## 4 Cost Calculation Method  
This section explains the method for estimating the cost of the system that is composed of the ESS and the WF operated by KE control. The cost during the lifespan of a WG is investigated in this analysis. Figure 5: Optimal kinetic energy control. We define the installation cost of WF, ESS, an opportunity loss of WF, ESS, and cost of power converter system (PCS) of the battery in order to calculate the total cost of the system composed of the ESS and the WF operated by KE control, as follows:  
Installation cost of WF:  
\[C_{\text{WF}}^{\text{I}}\left[\text{JPY}\right]=n\times P_{\text{n}}\left[\text {kW}\right]\times\text{WG price }\left[\text{JPY/kW}\right], \tag{21}\]  
Opportunity loss of WF:  
\[C_{\text{WF}}^{\text{O}}\left[\text{JPY}\right]=E_{\text{WF}}^{\text{loss}} \left[\text{kWh/h}\right]\times\text{Lifespan of WG }\left[\text{h}\right]\times\text{Wind power price }\left[\text{JPY/kWh}\right], \tag{22}\]  
Installation cost of ESS:  
\[C_{\text{ESS}}^{\text{I}}\left[\text{JPY}\right]=\frac{\text{Rated capacity }\left[\text{kWh}\right]\times\text{Battery price }\left[\text{JPY/kWh}\right]}{\text{Battery lifespan }\left[\text{h}\right]}\times\text{Lifespan of WG }\left[\text{h}\right], \tag{23}\]  
Opportunity loss of ESS:  
\[C_{\text{ESS}}^{\text{O}}\left[\text{JPY}\right]=E_{\text{ESS}}^{\text{loss }}\left[\text{kWh/h}\right]\times\text{Lifespan of WG }\left[\text{h}\right]\times\text{Wind power price }\left[\text{JPY/kWh}\right]. \tag{24}\]  
Cost of PCS:  
\[C_{\text{PCS}}\left[\text{JPY}\right]=\text{PCS price }\left[\text{JPY/kW}\right]\times\text{Rated power of ESS }\left[\text{kW}\right]. \tag{25}\]  
For simplicity, the maintenance cost is neglected. The energy loss \(E_{\text{WF}}^{\text{loss}}\) is defined by \(\int_{0}^{1\text{h}}[P_{\text{WF}}^{\text{MPPT}}(t)-P_{\text{WF}}^{\text{MA} }(t)(\text{or }P_{\text{WF}}^{\text{opt}}(t))]\text{d}t\) which implies the energy lost by the KE control. The rated capacity of the ESS [kWh] is determined by the rated power of the ESS that was obtained by the simulations. Although the cost of an ESS is determined by the rated capacity, it is constrained by the rated power of the ESS through the so-called C-rate [14; 26]. The C-rate is given by  
\[\text{C-rate}=\frac{\text{Rated power of ESS }\left[\text{kW}\right]}{\text{Rated Capacity of ESS }\left[\text{kWh}\right]}. \tag{26}\]  
In (24), \(E_{\text{ESS}}^{\text{loss}}\) is the integral of the charge/discharge loss \(\int_{0}^{1\text{h}}P_{\text{loss}}(t)\text{d}t\). Battery lifespan depends on the calendar life \(L^{\text{cal}}\) and cycle life \(L^{\text{cyc}}\). \(L^{\text{cal}}\) is the lifespan due to the deterioration over time when the ESS does not charge/discharge. By contrast, \(L^{\text{cyc}}\) is the lifespan due to the deterioration when ESS charges/discharges and it is influenced by the number of charge/discharge and the depth of discharge (DOD) [27]. Figure 7 shows the relation between the number of cycles (charge/discharge) and the DOD [28]. \(L^{\text{cyc}}\) can be estimated by the rain-flow counting [29; 30] and Figure 7 from the time responses obtained by simulation. Refer to [29; 30] for the detailed calculation method of \(L^{\text{cyc}}\).