# Multi-Objective Optimization of Hybrid Renewable Tri-Generation System Performance for Buildings
## 2 Constraints  
The technology constraints are generation limitations by the technology employed at each time step of the optimization (\(h,d\)). The energy generated from each technology in operation at part load conditions is between maximum and minimum capacity allowable at every time step (Equation (5)).  
\[PLR*Max\ \left(Tech_{Cap.}\right)\leq G\ _{Tech,h,d}\leq Max\ \left(Tech_{Cap.} \right)\ \forall Tech,h,d \tag{5}\]  
The energy generation and demand load constraints are the balance between the thermal energy supply and the thermal load at each time step (\(h\), \(d\)) as presented in Equation (6).  
\[\sum_{Tech}G_{h,d}(heat)\geq Dem_{h,d}(heat)\ \forall h,d \tag{6}\]  
All-time, the sum of electrical energy generation from all technologies and imported/export electricity, should satisfy the balance of electrical energy demand loads (Equation (7)).  
\[\sum_{Tech}G_{h,d}(elec.)+E(Purchase\ or\ Selling)_{h,d}=Dem(elec.)\ \forall h,d \tag{7}\]  
The relative magnitude of the energy generation (thermal and electric) is dependent on the energy demand loads and the technologies used to produce it. The generated electricity is directed to satisfy the hybrid system power needs with excess electricity exported to the grid at each time step. If the sum of generated electricity from different technologies in use is less than the electricity demand, the electricity sold to the grid is equal to zero (Equation (8)). The electricity exported to the grid is equal to the difference between the electricity generated and the sum of electricity demands for each time interval, as shown in Equation (9). At each time interval, when the electricity generated from all technology is less than the sum of the demand, the difference is imported from the grid, as shown in Equation (10).  
\[\text{If }G_{h,d}(elec.)\leq Dem_{h,d}(elec.)\rightarrow\ \Delta GT_{h,d}( elec.)=ES_{h,d}=0\ \forall h,d \tag{8}\]  
\[\text{If }G_{h,d}(elec.)>Dem_{h,d}(elec.)\ \rightarrow\ ES_{h,d}=\Delta GT_{h,d}( elec.)\ \forall h,d \tag{9}\]  
\[\text{If }G_{h,d}(elec.)<Dem_{h,d}(elec.)\ \rightarrow\ EPurchase_{h,d}=-\Delta GT_{h,d}( elec.)\ \forall h,d \tag{10}\]  
The capital cost is calculated according to Equation (11).  
\[C_{Cap.}=\sum_{Tech}Max\ (Cap._{Tech})*C_{Cap.(Tech)}*\frac{R}{1-\frac{1}{(1+R)^{ \
icefrac{{n}}{{\lambda}}}}}\ \forall\ Tech \tag{11}\]  
The grid electricity cost is calculated using Equation (12).  
\[C_{Elec.}=\sum_{d}\sum_{h}EP_{d,h}*EP_{price}\ \ \forall\ d,h \tag{12}\]  
The operation and maintenance costs are calculated based on annual fixed costs with a fraction value of the capital cost of each technology (Equation (13)).  
\[C_{OM}=\sum_{Tech}*C_{Cap.(Tech)}*fr \tag{13}\]  
The emission cost is calculated based on the tax charge per ton of emission (CAD 25/ton of CO2). The CO2 emission factor for grid electricity in Incheon (ROK) is 590 CO2e g/kWh. The generated energy by the PVT electricity reduces the CO2 emission with a value of MRCO2 (Equation (14)).  
\[C_{emission}=C_{emission\ per\ ton}*(\text{MP}_{\text{CO2}}-\text{MR}_{\text{ CO2}}) \tag{14}\]  
The revenue stream includes subsidies and incentives available for renewable technologies (Equation (15)).  
\[C_{Subs.}=\sum_{Tech}(sf_{Tech}*Cost_{cap.}(Tech))\ \forall\ Tech \tag{15}\]  
The cost of exported electricity to the grid is calculated according to Equation (16).  
\[C_{Selling}\ =\sum_{d}\sum_{h}ES_{d,h}*ES_{price}\ \forall\ d,h \tag{16}\]