## Problem Formulation
### Mathematical Description of BESS
#### 2.3.2 Constraints of BESS Accounting to the Scheduling  
The wind-energy union system participating in the scheduling aims to achieve the coordination with conventional units, that is, give full play to BESS on conventional units of the complementary role, while trying to avoid BESS frequent charge and discharge. Generally, the rated power and energy capacity are two key indicators in describing the sizing of BESS [25]. Under the condition of allowing wind power curtailment, the charge/discharge power of BESS can be illustrated as Formula (8):  
\[P_{s,t}=P_{wind,t}-P_{union,t}-P_{wloss,t} \tag{8}\]  
where, \(P_{s,t}>0\) refers to charging, and \(P_{s,t}<0\) refers to discharging.  
The net load of systems with wind-energy union system integration can be rewritten, as follows:  
\[P^{\prime}_{net,t}=P_{load,t}-P_{wind,t}-P_{s,t} \tag{9}\]  
The state of charge (SOC) of BESS at hour \(t\) can be expressed, as follows:  
\[\left\{\begin{array}{ll}S_{soc,t}=S_{soc,t-1}+\eta_{s}P_{s,t}\Delta t,&P_{s,t}>0\\ S_{soc,t}=S_{soc,t-1}+\frac{P_{s,t}}{\eta_{s}\Delta t},&P_{s,t}<0\end{array}\right. \tag{10}\]  
Set \(\lambda_{t}\) as the charge/discharge status of BESS at time \(t\), and \(\lambda_{t}\) can only have one state in each time period, that is:  
\[\lambda_{1},\lambda_{2},\cdots\lambda_{t}\in\{-1,0,1\} \tag{11}\]  
where, \(\lambda_{t}=0\) means the BESS being the idle float status; \(\lambda_{t}=1\) means the BESS being the discharging status; and, \(\lambda_{t}=1\) means the charging status.  
Subject to the limitations of the rated charge/discharge power and rated energy storage capacity of BESS, the constraints of BESS at time \(t\) are shown, as follows:  
\[\left\{\begin{array}{ll}|P_{s,t}|\leq\eta_{s}P_{cap}\\ 0\leq S_{soc,t}\leq S_{cap}\end{array}\right. \tag{12}\]  
Equation (12) limit the fluctuation range of wind power in adjacent scheduling intervals, which alleviates the additional peaking demand that is caused by wind power integration. If BESS can keep a certain reserve capacity for the uncertainty of wind output, it will further alleviate the additional spinning reserve requirement. Thus, constraints of BESS that adapting to the scheduling can be expressed as:  
\[\left\{\begin{array}{ll}-\eta_{s}P_{cap}+\Delta P^{dn}_{wind,t}\leq P_{s,t} \leq\eta_{s}P_{cap}-\Delta P^{up}_{wind,t}\\ \Delta P^{dn}_{wind,t}\Delta t\leq S_{soc,t}\leq S_{cap}+\ \Delta P^{up}_{wind,t} \Delta t\end{array}\right. \tag{13}\]  
It should be noted that when BESS provides spinning reserve for wind power forecast error, conventional units only need to reserve the load forecast error, that is, only \(\triangle P_{load,t}\) need to be retained in the Equation (3) given in Section 2.1.2.  
As shown in Formula (13), the BESS capacity for wind farm will be increased while considering BESS as reserve capacity for the wind uncertainty, which further increases the investment cost of the wind-energy union system. However, the equivalent cycle life of BESS can be expanded under this situation of shallow charge/discharge (detailed analysis shown in Section 2.3.1). It is equivalent to decrease the unit investment cost of BESS among the full cycle life to some extent. In addition, BESS providing the spinning reserve capacity will also receive more additional auxiliary services compensation benefit. Thus, how to reasonably consider the BESS reserve degree for the wind uncertainty is also the problem to be discussed in the following research.