Linyun JIA,Yuchun CHEN,Ronghui CHENG,Tin TAN,Kern SONG

a School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China

b Aero-engine Corporation of China, Shenyang Engine Research Institute, Shenyang 110066, China

KEYWORDS Acceleration and deceleration;Control schedule optimization;Steady-state reverse method;Transient-state reverse method;Variable cycle engine;Virtual power extraction method

Abstract Studies show that different geometries of a Variable Cycle Engine(VCE)can be adjusted during the transient stage of the engine operation to improve the engine performance.However,this improvement increases the complexity of the acceleration and deceleration control schedule. In order to resolve this problem,the Transient-state Reverse Method(TRM)is established in the present study based on the Steady-state Reverse Method (SRM) and the Virtual Power Extraction Method (VPEM). The state factors in the component-based engine performance models are replaced by variable geometry parameters to establish the TRM for a double bypass VCE.Obtained results are compared with the conventional component-based model from different aspects,including the accuracy and the convergence rate. The TRM is then employed to optimize the control schedule of a VCE. Obtained results show that the accuracy and the convergence rate of the proposed method are consistent with that of the conventional model. On the other hand, it is found that the new-model-optimized control schedules reduce the acceleration and deceleration time by 45%and 54%,respectively.Meanwhile,the surge margin of compressors,fuel-air ratio and the turbine inlet temperature maintained are within the acceptable criteria. It is concluded that the proposed TRM is a powerful method to design the acceleration and deceleration control schedule of the VCE.

1. Introduction

Variable Cycle Engine (VCE) is regarded as a promising propulsion system to empower supersonic cruise aircrafts.Considering superior characteristics of the VCE, it attracts many scholars so that it becomes a hot spot in the field of aeronautical propulsion. Different configurations of a VCE have been proposed so far to improve the steady-state and transient performance of the engine.However, studies show that diverse features remarkably increase the complexity of the VCE. This is especially more pronounced for designing the acceleration and deceleration control schedules, which are essential issues in designing the VCE.

Reviewing the literature indicates that different studies have been performed so far for designing the acceleration and deceleration control schedule of the VCE. Connolly et al.designed a fan speed controller for a VCE, while the VCE has a variable nozzle and Variable Guide Vanes (VGV) that are set on schedules with no active control. Montazeri et al.proposed a new design and the corresponding hardware implementation for the predictive control algorithm.Then they evaluated the performance of the hardware-in-the-loop for controlling a turbofan engine.Imani and Montazeriproposed a strategy to design linear regulators for Min-Max selector control to improve the transient limit protection of the gas turbine.Chen et al.proposed a new approach to establish a standard for evaluating the performance of components in variable cycle engines. Wang et al.investigated the control strategy and the influence of the guide vane angle on the performance of different tri-axial gas turbines.Kimand Gallaret al.analyzed the influence of optimum scheduling of variable guide vanes of compressors on a turbofan engine at design and off-design working conditions. Zhou et al.studied different models of VCEs and analyzed the performance of each model during the transition mode. Furthermore, Xie et al.proposed a complete distributed control framework for the VCE and verified its adaptability and robustness through the numerical simulation in the Simulink software.Zheng et al.studied the operating mechanism of the VCE and analyzed the influence of variable geometries on the matching and performance of the Adaptive Cycle Engine (ACE). Sun et al.introduced the inlet positive control scheme and developed an optimal control method for the acceleration stage in the propulsion system of supersonic cruises. Recently, Li et al.2studied adaptive control methods of VCEs and designed several adaptive controllers accordingly.

In the majority of reviewed researches,design procedures of the acceleration and deceleration control schedule of the engine are either ignored or directly determined as input parameters.A widely adopted method for designing the acceleration and deceleration control schedule of the engine is to solve a single variable dynamic functional optimization problem.When this method is further extended to design the acceleration and deceleration control schedule for VCEs,it becomes a matter of solving multivariable dynamic functional optimization problems.

The Virtual Power Extraction Method(VPEM)established by Chen et al.offers new ideas to design the transient control schedule for gas turbines. Since then, the VPEM has been remarkably developed so that it is applied in different types of gas turbines, including the turbojet,turbofan,turbopropturboshaftand variable cycle engines.Studies show that the VPEM is a powerful scheme to design the transient control schedule during the engine start,acceleration and deceleration processes.However, it is found that the VPEM can only be applied for fixed-geometry gas turbines.Kurzke and Halliwellemployed the steady-state power offtake method, which is similar to the VPEM, to analyze the control schedule of gas turbines.

Furthermore, Jia et al.established the Steady-state Reverse Method (SRM) for designing the multivariable steady-state control schedule of VCEs. The main feature of this method is to replace the state factors in the engine matching equation group with variable geometry parameters. Once the engine off-design balancing is performed, all variable geometry parameters can be calculated accordingly. This ensures excellent convergence and can be applied to clearly interpret the optimized results.

In the present study,it is intended to develop the SRM and VPEM to establish the Transient-state Reverse Method(TRM).To this end,state factors will be replaced with variable geometry parameters in the VPEM.

Contents of this article are divided into five sections as the following:The basic principle of the VPEM and TRM,and the establishment of the TRM model for a double bypass VCE are presented in Section 2.Then the proposed TRM model will be evaluated in Section 3 from different aspects, including the convergence rate and accuracy. Moreover, the acceleration and deceleration control schedule optimization model will be established in Section 4 and then it will be employed to optimize the VCE acceleration and deceleration control schedule.Finally, the designed control schedule and the relevant transient performance will be compared with conventional methods to show the feasibility and advantages of the TRM.

2. Establishment of the transient-state reverse method

In this section, it is intended to review the VPEM and SRM and combine them to establish the TRM for a double-bypass VCE. Fig. 1 illustrates the configuration of the doublebypass VCE with the Core Driven Fan Stage (CDFS).

Fig.1 indicates that the main geometrical parameters of the VCE are the fan stator angle α, CDFS stator angle α,compressor stator angle α,mode select valve area A,Forward Variable Area Bypass Injector (FVABI) area A, Rear Variable Area Bypass Injector (RVABI) area A, throat area of the high-pressure turbine nozzle guide vane A, throat area of the low-pressure turbine nozzle guide vane Aand the nozzle throat area A.

2.1. Reviewing the SRM

The SRM is a component-based performance simulation model for gas turbines with variable geometrical parameters,such as VCE. Studies show that parameters defining the VCE state can be divided into two main categories as the following: The first category Xcontains the state factors,including n, n, T, β, β, β, πand π. On the other hand, the second category Xconsists of geometrical parameters, including α, α, α, A, A, A, Aand A.The conventional off-design models consider Xas inputs,while the state factors in Xare variables in the off-design equation group. The SRM can be established by substituting the variable geometry parameters for state factors in the offdesign point of the engine. In comparison, the known inputs and variables of the equation group for the SRM and conventional models are exchanged. This feature gives the SRM the advantage of knowing the state factors such as n, n, T,β, β, βbefore the off-design simulation, which improves the convergency of off-design models and avoids the over-speed, over-temperature or surge.

Fig. 1 Configuration of double-bypass VCE with CDFS.

The substitution must be conducted in accordance with the sensitivity matrix between the state factors and the variable geometry parameters. Different SRM models have been proposed so far,where five models for the CDFS VCE in the double bypass mode are established in the study of Jia et al..Table 1 presents the controlled parameters, the variables and the error equations of these models.

In Table 1, nand nrepresent the rotational speeds of the two rotors, while β, βand βindicate the β values of the fan, CDFS and the compressor, respectively. Moreover, π,πdenote pressure ratios of the high-pressure and lowpressure turbines, respectively. These models have the same error equations i.e. Y=[δW, δW, δL, δL, δp, δW,δW]. Where δWand δWrepresent error terms of the mass flow balance at the inlet of high-pressure and lowpressure turbines, respectively. Moreover, δLand δLare error terms of the power balancing of the two rotors. Meanwhile, δpindicates the error term of the pressure balance at the mixer entrance. Finally, δWand δWdenote the error term of the mass flow balance at the FVABI and the nozzle throat, respectively.

For the single-bypass mode, δWshould be replaced by the error term of the mass flow balance between the fan and the CDFS δW,while the controlled parameters and the variables remain the same.

When performing the engine off-design balancing, controlled parameters are given as the input, while variables and error equations Y form a 7-dimensional nonlinear system of equation. Then the Newton-Raphson iterative method is applied to solve this system of equation. The number of substitutes increases gradually from model No.1 to model No.5, where the later model simultaneously controls n, n, T, β, β, β, while A, A, A, A,αcan be calculated through the engine off-design balancing. Consequently, the operating point of the VCE is determined by the controlled parameters, which ensures a reasonable convergence rate and prevents the overs-peed,over-temperature and the surge.

2.2. Reviewing the VPEM

The VPEM is a gas turbine engine steady-state performance simulation model which is able to simulate the transient performance of gas turbines. In this regard, virtual power extractions are performed on both high-pressure and low-pressure rotors, which changes the engine operating point and unbalanced power on both rotors. Moreover, the VPEM is well developed so that transient parameters such as β, Tand the Fuel-Air Ratio at combustor outlet FARduring the transient state can be given as the input to calculate the engine operating point and unbalanced power. Table 2 presents the fixed-geometry parameters of the VPEM for a CDFS VCE.It is observed that the power balance of the compressors and turbines,including δLand δL,are removed from error equations. Therefore, the unbalanced power approximately equals the acceleration/deceleration power of the rotor. Rotational speeds nand nare the controlled parameters to determine the engine rating. The correlation between nand ncan be approximately obtained through the steady-state throttling.Moreover,β,Tand FARare given as the controlled parameter to determine the surge margin of the compressor, turbine inlet total temperature and the fuel-air ratio, respectively.These parameters are key constraints for the transient maneuver of the gas turbine.

Consequently, the dimension of the system of the equation for the engine off-design balancing reduces to 5.For the singlebypass mode,δWshould be replaced by the error equation of the mass flow balance between the fan and the CDFS δW, while the controlled parameters and the variables remain the same.

In order to design the transient control schedule,the throttling performance of the relevent engine is initially calculated to find the correlation between nand n. Then, n, nand β,Tor FARare given as inputs to perform the VPEM calculation,which deduces the fuel mass flow rate Wand the engine acceleration ratio. Finally, the obtained Wor acceleration rate is described as the transient control schedule, which is ready to control the transient maneuver of the engine.

Two simplifications are considered in the VPEM, which cause deviation from the actual problem. The first simplification is the assumed correlation between nand nof the steady-state throttling.In the other simplification,the transient phenomenon,such as the volume effect is ignored.Chen et al.proved that deviations caused by the volume effect are conservative and deviations caused by these approximations are within the acceptable range. Moreover, the deviation caused by ncan be eliminated by replacing nin the VPEM with thetransient-state simulation and repeating the VPEM calculation procedure.

Table 1 Settings of parameters in SRM model for a CDFS VCE (Double-bypass mode).

Table 2 Setups of different VPEM models for a double bypass VCE (Double-bypass mode).

It is worth noting that variable geometrical parameters are not considered in the VPEM. Therefore, the VPEM can only be applied to design the control schedule for fixed geometry engines or VCEs, where the control schedule of variable geometrical parameters is already known.

2.3. The TRM for a double-bypass VCE

The main idea of the TRM model is to find substitutes for variable geometrical parameters in state factors in the fix-geometry VPEM.Consequently,the TRM inherits all the simplifications and calculation procedures of the VPEM.

Substituted parameters of the TRM should follow the same principles as the SRM.In other words,each variable geometrical parameter must have a strong correlation with the replaced state factor. Otherwise, the established model will not converge. For example, the TRM with variable Aand Acan be established by substituting the Awith β.However, it will not converge, because Ais poorly related to β. In other words, it is a great challenge to reach a given βvalue by only adjusting A.

Studiesshow that in a double-bypass VCE, Ahas a significant influence on the gas generator operating point β.Meanwhile, Ahas a significant influence on the operating point of the fan β,while it does not affect the operating point of the compressor. It is found that Acan change the power distribution between the high-pressure and low-pressure turbines. Moreover, αis an effective indicator to adjust the operating point of the CDFS. On the other hand, αand αvary in accordance with their corrected rotational speeds.Furthermore, FVABI is automatically adjusted according to the CDFS pressure ratio.It is worth noting that the control schedule of the FVABI is discussed in details in Ref. 32. For the RVABI,Jia et al.has proved that Ahas a negligible impact on VCE transient performance. Accordingly, Ais not optimized during the acceleration and deceleration. It should be indicated that the control schedule of Aduring the transient state is the same as that of the steady state.

In the TRM,parameters A,Aand αsubstitutes for β,βand β,respectively.Moreover,Ais applied as an independent variable, which will be optimized through the optimization model. Based on the abovementioned substitutions,parameters of the TRM for the VCE are listed in Table 3.It should be indicated that equation Y is expressed as [δW,δW,δp,δW,δW]in the double bypass mode,while it changes to [δW, δW, δp, δW, δW] in the single bypass mode. Furthermore, δLand δLare removed from error equation Y similar to that in the VPEM.

When n,n,β,β,and βare given,the TRM actually defines the operating point of the compressor prior to the calculation, which ensures that no surge occurs during the transient-state. On the other hand, when Tis given, no over-temperature occurs during the acceleration. Since either Tor Wcan be applied to determine the outlet conditions of the combustor, the value of the parameter Tin the model is set equivalent to that of W.On the other hand,when Wis the controlled parameter instead of T,the fuel-air ratio of the combustor can be simply controlled, which ensures no rich/lean flameout. Once the engine reaches the steady-state offdesign balancing, different design parameters, including the additional power, fuel mass flow and geometrical parameters can be calculated through the TRM.

The TRM offers an idea to design the transient control schedule for VCEs, including the control schedule for the mode transition. However, the mode transition process, and the acceleration and deceleration processes are so different that the TRM for acceleration and deceleration process needs some major modifications to fit the mode transition process.Among so many details in the model,the present work mainly focuses on the design method of acceleration and deceleration control schedule. It should be indicated that the control schedule design method for mode transition employing the TRM related methods are discussed by Jia.

3. Validation of the TRM

Evaluating the performance of the TRM can be conducted in two levels.In the first level,the simulation accuracy of the conventional steady-state model of the VCE is evaluated. Therefore, simulation results of the VCE are compared with the performance data of the GE21 engine.In the second level,obtained results from the TRM are compared with that of the conventional steady-state simulation model.

3.1. Validation of the steady-state simulation

In order to evaluate the performance of the steady-state simulation,the performance of the GE21-J11B4 engine is studied in this section. Fig. 2 illustrates the comparative results between the steady-state performance simulation model and the data of GE21 enginefor the installed Specific Fuel Consumption(SFC) of the subsonic cruise.

It should be indicated that this engine is a double bypass type VCE, where the main design parameters are presented in Table 4.

It is observed that the relative error between the obtained results from the steady-state simulation and the experimental data for GE21-J11B4 engine is less than 1%.

3.2. Validation of the TRM

Both the TRM and conventional schemes are componentbased performance models. Therefore, in order to evaluate the performance of these schemes, it is necessary that both models should converge to the same operating point at the same operating condition.

Table 3 Parameters of TRM for double bypass VCE (Double-bypass mode).

Fig. 2 Model validation with the GE21 data.

The validation process is divided into three main steps. In the first step, it is intended to calculate the steady-state throttling performance of the VCE through the conventional model and obtain the state factors,variable geometry parameters and performance parameters in the throttling process,accordingly.Moreover,the main purpose of the second step is to implement the previously calculated state factors(e.g.n,n,β,β,β,T) as inputs, and use the TRM to re-calculate these factors.Finally, the calculation results of two methods are compared in the third step. Relative errors of these parameters represent simulation errors of the TRM, while the number of iterations denotes the convergence speed of the method.

Table 5 presents the state factors in the throttling process,which is obtained from the first step of the validation.Table 6-8 present relative errors between the conventional model and three TRM-based models, as well as steps of each off-design balancing iteration. Results show that the maximum relative error of variable geometry parameters is 7.98×10, while that of the additional power is 1.96×10.Since relative error of the component-level model is 1.0×10, it is interpreted that the TRM does not affect the accuracy of the component-based performance model.The number of iterative steps in the TRM is within the range of 3 and 10,which is similar to that of the conventional model.Based on the performed analysis, it is concluded that the computational accuracy and convergence speed of the TRM are consistent with that of the conventional model.

4.Optimizing the control schedules of the VCE acceleration and deceleration

4.1. The optimization model

The main task of establishing the optimization model is to determine the optimization variables,optimization target,constraint conditions and the optimization algorithm.

In this section,the model No.3 in Table 3 is selected to perform the optimization.The optimization variables are the state factors(i.e.T(or W),β,βand β)and A.Moreover,rotational speeds nand nare given as the input to determine the engine rating.

Studies show that during the transient maneuver,state factors actually stand for constraints. Subsequently, it is possible to avoid the over-temperature,rich/lean flameout or the surge.The upper boundary of the optimization variable Tis set as the maximum allowable inlet temperature of the turbine.Moreover, the upper boundary of β, βand βis determined by the minimum surge margin limitation of the compressor, fan and CDFS, which usually depends on the corrected rotational speeds. During the deceleration process,the lower boundary of Wcan be determined through the lean flameout FAR.

Based on the abovementioned descriptions, the only remained constraint is the Mach number at critical sections,which refers to the Mach number at FVABI (Maand Ma) and the Mach number at RVABI (Ma). By limiting the Mach number at these sections, the reverse flow or critical flow can be prevented on key sections.

Table 9 presents set parameters of the established optimization model for the acceleration and deceleration control schedule.

The main objective of the optimization is to minimize the transient time, which leads to the maximum acceleration power during the acceleration and the minimum deceleration power during the deceleration. The objective function Pconsists of the additional power of two rotors.It is worth noting that this function is positive for the acceleration,while it is negative for the deceleration.As a result,the greater the objective function P, the faster the acceleration, while lower Presults in faster deceleration. However, it is not reasonable to simply define Pby summing the additional power of two rotors. Therefore, the weight coefficients ωand ωare introduced accordingly. Then the objective function Pcan be mathematically expressed in the form below:

Table 4 Main design parameters of the GE21-J11B4 engine.

Table 5 State factors of VCE during double-bypass mode throttling (Sea level, ISA).

Table 6 Validation of variable Anb,h TRM (model No. 1 in Table 3).

Table 7 Validation of variable Anb,h, A8 TRM (model No. 2 in Table 3).

Table 8 Validation of variable Anb,h, A8,αCDFS TRM (model No. 3 in Table 3).

where, Pand Pare additional powers on two rotors.These powers are positive for the acceleration, while they are negative for the deceleration. Moreover, ωand ωare the weight coefficients for the two rotors.

The Hooke_Jeeves optimization algorithm is utilized for optimizing the VCE acceleration and deceleration control schedule. It should be indicated that this is a direct searching algorithm that searches for the minimum of a nonlinear function with no need to the function derivatives. It is based on a heuristic that suggests a descent direction using the values of the function calculated in some of previous iterations.The Hooke_Jeeves optimization algorithm is established in the Isight optimization software, which is employed in the next section to perform the optimization.

The abovementioned optimization model is used to obtain the Por the Pthrough the optimization for given nand neach time.The optimization result of variable geometry parameters and Wat multiple npoints jointly constitutes the control schedule of the engine acceleration or deceleration.

4.2. Designing the acceleration control schedule of the VCE

In order to optimize the acceleration and deceleration control schedule for the VCE, the Isight optimization software isapplied. To this end, design parameters of the relevant CDFS VCE at the standard condition are set as follows:T=1618 K, π=2.8, π=1.2, π=5.6, Bypass Ratio of the first bypass (BPR)=0.2 and Bypass Ratio of the second bypass (BPR)=0.3.

Table 9 Parameter settings of optimization model based on TRM of a double bypass VCE.

It should be indicated that the acceleration control schedule optimization is performed at the sea level ISA within the range of n=[64%, 100%]. The constraints are as follows:T≤1850 K, the Surge Margin of the fan (ΔSM)≥8%, the Surge Margin of the CDFS (ΔSM)≥10% and the Surge Margin of the compressor (ΔSM)≥10%. Fig. 3 shows the obtained optimization result. It contains the correlation between nand other parameters such as the main state factors(T, β, βand β), main variable geometry parameters(A, A, A, Aand α) and the fuel-air ratio (W/p).

Fig. 3(a) indicates that Treaches the maximum value of 1850 K during the acceleration process. Moreover, βand βmove along their lower boundary,which leads to a high surge margin,high mass flowrate and low-pressure ratio of the fan and CDFS. These results are in excellent consistency with the theoretical analysis, where higher Tand mass flowrate lead to higher turbine power, while the lower pressure ratio results in lower power for the compression. Consequently,the maximum acceleration power, which equals the turbine power minus the compressional power, can be obtained.

Fig.3(b)shows the variable areas in the form of percentage of the corresponding design value. It is observed that as the rotational speed increases, Aand Agradually decrease,while Aincreases. Meanwhile, W/pgradually decreases,while αand Ainitially increases and then decreases.The maximum value of Areaches 133% of its design value,while the largest Areaches 114%of its design value.Meanwhile Avaries within the range of[86%,107%]of its design value, while αis adjusted in the range of [-45°, -11°].Avaries in the range of [120%, 250%] of its design value,which is a challenge for FVABI design.

Then the control schedule of variable geometries and Ware given as input to simulate the acceleration performance of the VCE. Figs. 4 and 5 present the obtained results. Moreover, the fixed geometry control schedule designed by the VPEM is presented and results are compared with that of the variable geometry control schedule designed by the TRM.

Fig. 4(a) indicates that when the TRM is applied to optimize the VCE acceleration control schedule, the accelerating time that the gas generator requires to reach 99%of its design speed decreases from 6.7 s to 3.7 s. In other words, the required acceleration time reduces by 45%. Furthermore,Fig. 4(b) shows that after performing the optimization, the surge margin of the CDFS during the acceleration exceeds that of the steady-state throttling.Fig.4(c)and(d)show that as the mass flow rate increases,the surge margin of the both gas generator and fan increases too. It is found that in some regions,obtained surge margins are even higher than those of the steady-state flow. Fig. 4(e) and (f) indicate that Wand Tare basically the same in the acceleration process. This further indicates that the TRM does not improve the acceleration performance by increasing Wor T,but by varying the geometrical parameters to adjust the operating point of components and the generated power of the turbomachine.

The variation of the operating line shows different laws as it is in a fixed geometry engine.For a fixed geometry engine,it is observed that the closer the operating point to the surge line,the faster the engine accelerates.However,this is not necessary for VCEs. This is comprehensively described as the following:

The additional power (P) of an engine rotor is the key factor to determine how quickly the engine accelerates or decelerates. The additional power of the single-spool turbojet engine can be mathematically expressed as the following:

where, Wis the mass flow rate of the compressor, Crepresents the constant pressure specific heat, FARstands for the Fuel-Air Ratio (FAR) at combustor outlet. It should be indicated that subscripts ‘‘g” and ‘‘a” stand for the gas and air,respectively.Moreover,T,ηand ηdenote the inlet temperature of the compressor,isentropic efficiency of the turbine and the compressor, respectively. Furthermore, πand πdenote the pressure ratio of the turbine and compressor, respectively.Pis the power extracted by engine accessories and the aircraft.

Eq. (2) indicates that as W, Tand πincrease or πdecreases, the additional power Pincreases.

It is concluded that for a gas turbine with a fixed-geometry,the only way to increase Pis to increase the fuel supply of the combustion chamber,thereby obtaining a higher T.However, increasing the temperature Talso enhances the thermal throttling,which decreases the surge margin.In this case,Wdecreases, while πincreases, which is not conducive to increase the additional power P.Under these circumstances,the compressor is prone to the surge.

In the turbojet engine with variable Aand A, these factors can be adjusted to change the additional power. If Aincreases during the engine acceleration, πand Wincrease,while πdecreases. In this case, the additional power Pincreases. If the NGV throat area (A) increases, Wincreases, while both πand πdecrease.

Fig. 5 shows that the distribution of Tchanges in accordance with the designed schedule of the TRM. However, it is observed that there are still deviations between results obtained from the TRM optimization and the dynamic simulation.Similar to the VPEM,these deviations are mainly originated from deviations of nand the volume effect during the transient maneuver. Chen et al.proved that deviations caused by the volume effect are conservative.This can be physically interpreted that the engine actually works with higher Surge Margin (ΔSM) and lower T. Moreover, deviation caused by ncan be eliminated by replacing nin the optimized TRM, which can be conducted through the dynamic simulation and repeating the optimization procedure. Since deviations in this case are not so remarkable, improvements in this area are not covered in the present study.

Fig. 3 Results obtained from optimized acceleration control schedule.

Fig. 4 Obtained results for acceleration performance of VCE.

Fig. 5 Optimization results from transient-state simulation for T4.

4.3. Designing the deceleration control schedule of the VCE

In this section, it is intended to optimize the deceleration control schedule within the range of n=[64%, 100%] at the sea level ISA. Moreover, constraints are set as follows: FAR-≥0.01, ΔSM≥8%, ΔSM≥10% and ΔSM≥10%.

Fig. 6 shows obtained results from the optimized deceleration control schedule from different aspects, including the distribution of the state parameters (T, β, ββ), variable geometric parameters (A, A, A, A, α) and the fuel-air ratio.

Fig.6(a)indicates that Tis maintained within the range of[900 K, 1000 K] during the deceleration process so that the fuel-air ratio is set to 0.01. Under this circumstance, βand βmove along the upper boundary, which is limited by the minimum surge margin of the fan and the CDFS. Then,the pressure ratio and the power of the compressor increase so that the additional power reduces accordingly. It should be indicated that for the deceleration process, the additional power is negative. In fact, the smaller the additional power,the faster the deceleration process.

As shown in Fig. 6(b), Ais always maintained in the maximum opening position, while Ais smaller than its design value to reduce π. Meanwhile, as Aincreases and Tdecreases, the surge margin of the compressor increases. In the present study, Ais set to 0.8 of the corresponding design value to reduce πand the deceleration power of the low-pressure rotor.Moreover,αis maintained near its minimum value to reduce the CDFS pressure ratio and the mass flow rate. Since FARis always around its minimum value during the throttling, W/pdoes not change significantly.

All variable areas are given as a percentage of the corresponding design value.Based on the obtained results,the ratio of the parameter Ato the corresponding design value remains around 120%, while the ratio of parameter Aand Ato the corresponding design value vary within the range of [75%, 84%] and [85%, 100%], respectively. Furthermore,it is found that αvaries within the range of [-45°,-38°]. Aincreases from 65% to 120% of its design value as ndecreases. This increment may be attributed to the decrease of π.

Fig. 7 illustrates the obtained deceleration performance of the VCE,when the optimized control schedule is implemented.Moreover,the fixed geometry control schedule designed by the VPEM is presented and results are compared with that of the variable geometry control schedule designed by the TRM.

Fig. 7(a) shows that in the optimized control schedule, the deceleration time reduces from 10.7 s to 4.9 s. It is found that when the TRM is applied for the optimization, the deceleration time reduces by 54%. Furthermore, Fig. 7(b) indicates that at the beginning of the deceleration process,Wdecreases rapidly, while FARremains at its lowest value. Then, the air flows into the combustion chamber, but the flowrate continuously decreases. Under this circumstance, Wreaches to the desired target value and remains at this point, while FARincreases gradually. Fig. 7(c) shows that ΔSMsignificantly increases. Fig. 7(d) shows that the fan operating point moves along the surge boundary. This is mainly attributed to the reduction of A. Fig. 7(e) shows that Wand Tare basically constant in the fixed-geometry VPEM and TRM. This further demonstrates that the TRM does not improve the engine deceleration performance by reducing Wor T, but by varying geometric parameters to adjust the operating point of components and the power of the turbomachine. Fig. 7(f) shows that the fuel-air ratio changes in accordance with the designed schedule and the minimum fuel-air ratio is not exceeded during the deceleration process. The only exception occurs after restricting W, indicating that the TRM can strictly control the fuel-air ratio during the VCE deceleration process,thereby preventing the lean flameout phenomenon.

Fig. 6 Results obtained from optimized deceleration control schedule.

Fig. 7 Obtained results for deceleration performance of the VCE.

5. Conclusion

In the present study, the TRM is established and applied to optimize the control schedule during the acceleration and deceleration process of a double-bypass VCE. Based on the obtained results, the following conclusions are made:

(1) By replacing the state factors with variable geometry parameters in the VPEM of the VCE, the TRM can be established. It is found that the simulation accuracy and the convergence rate of the TRM are consistent with that of the conventional component-based performance model.

(2) In the TRM,state factors,including n,n,β,β,βand T, are given as input. These factors can be applied to define the operating point of the VCE prior to the calculation.Moreover,the additional power,fuel mass flow and variable geometries are calculated by the off-design balancing of the engine.

(3) The optimization model of the acceleration and deceleration control schedule of the VCE is established based on the TRM. It is found that the optimization space of variables β,β,β,Tand Wdetermine the variation range of ΔSM, Tand FAR. Therefore, it is ensured that no surge, flameout or over-temperature occurs during the acceleration and deceleration processes.

(4) Compared with the fixed geometry acceleration control schedule, the TRM optimization reduces the acceleration time by 45%, while it improves the surge margin of compressors and avoids the over-temperature.

(5) Compared with the fixed geometry deceleration control schedule, the TRM optimization reduces the deceleration time by 54%, while it limits the surge margin of compressors and avoids the lean flameout.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are grateful to Prof. Yasong Sun for discussions and kind suggestions. This study was supported by the Aviation Power Foundation of China (6141B09050382).