## 3 Multi-Model Modeling of the CO\({}_{2}\) Capture Process
### Identification of the Local Linear Models  
As mentioned above, the flexible operation requires the CO\({}_{2}\) capture system to adapt quickly to the variation of blue gas and the CO\({}_{2}\) capture rate demand. Therefore, the local linear models are identified at different CO\({}_{2}\) capture rates (Group I) and different mass flows of blue gas (Group II), \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controlled Variable** & **Manipulated Variable** & **Set Point Value** & \(K_{p}\) & \(K_{I}\) \\ \hline Absorber sum liquid level & Absorber sum outlet liquid mass flow & 1.25 m & 200 & 0.10 \\ Stripper sum liquid level & Stripper sum outlet liquid mass flow & 1.25 m & 200 & 0.10 \\ Condenser temperature & Cooling water mass flow & 313.15 K & 80 & 0.06 \\ Condenser pressure & Condenser outlet gas mass flow & 1.69 \(\times\) 10\({}^{5}\) Pa & 5.5 & 0.15 \\ Condenser liquid level & Condenser outlet liquid mass flow & 0.25 m & 100 & 0.10 \\ Reboiler pressure & Reboiler outlet gas mass flow & 1.79 \(\times\) 10\({}^{5}\) Pa & 5.0 & 0.15 \\ Reboiler liquid level & Reboiler outlet liquid mass flow & 0.25 m & 100 & 0.10 \\ Make-up tank liquid level & Tank outlet liquid mass flow & 1 m & 20 & 0.10 \\ Mass fraction of MEA & Make-up MEA mass flow & 0.307 & 10 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Set points and tuning parameters of the decentralized PI. \(K_{p}\): proportional coefficient, \(K_{I}\): integral coefficient. Figure 3: Overall control structure of the CO\({}_{2}\) capture process. PI: proportional integral. covering the major operating range of the CO\({}_{2}\) capture process. In Group I, the load of the power plant is fixed to 100% load, which can generate 0.13 kg flue gas per second, and the CO\({}_{2}\) capture rate varies from 50% to 95%. In Group II, the CO\({}_{2}\) capture rate is fixed to 80%, while the flue gas changes from 0.13 kg/s to 0.07 kg/s. The selected operating points in Group I and Group II for nonlinearity analysis are listed in Table 3.  
For each selected operating point, the subspace identification approach (SID) is employed to obtain the local linear model with a state-space form. Compared with the conventional transfer function model, because the state-space model has a simple expression for the multi-input multi-output system, it is very convenient to describe the CO\({}_{2}\) capture system. Moreover, the SID algorithm uses numerically efficient methods, such as QR factorization and singular value decomposition, which do not involve any nonlinear optimization techniques and thus avoids the computational issues of the conventional prediction error methods (PEM) for the identification of MIMO systems [41]. The major procedures of SID are summarized as follows:  
1. [leftmargin=*]
2. Stimulate the CO\({}_{2}\) capture process with a random identification signal, which fluctuates about \(\pm 2\%\) of the steady-state input value around the selected operating point so that the CO\({}_{2}\) capture rate varies about \(\pm 5\%\) in absolute value and the reboiler temperature varies within \(\pm 0.5\) K. The gCCS simulator can provide data at every second; however, considering the slow dynamics of the MEA-based post-combustion CO\({}_{2}\) capture process, the sampling time is selected as 30 s. The input signal is designed to change every 3000 s since a fast identification signal change is not suitable for capturing the dynamics and the steady state of the CO\({}_{2}\) capture process. 3. Construct the Hankel Matrixes using the collected input-output data, and then partition the Hankel Matrixes into past and the future block matrixes. 4. Combine the partitioned Hankel Matrixes into a new data matrix and perform QR factorization on them. 5. Calculate the subspace matrix from the factorization results of step (3) and perform singular-value decomposition (SVD) on the subspace matrix. Extract the system matrixes from the subspace matrices. More details of SID are introduced in reference [42] and not repeated here.