## 2 Optimization Model
### Objective Functions
#### 2.1.2 Economic Objective  
The costs include operational costs and environmental costs. Operational costs cover electricity expenses, startup and shutdown-related costs, and maintenance costs. \[F_{2}=min(C_{ECO}+C_{sco}) \tag{2}\]  
where \(C_{ECO}\) is operational cost and \(C_{sco}\) is environmental cost. The operational cost (\(C_{ECO}\)) is given as follows:  
\[C_{ECO}=\sum{{}_{t\in T}}(\tau C_{E}(t)+\tau C_{OM}(t)+\tau C_{SC}(t)) \tag{3}\]  
where \(C_{E}(t)\), \(C_{O}(t)\), and \(C_{S}(t)\) are cost of electricity consumption, maintenance, startup and shutdown of machine in time period \(t\), respectively [34]. \[C_{E}(t)=C_{E}^{pen}(t)+C_{E}^{dt}(t)+C_{E}^{ht}(t) \tag{4}\]  
\[C_{OM}(t)=C_{OM}^{pen}(t)+C_{OM}^{dt}(t)+C_{OM}^{ht}(t) \tag{5}\]  
\[C_{SC}(t)=C_{SC}^{pen}(t)+C_{SC}^{dt}(t)+C_{SC}^{ht}(t) \tag{6}\]  
where \(C_{E}^{pen}(t)\), \(C_{E}^{dt}(t)\), and \(C_{E}^{ht}(t)\) are electricity costs of ventilation system, dehumidification, and radiant heating, respectively. \(C_{OM}^{pen}(t)\), \(C_{OM}^{dt}(t)\), and \(C_{OM}^{ht}(t)\) are maintenance costs of ventilation system, dehumidification, and radiant heating, respectively. \(C_{SC}^{pen}(t)\), \(C_{SC}^{ht}(t)\), and \(C_{SC}^{ht}(t)\) are startup and shutdown costs of the ventilation system, dehumidification, and radiant heating, respectively. The ventilation system supplies air at a specified temperature and has a cooling unit. When the outdoor temperature exceeds the setpoint of the ventilation system, the cooling function is activated. When the outdoor temperature is below the setpoint, cooling is not applied. \[C_{E}^{pen}(t)=\left\{ \begin{matrix}\frac{Q_{max}}{C_{CO}}\times C_{ele}(t),\, \theta_{ex}(t)\geq\theta_{z}^{ven}\\ C_{ele}(t)\times P_{ven}\times S_{ven}(t),\,\theta_{ex}(t)<\theta_{z}^{zen} \end{matrix} \right. \tag{7}\]  
\[C_{E}^{dt}(t)=C_{ele}(t)\times P_{dh}\times S_{dh}(t) \tag{8}\]  
\[C_{E}^{ht}(t)=C_{ele}(t)\times P_{ht}\times S_{ht}(t) \tag{9}\]where \(Q_{ven}\) is the cooling load provided by the ventilation system. \(COP\) is the coefficient of performance of the ventilation system. \(C_{ele}(t)\) is the electricity price at time \(t\). \(P_{ven}\), \(P_{dh}\), and \(P_{ht}\) are the power of the ventilation system, dehumidifier, and radiation heaters, respectively (kW). \(S_{ven}(t)\), \(S_{dh}(t)\), and \(S_{ht}(t)\) are binary variables for controlling the on/off state of the ventilation with a cooling unit, dehumidifier, and radiation heaters, respectively. \[C_{OM}^{ven}(t)=C_{om-ven}(t)\times C_{E}^{ven}(t) \tag{10}\]  
\[C_{OM}^{ht}(t)=C_{om-dh}(t)\times C_{E}^{dh}(t) \tag{11}\]  
\[C_{OM}^{ht}(t)=C_{om-ht}(t)\times C_{E}^{ht}(t) \tag{12}\]  
where \(C_{om-ven}(t)\), \(C_{om-dh}(t)\), and \(C_{om-ht}(t)\) are cost coefficients of maintenance of ventilation, dehumidifier, and radiation heaters, respectively (CNY\(\cdot\)kWh\({}^{-1}\)). \[C_{SC}^{ven}(t)=C_{sc-ven}(t)\times C_{E}^{ven}(t) \tag{13}\]  
\[C_{SC}^{dh}(t)=C_{sc-dh}(t)\times C_{E}^{dh}(t) \tag{14}\]  
\[C_{SC}^{ht}(t)=C_{sc-ht}(t)\times C_{E}^{ht}(t) \tag{15}\]  
where \(C_{sc-ven}(t)\), \(C_{sc-dh}(t)\), and \(C_{sc-dh}(t)\) are cost coefficients of startup and shutdown of ventilation, dehumidifier, and radiation heaters, respectively (CNY\(\cdot\)kWh\({}^{-1}\)). \(\forall t\in T\). \(\tau\) is a time interval (this study took 1 h (h)). Environmental costs are driven by energy consumption, which, in the context of ice sports venues, primarily results from electricity usage. The most important emissions considered in the power generation industry, due to their effects on the environment, are SO\({}_{2}\), CO\({}_{2}\), and NO\({}_{\mathsf{x}}\), which are modeled technically by the polynomial functions for the SO\({}_{2}\) emission, an emitted pollution model in tons per MW for CO\({}_{2}\), and an exponential function for NO\({}_{\mathsf{x}}\) emissions [35; 36].