## 4 Optimization Design of Acceleration Control Plan
### Accelerated Process Control Plan Design of VCE  
Acceleration is the process of an engine from one thrust level to another. An important index to measure the performance of the acceleration process is acceleration time, which is mainly determined by the residual power of the engine. The main control variables of the acceleration process are the fuel flow and the nozzle area, among which the fuel quantity is the biggest factor affecting the engine acceleration performance. Increasing the fuel flow can increase the gas temperature in front of the turbine (\(T_{t4}\)), thus increasing the power capacity of the turbine. However, it should be noted that \(T_{t4}\) is limited by the strength of turbine blades and oil-rich flameout of the combustion chamber, and the sharp increase in fuel quantity will also cause engine speed to exceed limit and pose a certain threat to the mechanical load of the rotor. On the other hand, the acceleration process will lead to the decline of the compressor stability margin, so the compressor stability margin is often increased by opening the nozzle, deflating air, adjusting the guide vane of the compressor, etc. Select the starting point of the accelerated process state \(x_{0}\) and the control quantity \(u_{0}\):  
\[\left\{ \begin{array}{l}x_{0}:n_{l}=67.87\%,\;n_{h}=84.37\%\\ u_{0}:W_{f}=0.33\;\mathrm{kg/s},\;A_{8}=0.088\;\mathrm{m}^{2}\end{array} \right.. \tag{38}\]  
Set the terminal state quantity \(x_{T}\) and terminal control quantity \(u_{T}\):  
\[\left\{ \begin{array}{l}x_{T}:n_{l}=99.90\%,\;n_{h}=98.50\%\\ u_{T}:W_{f}=0.80\;\mathrm{kg/s},\;A_{8}=0.096\;\mathrm{m}^{2}\end{array} \right.. \tag{39}\]  
The constraints in the optimization process are as follows:  
\[\left\{ \begin{array}{l}n_{l}<100\%\\ n_{l}<100\%\\ T_{t4}<1850\;\mathrm{K}\\ \mathrm{SM}_{\mathrm{CDFS}}>5\%\\ \mathrm{SM}_{\mathrm{HPC}}>5\%\\ W_{f}<0.84\;\mathrm{kg/s}\\ A_{8}<0.1\;\mathrm{m}^{2}\\ \dot{W}_{f}<0.25\;\mathrm{kg/s}^{2}\\ \dot{A}_{8}<0.01\;\mathrm{m}^{2}/\mathrm{s}\end{array} \right., \tag{40}\]  
where the last two terms represent the limit on the rate of change of the control quantity. The optimization objective is the shortest acceleration time:  
\[\mathrm{min}J=\int_{0}^{T}dt. \tag{41}\]The AEMM is used as the optimization model of the accelerated process. Then the problem is modified to apply it to the target shooting method. Divide acceleration process into \(m\) intervals and the optimization variables \(y=\left(S_{0}^{T},S_{1}^{T},\ldots,S_{m}^{T},P_{0}^{T},P_{1}^{T},\ldots,P_{m}^{T },T\right)^{T}\) can be obtained. The starting and ending constraints can then be translated into the following equality constraints:  
\[F_{1}(y)_{1}=\left\{ \begin{array}{c}S_{0}-x_{0}\\ S_{m}-x_{T}\end{array} \right.=0. \tag{42}\]  
In addition, the state quantity of the starting point and the end point of each subinterval should be guaranteed to match. \[F_{1}(y)_{2}=x(S_{i},u_{i},T)-S_{i+1}=0,\;i=0,1,\ldots,m-1. \tag{43}\]  
These two expressions together form equality constraints \(F_{1}(y)\). Note that the fourth-order Runge-Kutta method is used here to calculate the state quantities at the end of each interval. In the process, the restrictions on the high and low spool speed, the gas temperature in front of the turbine, the surge margin, the change rate of the control quantity, etc. are also easily expressed in the form of \(F_{2}(y)<0\). For example, the limit on the rate of change of fuel flow can be written as:  
\[F_{2}(y)_{1}=\frac{(P_{i+1,W_{f}}-P_{i,W_{f}})\cdot T}{m}-0.3<0,\;i=0,1,\ldots,m-1, \tag{44}\]  
where \(P_{i,W_{f}}\) represents the parameterized results of fuel flow at each node. The optimization goal is transformed into:  
\[\text{min}F_{3}(y)=T. \tag{45}\]  
At this point, the problem has been transformed into an NLP problem as shown in Equation (32). Solve the above problems, and the results are shown in Figures 16-18, where CDFS represents the core drive fan stage of VCE. Figure 16: Changing curve of speed during acceleration. (**a**) Low spool speed. (**b**) High spool speed.