# Multi-Objective Optimization of Hybrid Renewable Tri-Generation System Performance for Buildings
### Simulation, Modeling, Optimization and Analysis Methodology
#### 1.2.2 Optimization Model  
A General Algebraic Modeling System (GAMS) [61] platform is used to optimize system performance. Due to model complexity and nonlinear constraints, nonlinear programming was applied. The developed programming code utilizes a multi-objective algorithm with an optimization model of 1.5 million equations and 245,299 variables.  
The hybrid trigeneration system has multiple power flows and a converter (multiplication factor) was applied to the input power (\(P\)) to produce outlet energy flows (\(L\)), as shown  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Value** & **Unit** & **Comment** \\ \hline Infiltration & 0.15 & ACH & \\ Window to area ratio & 35 & \% & Vertical glazing of wall \\ Ventilation & 10 & L/s. person & \\ Internal equipment load & 8.07 & W/m\({}^{2}\) & \\ Internal lighting load & 10.76 & W/m\({}^{2}\) & \\ Roof U value & 0.27 & W/m\({}^{2}\) k & \\ Exterior walls U value & 0.43 & W/m\({}^{2}\) k & \\ Opaque doors U value & 3.97 & W/m\({}^{2}\) k & \\ Slab on grade Floors & 0.934 & W/m k & \\ Windows U value & 3.12 & W/m\({}^{2}\) k & Metal framing (SHGC 0.4) \\ Floor to ceiling height & 2.7 & M & \\ Attic height & 1.5 & M & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Single residential building specification.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**System** & **Signal** & **Heating Mode** & **Cooling Mode** \\ \hline AWHP & ON & T\({}_{\text{TanRB}}\leq 40\)\({}^{\circ}\)C & T\({}_{\text{room}}\leq\) T\({}_{\text{set}}-0.5\)\({}^{\circ}\)C \\  & OFF & T\({}_{\text{TanRB}}\geq 45\)\({}^{\circ}\)C & T\({}_{\text{room}}\geq\) T\({}_{\text{set}}+0.5\)\({}^{\circ}\)C \\ PVT (preheating) & ON & T\({}_{\text{PVT}}-\) T\({}_{\text{GAHX}}\geq 5\)\({}^{\circ}\)C & N/A \\  & OFF & T\({}_{\text{PVT}}-\) T\({}_{\text{GAHX}}\leq 2\)\({}^{\circ}\)C & N/A \\ \hline \hline \end{tabular}
\end{table}
Table 2: Control strategies for hybrid trigeneration system.  
in Equation (1), where \(\eta_{oi}\) is the input power flows efficiency matrix and \(F_{oi}\) is the converter units' factor. The elements of the \(\eta_{oi}\) matrix are zero when no conversion exists between the input and output energy flows. In case there is a transfer of \(P_{i}\) to \(L_{0}\), the coupling factor presents the technology's single efficiency. The coupling factor is equal to the product of the technology's efficiencies in the case of more than one energy conversion technology.  
\[\left[\begin{array}{c}L_{1}\\ \cdot\\ L_{o}\cdot\\ \cdot\\ L_{O}\end{array}\right]=\eta_{oi}*F_{oi}\left[\begin{array}{c}P_{1}\\ \cdot\\ \cdot\\ P_{i}\\ \cdot\\ P_{I}\end{array}\right],\text{ where }\eta_{oi}\ =\ \left[\begin{array}{ c c c}\eta_{11}&\cdot&\eta_{11}\\ \cdot&\eta_{oi}&\cdot\\ \eta_{OI}&\cdot&\eta_{OI}\end{array}\right] \tag{1}\]  
In addition, at each time step, the input energy for each technology is constrained by the limitation of minimum and maximum capacity as per Equation (2).  
\[P_{min}\ \leq P_{i}\leq P_{\max} \tag{2}\]  
The transferred energy (\(E_{T}\)) from the hybrid system is calculated as per Equation (3). where \(E_{o}\) is the energy output transfer from the system to the loads and \(N\) is the total number of technologies.  
\[E_{o}\ =\sum_{N}E_{T} \tag{3}\]