# Hierarchical Coordinated Predictive Control of Multiagent Systems for Process Industries
## 4 Case Study: Multisource High-Power Microwave Heating Process  
### Microwave Heating Process and Device  
The process of heating materials using microwaves relies on the direct interaction between microwaves and internal polarized molecules, and the governing equations for this process are shown in Appendix B[33]. The governing equations of the microwave heating process indicate that the nonlinear dynamics of the microwave heating comes mainly from the complex physical properties of heated material and complex electromagnetic field. As a result, multisource microwave heating is a highly coupled process and is very complex to control. The established microwave heating device is shown in Figure 2. It consists of a rectangular heating cavity, six microwave sources with a WR340 waveguide and five temperature sensors. The heated material is placed in the middle of a piece of polycrystalline mullite fiberboard, which is placed on the bottom center of the heating cavity. The size of the established microwave heating device and the number of microwave sources are labeled in Figure 2. Each microwave source converts the input electric power (IEP) into microwave energy and feeds the microwaves into the heating cavity through the related waveguide to heat the material. The heated material is an alumina ceramic block, and its size is labeled in Figure 2. The physical parameters [34] are shown in Table 1. Five temperature sensors were used to measure the temperature of the heated material, and the arrangement of the temperature measurement points is shown in Figure 3. The influence of polycrystalline mullite fiberboard on the microwave heating process can be ignored because of its characteristics, such as high-temperature resistance, heat preservation and microwave-transparent. Corresponding to the proposed HCPC, there are three types of constraints in the microwave heating process: (i) constraints on the total IEP of the microwave sources, (ii) constraints on the IEP increments of the microwave sources and (iii) constraints on the temperature of the material. The microwave frequency was set to 2.45 GHz, and the initial temperature \(T_{0}=20\)\({}^{\circ}\)C was used for trial heating to obtain the relevant parameters of the open-loop and closed-loop prediction models. The obtained energy-conversion efficiency of each microwave source, the steady-state temperature rise per unit power (i.e., the increase in steady-state temperature that can be induced by a 1 W change in the IEP) at each temperature measurement point and the average steady-state temperature rise of all the temperature measurement points are shown in Table 2. Note that a higher steady-state temperature rise represents a greater heating capacity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Real part of the complex permittivity & \(\epsilon^{\prime}\) & 9.8 & \(-\) \\ Loss factor & \(\epsilon^{\prime\prime}\) & \(3.7276\times 10^{-3}\times 10^{7/700}\) & \(-\) \\ Thermal conductivity & \(\kappa\) & 29.7 & W/(m\({}^{\circ}\)C) \\ Density & \(\rho\) & 3970 & kg/m\({}^{3}\) \\ Heat capacity & \(C_{\rm P}\) & 774.9 & J/(kg\({}^{\circ}\)C) \\ Convective heat transfer coefficient & \(h\) & 10 & W/(m\({}^{2.\circ}\)C) \\ Surface emissivity & \(\varphi_{\rm se}\) & 0.599 & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the heated material. Figure 3: Arrangement of temperature measurement points: (**a**) Top view. (**b**) Side view. Figure 2: Geometric model of the microwave heating device. Let the cost coefficient of each agent and the payoff coefficient of each temperature measurement point be 1. The normalized cost of the unit power corresponding to each microwave source agent, as shown in Table 2, is obtained by substituting the cost coefficients, payoff coefficients and steady-state temperature rise of unit power, as shown in Table 2, into the normalized cost (22c), taking 1W as the IEP increment of each microwave source agent.