system and PV system play a critical role in reducing carbon emissions and system operational costs, which accelerates the pace of the installation of renewable energy generators. However, the installation of renewable generators can increase system uncertainty, which may affect the stable operation of a power system. Therefore, it is better to clearly analyze the key factors that can affect the output power of renewable energy generators. As reported by [27], the output power of a wind turbine system (\(P_{W}\)) is mainly constrained by actual wind speed (\(v\)), blade area (\(A\)), wind turbine system efficiency (\(\eta_{W}\)), and air density (\(\rho\)), and its mathematical expression can be represented as follows:  
\[P_{W}=\left\{ \begin{array}{ll}0&v<v_{in}\;\;\text{or}\;v>v_{out}\\ 0.5\times\rho\times A\times\eta_{W}v^{3}&v_{in}\leq v\leq v_{out}\end{array} \right. \tag{1}\]  
where \(v_{in}\) and \(v_{out}\) are the cut-in speed and cut-out speed of a wind turbine system, respectively. For a distributed wind turbine system, the cut-in speed is about 3 m/s, and is saturated near 10 m/s. As for the PV system, its output power (\(P_{PV}\)) is strictly influenced by the ambient temperature (\(T_{amb}\)) and equivalent solar irradiance (\(R\)), which can be expressed as [28]:  
\[P_{PV}=P_{PV,STC}\times\frac{R}{R_{STC}}\times(1-\gamma\cdot(T_{a}-T_{r})) \tag{2}\]  
Figure 1: The structure diagram of a distributed power system. where:  
\[T_{a}=T_{amb}+\frac{R}{R_{STC}}\times(T_{NOC}-20) \tag{3}\]  
In Equations (2) and (3), \(P_{PV,STC}\) is the maximum power under the standard test condition, \(R_{STC}\) represents the solar irradiance level under the standard test condition, \(\gamma\) is a coefficient, and \(T_{r}\), \(T_{a}\), and \(T_{NOC}\) are the reference temperature of the PV cells, the actual temperature of PV cells, and the temperature of PV cells under the normal operating condition, respectively. It can be concluded from Equations (1) and (2) that the output power of a wind turbine system and a PV system is primarily limited by the wind speed and the solar irradiance. However, the strong uncertainty of wind speed and solar irradiance can break the balance between power supply and demands, which can finally lead to power system outage. Therefore, many scholars have tried to develop complicated mathematical models to predict wind speed and solar irradiance in a more accurate way when sizing energy storage system capacity or optimizing system power flow, but this can increase the complexity of computation to some extent. To ease the process of predicting wind speed and solar irradiance, this paper first sizes hybrid energy storage system capacity by not considering the uncertainty of renewable energy. Then, it defines the maximum wind speed volatility (\(v_{v}\)) and solar irradiance volatility (\(R_{v}\)) to represent the uncertainty of renewable generation. Finally, it uses these two variables to evaluate the anti-interference capacity of the proposed storage system (i.e., maximum acceptable renewable energy resource volatility for the designed hybrid energy storage system). In other words, this paper first sizes the energy storage system capacity, and then evaluates the influence of renewable generation uncertainty on the proposed storage system. The mathematical expressions of wind speed volatility and solar irradiance volatility are defined as follows:  
\[v_{v}=random\|(-\alpha_{w}\times v_{h},\alpha_{w}\times v_{h})\| \tag{4}\]  
\[R_{v}=random\|(-\alpha_{s}\times R_{h},\alpha_{s}\times R_{h})\| \tag{5}\]  
In Equations (4) and (5), \(\alpha_{w}\) and \(\alpha_{s}\) are coefficients of wind speed volatility and solar irradiance volatility, respectively, while \(v_{h}\) and \(R_{h}\) represent the historical wind speed and historical solar irradiance, respectively.