## 2 Optimization Model
### Thermal Model
#### 2.3.2 Temperature Variation Model  
The temperature distribution inside ice sports venues is a complex phenomenon. To simulate the temperature changes in the spectator stands, many factors are considered, such as the ventilation system, the operation of radiant heating systems, heat loss through walls, air leakage, lighting, audience-generated heat, and the impact of heat loads such as radiative heat transfer between different areas [40; 41]. The temperature variation model is given as follows:  
\[\theta_{z}(t)=\theta_{z}(t-1)+\theta_{ht}(t)+\theta_{em}(t)+\theta_{ven}(t)+ \theta_{light}(t)+\theta_{and}(t)+\theta_{ref}(t) \tag{28}\]  
where \(\theta_{z}(t)\) is the ambient temperature at time \(t\) (K). \(\theta_{z}(t-1)\) is the ambient temperature at time \(t-1\) (K). \(\theta_{ht}(t)\) is the effect of radiant heating on temperature during an interval (K). \(\theta_{en}(t)\) is the effect of outdoor air heat conduction through the envelope on temperature during an interval (K). \(\theta_{ven}(t)\) is the effect of cooling air conditioning operation on temperature during an interval (K). \(\theta_{light}(t)\) is the effect of lighting on temperature during an interval (K). \(\theta_{aud}(t)\) is the effect of body heat dissipation on temperature during an interval (K). \(\theta_{ref}(t)\) is the effect of ice temperature through thermal radiation on temperature (K). \(\theta_{ht}(t)\) is given as follows:  
\[\theta_{ht}(t)=\frac{P_{dh}S_{ht}(t)\tau\times 3.6}{m_{z}\times C_{Z}} \tag{29}\]  
where \(m_{z}\) is the mass of the air in the spectator stands (kg). \(C_{Z}\) is the specific heat capacity of air (commonly taken as 1.005 kJ/(kg\(\cdot\)K)). \(m_{z}\) is given as follows:  
\[m_{z}=V\times\rho \tag{30}\]  
where \(V\) is the space volume (m\({}^{3}\)). \(\rho\) is the air density (kg/m\({}^{3}\)). \(\theta_{en}(t)\) is given as follows:  
\[\theta_{en}(t)=\frac{(UA+\dot{m}C_{Z})(\theta_{ex}(t)-\theta_{z}(t-1))\tau}{m _{z}\times C_{Z}} \tag{31}\]  
where \(U\) is the heat transfer coefficient of the building envelope (kW/(m\({}^{2}\cdot\)K)). A is the surface area of the building envelope (m\({}^{2}\)). \(\dot{m}\) is the air leakage flow rate (m\({}^{3}\)/h). \(\theta_{ex}(t)\) is the outdoor air temperature at time \(t\) (K). \(\theta_{ven}(t)\) is given as follows:  
\[\theta_{ven}(t)=\frac{Q_{ven}\times\tau}{m_{z}\times C_{Z}} \tag{32}\]  
where \(Q_{ven}\) is the heat transferred by the ventilation system (K). \(Q_{ven}\) is given as follows:  
\[Q_{ven}=\dot{v}\times C_{Z}\times S_{ven}(t)(\theta_{z}^{gen}-\theta_{z}(t-1)) \tag{33}\]  
where \(\theta_{z}^{gen}\) is the setpoint temperature for ventilation system (K). \(\dot{v}\) is the air mass flow rate (kg/h). \(\theta_{light}(t)\) is given as follows:  
\[\theta_{light}(t)=\frac{P_{light}\times\tau}{m_{z}\times C_{Z}} \tag{34}\]  
where \(P_{light}\) is the total power of the light in the spectator stands (kW). \(\theta_{aud}(t)\) is given as follows:  
\[\theta_{aud}(t)=\frac{q_{rad}\times N_{aud}(t)\times N_{max}}{m_{z}\times C_{Z}} \tag{35}\]  
where \(q_{rad}\) is the heat generated by indoor occupants (W/(h\(\cdot\)Person)). \(N_{aud}(t)\) is the schedule of audience occupancy as a percentage (%). \(N_{max}\) is the maximum capacity of the audience (Person). According to the Stefan-Boltzmann thermodynamic law, the total energy radiated per unit area from the surface of a black body per unit time is proportional to the fourth power of the black body's thermodynamic temperature T. Thermal radiation originating from the near-ice area, ceiling, and spectator stands contributes to the temperature rise in the spectator stands. To better calculate the thermal radiation between these areas, the near-ice area is designated as zone X, the ceiling area as zone Y, and the spectator stands as zone Z. The influence of the thermal radiation from the ice temperature on the temperature of the spectator stands (zone Z) is calculated using shape factors (F) and surface radiation quantities (Q).