## 4 Simulation Results  
In this section, the proposed adaptive constrained control method based on the Wiener-type neural network presented in the above sections is used to obtain safe fuel utilization and make the SOFC system satisfy the operating constraints under the condition of changing load \(I\). At the same time, the voltage output \(V_{dc}\) is guaranteed to be steady. The effective and safe power source will provide stable electric energy for plants; thus, we must guarantee that output voltage \(V_{dc}\) is the desired constant value as much as possible, while the external current load may have a negative impact on \(V_{dc}\) for the SOFC for some time. In conventional working conditions, the current load \(I\) is set as 300 A, and our expected stable electric tension \(V_{dc}\) can be set as 332.8 V. The desired fuel utilization is \([\rho_{\min},\rho_{\max}]=[0.7,0.9]\)[20]. The maintained operating constraints are magnitude saturation \(\left[q_{f\min},q_{f\max}\right]=[0.6,1.2]\text{mol/s}\) and rate saturation \(\left[q_{f\min},q_{f\max}\right]=[-0.7,0.7]\text{mol/s}^{2}\)[20]. For the simulations, the sample time is \(T=1s\), and we choose orders of \(n_{a}=2\), \(n_{b}=1\), \(n_{c}=2\) and \(p=2\) for the Wiener-type neural network. Notice that open-loop input-output data sequences can obtain the initial values \(a_{i}(0)\), \(b_{j}(0)\), \(c_{l}(0)\) and \(\sigma_{m}(0)\). Moreover, open-loop input-output tested data samples are achieved by exciting the open-loop SOFC control system using given sinusoidal signals \(0.7823+0.3\sin(0.5t)\sin t\) and \(300+50\sin(0.03t)\sin(0.04t)\) for the fuel and the current demand (i.e., \(q_{f}\) and \(I\)), respectively. \(a_{i}(0)\), \(b_{j}(0)\), \(c_{l}(0)\) and \(\sigma_{m}(0)\) can be obtained as:  
\[\left[ \begin{array}{cc}a_{1}&a_{2}\\ b_{0}&b_{1}\\ c_{1}&c_{2}\\ \sigma_{1}&\sigma_{2}\end{array} \right]=\left[ \begin{array}{cc}-0.4573&-0.22 86\\ -0.0018&0.0018\\ 0.1278&0.1278\\ 1.6567&-0.6735\end{array} \right]\]  
For the proposed control law, \(\alpha\) is chosen as mathematical constant e. The learning rates are selected as \(\eta_{w1}=[0.7,0.7,0.8,0.8]^{T}\) and \(\eta_{w2}=[15,10,0.1]^{T}\). The initial values of the proposed controller are the same as the traditional PID parameters \(K_{p},K_{i},K_{d}\), i.e., \(k_{1}(0)=K_{p}=10\), \(k_{2}(0)=K_{i}=2\) and \(k_{3}(0)=K_{d}=0.01\). The simulation results are given from Figures 6-10. Firstly, the load current is assumed to change as the curve in Figure 6A with three step signals combined. Under the most stringent step signal conditions, the proposed control algorithm presents a favorable control performance, which is shown in Figure 6B. It shows that once there exists a large mutation of the load current, the corresponding oscillations occur on the stack output voltage, yet disappear soon under the proposed control method,  
Figure 5: Structure of the presented control strategy for SOFC. and the voltage tends to be stable. The curves of the natural gas flow rate (i.e., the input) and the acceleration are drawn in Figure 7. As can be seen in Figure 7A, the input magnitude constraint generates a saturation phenomenon around Step 200, and the acceleration is within the confining scope. Once the constraints occur, the proposed anti-windup compensator takes effect, as shown in Figure 8A, at the same time to compensate the performance. The Jacobian information and fuel utilization are depicted in Figure 8B,C, which indicates that the fuel utilization is satisfactory within the excellent range under the proposed control strategy. The curves of the exponential PID parameters and the Wiener-type neural network weight coefficients are presented in Figures 9 and 10, respectively. Moreover, the conventional PID controller is adopted in the simulation in contrast with the proposed control method. The comparisons focus on the following points:  
1. In Figure 6B, it can be vividly seen that the voltage under PID control has a greater oscillation amplitude and slower convergence speed than the proposed control method.