Transient Optimization of a Gas Turbine Engine

Transient Optimization of a Gas Turbine

Engine

*

Jonathan L. Kratz

NASA Glenn Research Center, Cleveland, Ohio, 44135, U.S.A.

Gas turbine engines are the primary power plant s for modern commercial aircraft .

Transients prompted by significant changes in thrust or power demand are common and unavoidable . Extreme transient scenarios such as those associated with a go - around during a landing attempt are possible and must be accounted for in the design of the engine and its controller . Engine transients tend to cause a reduction in compressor operability margin , which must be addressed by the engine control system and accounted for in the engine design to prevent events such as compressor stall/surge and combustor blow out . Transient ope rability concerns typically lead to compromises in the engine design that sacrifice efficiency and/ or limit responsiveness . Transient operability is typically managed by logic that limits the fuel flow command . If this logic is not optimized, then the pote ntial for valuable performance could be lost. This study presents a strategy for optimizing the transient limit logic and proposes a strategy for updating the control logic over the lifespan of the engine . The r esults demonstrate significant improvements in transient operability . For example, of the results at sea level static conditions demonstrated a 31% reduction in t he usage of the high pressure compressor operability stack during a snap acceleration transient. Furthermore , a reinforcement learning algorithm is demonstrate d to modify the transient logic as the engin e degrades to minimize response time while respecting a prescribed compressor operability margin limit . A simple demonstration of the rei nforcement learning algorithm resulted in a thrust response time reduction of ~11.8% .

I. Introduction Most modern commercial aircraft are powered and propelled by gas turbine engines. These workhorses of the aviation industry will remain crucial components of aircraft propulsion system s , e ven as new concepts such as electrified aircraft propulsion (EAP) are pursued . M any EAP con cepts , particularly large aircraft concepts , retain the use of gas turbine engines through hybrid gas - electric propulsion. The point is that gas turbine engines will remain the predominate powerplant of the aviation industry for many years to come .

One challenge of gas turbine engines is maintain ing operability during engine power transients and throughout a vast operating envelope . The design of transient control logic takes up nearly 75% of the total time dedicated to engine control system developm ent [ 1 ]. Engine power transients occur when there is a change in thrust or power demand , often associated with movement of the throttle position by the pilot. T ransients result in a change in the shaft speeds of the gas turbine engine. G as turbine engines often have two shafts that are referred to as the low - pressure shaft (LPS) and high - pressure shaft (HPS). Each shaft is attached to a compressor and a turbine that app ly opposing torques .

During an acc e leration or deceleration, there is a temporary imbalance in torques . This is the result of a change in fuel flow that increases or decreases the amount of energy in the flow path of the turbine and the amount of power that it extracts from the flow to driv e the compressor. The non - zero net torque cause s a change in shaft speeds . However, the shafts have a considerable amount of inertia , and thus the change in speed is not instantaneous. A mismatch in the internal engine air flow and shaft speed s results in off - incidence flow impinging on the compressor blades . If t he off - incidence flow is severe enough, compressor stall or surge can occur . For a two - spool engine , the tendency is for the high - pressure compressor ( HP C ) operability to degrade during accelerations and the low - pressure compressor ( LPC ) operability to degrade during decelerations.

* Research Engineer, Intelligent Control & Autonomy Branch, AIAA senior member.

In theory, the transient operability issue could be mitigated by changing the throttle very slowly such that a quasi - steady - state condition is maintained. However, this is not possible from a practical perspective. Aero - engines must be able to accelerate and decelerate relatively quickly to meet Federal Aviation Administration ( FAA ) and user requirements. FAA regulations state that a commercial engine must b e able to accelerate from 15% thrust to 95% thrust within 5 seconds for relevant flight conditions [ 2 ] . For so me aircraft, especially military aircraft , there are user - imposed requirements to respond even faster . O perability issues must be dealt with thro ugh the engine design and its control logic. The transient control logic i s typically found in the form of a fuel flow rate limiter the enforces a shaft acceleration / deceleration ( Ṅ ) limit or a ratio unit ( RU ) limit in a max - min switch logic structure [ 3 ] or command governor (CG) logic [ 4 ] . The ratio unit is the ratio of the fuel flow rate and the HPC static discharge pressure ( w /p ) .

f s3 Additionally, more advanced control schemes have been studie d , including the use of Model Predictive Control (MPC) [ 5 , 6 ] and L inear Parameter Varying control [ 7 ].

Regardless of the controls approach, t he control logic can only manage the issue, not eliminate it . As a result, the engine must be designed to account for some degree of v ariability in operability as the engine goes through transients.

Th us , transient operability places constraints on the engine design that forces compromises in engine performance .

For applications that desire fast responsiveness, this can limit acceleration and deceleration rates . In addition, constraint s placed on the engine design can impact efficiency metrics such as fuel burn. Ref. [ 8 ] states that r oughly one third of the compressor operability allowance can be attribute d to transient operat ion while Ref. [ 9 ] states that about half of the stall margin can be devoted to transient operations . Furthermore , high duty compressors tend to achieve maximum efficiency close to the stall line [ 8 ] . The highest efficiency of an LPC may occur near or below 10% stall margin (SM) and the shifting of the operating line demanded to maintain compressor stability during transients could cost more than 3% in compressor efficiency [ 8 ]. The ability to operate closer to stall could influence the design itself, resulting in different perfor mance map s and a lighter design . Ref. [ 10 ] discusses trades between a traditional optimal design and a robust optimal design , which must account for various uncertainties and a wide range of operation that include s transients . Inevitably , sacrifices are made in the design to assure safe and reliable operability.

The methodology for control design could be inherently sub - optimal in application . The NASA - developed Tool for Turbine Engine Closed - loop Transient Analysis (TTECTrA) [ 11 ] is a nice tool for preliminary transient control design and dynamic system analysis. It can be used to design transient control schedules . However, it does so assuming a fixed fuel flow input profile , which happen s to be a linear ramp. The software utilizes an iterative solver to adjust the slope of the ramp until stall margin and response time constraints are met. The fixed form of the fuel flow input will almost certainly result in a sub - optimal solution. There is plenty of liter ature and patents related to optimizing the transient control logic . Much of the focus of the literature is on military application s for which engine responsiveness is of great importance . Ref. [ 12 ] and [ 13 ] are examples of patent s in this subject area. The former adapts bleed valve and Ṅ schedules based upon estimated combustor discharge temperature to minimize engine response time without stalling the engine. The latter has the same goal but utilizes a simulated compressor stall l imit signal that is converted to a desired burner pressure limit . That limit is regulated via control either directly or through a proxy measurement.

Ref. [ 14 ] applies a method referred to as an extrapolation approach to design the transient limit schedule . This method does not require a dynamic model, in contrary to most other methods, but it is known to be less accurate. Ref. [ 15 ] performs an optimization of the transients to reduce response time under operability constraints . It also avoids the need for a transient model but tends to be more accurate than the prior mentioned method . It utilizes a noteworthy method referred to as the virtual power extraction method (VPEM) that applies power extraction to the engine via steady - state simulations to shift the operation of the engine and emulate transient conditions. Ref erence [ 16 ] proposes the use of a variable replacement method along with particle swarm optimization to improve transient control performance by modifying the gains of a proportional integra l (PI) Ṅ transient controller . Ref. [ 17 ] applies a genetic algorithm to tune the gains of a max - min switch logic controller that includes transient limiters with an objective function that considers the thrust response time and transient fuel consumption . Both approaches are limited as they assum e static Ṅ limit set - point determination . Ref. [ 18 ] uses a genetic algorithm to create open - loop fuel flow and nozzle area command s that minimize the thrust response time. The study covered in this paper has similarities to Ref.

[ 18 ] but differs in several ways including differences in the problem formulation and det ails about the genetic algorithm, extension of the results to a traditional control schedule, application to a commercial engine, and extension of the strategy toward life - cycle optimization. These are just some examples of the literature o n this rich topic of research.

Even if an “ optimal ” acceleration/deceleration controller is developed and employed on a new engine, it will likely be or become sub - optimal. Reasons for this include factors s uch as engine - to - engine variations, differences in operation ( e.g., varying levels of engine power extraction), secondary effects such as heat soakage, and shifts in performance as the engine ages. In addition, the logic is typically designed with worst ca se conditions in - mind, and therefore the potential for a faster response or better performance characteristics could be squandered for any given transient that isn’t the worst case. The concept of digital twin could help to improve this situation in the fu ture. A digital twin is a virtual representation of a connected physical asset [ 19 ] , i n this case a gas turbine engine. Gas turbine engines are prime candidates for digital twin given the ir abundance of data . Ref. [ 20 ] approximates that 20TB of data can be collected from an engine per hour . The definition provided by AIAA and the Aerospace Industries Association (AIA) in Ref. [ 19 ] is “A set of virtual information constructs that mimic the structure, context and behavior of an individual/unique physical asset, or a group of physical assets, is dynamically updated with data from its physical twin throughout its life cycle and informs decisions tha t realize value.” N umerous potential applications of a digital twin exist including enhancing operational performance through controls. This may include using the digital twin to update control schedules and tuning control gains. It can also extend to usin g the digital twin model within the control logic, otherwise referred to as model - based engine control (MBEC) . Ref. [ 21 ], [ 22 ], [ 23 ] and [ 24 ] describe MBEC strategies that utilize a tracking filter for tuning the model to the real system and utilizing the model outputs for controls. Ref.

[ 25 ] is an example of using machine learning to update the digital twin. A recent application of a digital twin is demonstrated by a software tool released by General Electric to optimize gas turbine operations [ 26 ]. The so ftware utilizes artificial intelligence to build a machine learning digital twin model. The model is used to determine the optimal flame temperature and fuel splits that minimize emissions and acoustics. This technology along with the prior work mentioned above provide s promising signs of the near - term technology readiness of digital twins to influence engine controls.

T he application in this paper is for a commercial single - aisle engine , for which the operations are constrained and predictable compared to military applications, minimizing component life usage during takeoff and landing transients , and minimizing cruise fuel consumption are among the primary concerns [ 18 ]. Thus, the goal of this study is to optimize acceleration and deceleration control logic to minimize variations from the steady - state operating line such that the engine design can be improved to enhance efficiency and the transie nts will be less harsh in terms of the metrics such as compressor operability and peak temperatures. The study considers various optimizations and simulations to draw a variety of conclusions . The topics investigated include: the impact o n compression system operability and peak operating temperatures th at contribute to engine deterioration , and the ability to generalize the form of an optimal solution f or a given flight condition to other flight conditions . A strategy for updating the tran sient limit logic throughout the engine lifespan is also explored.

The approach involves the use of a nonlinear model of a conceptual advanced geared turbofan and its controller to simulate various engine transient s . A genetic algorithm optimizer is given control of the control inputs along with the objective to optimize a measure of compressor operability and to achieve a given response time . The results are used to derive an RU limit schedule . Assuming the model to be a digital twin of a specific physical engine , additional optimizations are conducted at different health states of the engine , and those results are used to guide a reinforcement learning algorithm to gradually update the control logic as the engine ages.

The rest of the paper is organized as follows. Section II provides a brief overview of the Advanced Geared Turbofan 30,000lb (AGTF30) engine model [ 27 ] , which is the plant considered in this study. Section III describe s f the genetic algorithm employed in the study and Section IV covers the transient optimization procedure and results.

Section V comments on extending the optimization to updating the control schedules using a digital twin. Finally, Section V I provides a sum mary .

II. The AGTF30 Propulsion System The AGTF30 is a model of a conceptual two - spool geared turbofan capable of producing ~30,000 lb of thrust at sea level f static (SLS) conditions. The engine is envisioned for a single - aisle commercial transport application. The AGTF30 is meant to be representative of technology available in 2035 and includes features such as a compact gas turbine core and a variable area fan nozzle. The engine model is coded in the MATLAB/Simulink® environment using the NASA - developed Toolbox for Modeling & Analysis of Thermodynamic Figure 1 . Diagram of the AGTF30 conceptual engine Systems (T - MATS) [ 28 ] . T - MATS provides the building blocks for creating a 0 - D (component level) model of a gas turbine engine utilizing turbomachinery performance maps, thermodynamic relations, actuator models, and more. The AGTF30 is represented in Fig. 1 . The model includes a realistically performing full - flight envelope controller that was develop ed in Ref. [ 27 ].

This controller includes schedules for the variable area fan nozzle and variable bleed valve. It also includes closed - loop gain - s cheduled PI controllers for the fuel flow rate that include a nominal corrected fan speed controller and various limit controllers. Among the limit controller s are limiters for over - speed and over - temperature conditions. The model also include s acceleration and deceleration limit logic, which is employed as a maximum RU limit schedule for acceleration and a minimum RU limit schedule for deceleration. The various fuel flow rate commands go through a max - min decision tree to decide which command t o use. Health parameters are used within the engine model to set the health state of the turbomachinery components. H ealth parameters are modifiers that sh i ft the flow capacity and efficiency of the compressors and turbines based on degradation. The degrad ation model that relates the health parameters to engine life was taken from another engine model known as the Commercial Modular Aero - Propulsion System Simulation 40,000lb engine model [ 29 ].

f III. The Genetic Algorithm Genetic algorithms are optimization sch emes built upon the biological principles of natural selection and fitness [ 30 ]. Genetic algorithms tend to be less likely to get stuck at local minima/maxima than gradient based methods and are substantially more efficient than brute force methods. A population is comprise d of numerous solution realizations, each with different parameters . Each individual solution is evaluate d based upon a fitness function. Th e members of the population compete for survival into the next generation and for participation in reproduction. The genetic algorithm utilized in this study has the primary components of elitism, carry - over (replication), reproduction (crossover), and imm igration . The primary sub - components of the genetic algorithm are selection , mutat ion, and duplication removal . Each of these components and sub - components will be described in the following paragraphs.

The sub - components are described first to set the foundation for describing the components .

The two s election metho d s used in this application were random and rank - biased selection. In random selection, all members have the same probability of being chosen. Rank - based selection utilizes the p areto distribution [ 31 ] and allows the user to specify parameters that define the exact shape. For instance, the 80 - 20 rule [ 31 ] can be applied by specifying that the probability of selecting a member from the top 20% will be 80% .

Mutation create s a modified version of a member of the population. It will select the number of parameters to mutate based on specified probabilities and then will randomly select parameters to mutate. Finally , those parameters are mutated within specified bounds using a random distribution.

Duplicate removal applies whenever a duplicate shows up in the population. This feature will remove the duplicate and repl ace it with new member that is generated within the specified parameter bounds using a random number generator.

Elitism involves advancing a set number of the most fit individuals to the next generation. Elitism seeks to preserve the best solutions and enable them to take part in finding better solutions through the functions of reproduction and mutation. In addition to advancing the elite, mutated variants of the elite may also be added to the next generation.

This action promotes diversity but also exploits the high fitness of the elite. Inputs include the number of top members of the population to include in the elite, the number of the elite to mutate, the bounds for mutation, and the probability of mutating any number o f parameters up to the full number of parameters .

The carry - over component of the algorithm selects members of the population , outside of the elite, to advance to the next generation . Mutation can apply as these members are carried to the next generation . The inputs include the number of members to carry - over, the method of selection , inputs associated with the method of selection, and inputs associated with mutation including the b ounds of mutation, the probability of a mutation occurring, and the probabi lity of any number of parameters being mutated.

Reproduction consist s of the combining of two members of the population to produce one or more new member s of the population in the next generation. The offspring will derive its parameters from its parents. There are 3 methods for assigning parameters. The first is to inherit the parameter from one of the parents. The second is to average the parameters of the parents. The final is to randomly select a value for the parameter between the value s of the parameter for the two parents. The probability of each method being used can be specified. In this application, each method had an equal chance of use. The parents are chosen based on the specified selection method, as is the number of offspring that a pai r of members will produce. The combined fitness of the parents can be utilized to determine the number of offspring. Limits can be set for the number of time s a single member of the population can participate in reproduction.

In this application the number of offspring is specified . After the reproduction function is carried out, the offspring can be mutated given inputs about the probability of mutation, the bounds of mutation, and the probability of mutating any number of parameters.

Immigration refers t o the introduction of new members to the population that will appear in the next generation.

The new member s are generated with in the specified parameter bounds using a random number generator. This feature helps to explore the solution space more thoroughly .

In this application, the members of the population define a fuel flow command input profile for the transient. The profile is defined by 8 points between the minimum and maximum fuel flow rate. The time at each data point in the profile is con stant . The variables in the optimizer include 9 values between 0 and 1 , Y , that are used to define the change in fuel flow between each of the 8 data points . The fuel flow rate value i s a function of Y .

𝑌 𝑖 𝑤 = 𝑤 + ( 𝑤 − 𝑤 ) ( 1 ) 𝑓 , 𝑖 𝑓 , 𝑖 − 1 𝑓 , 𝑚𝑎𝑥 𝑓 , 𝑚𝑖𝑛 ∑ 𝑌 𝑗 𝑗 = 1 In the equation above, w is the fuel flow rate, and i and j refer to the indices of the time interval that each fuel flow f change occurs over . The variable i differs from j by referring only to the time interval for which the fuel flow rate change is b eing calculated in the equation. T he subscripts “ max ” and “ min ” refer to maximum and minimum fuel flow rate. This approach guarantees the fuel flow rate input remains monotonically increasing or decreas ing between the minimum and maximum fuel flow rate . The population was initialized using a random number generator for a population of 4 5 and the genetic algorithm was r un for 50 - 200 generation s. For each generation, the fitness of new members was evaluated. This entailed running a transient simulation and calculating fitness for each member. The fitne ss , f , is defined in Eq. ( 2 ) where TSU is the transient stack usage , t is thrust response time, and t , is the thrust r r,target response target value.

1 1 𝑓 = + ( 2 ) 𝑇𝑆𝑈 10 𝑚𝑎𝑥 ( 𝑡 − 𝑡 , 0 ) + 1 𝑟 𝑟 , 𝑡𝑎𝑟𝑔𝑒𝑡 The thrust response time for an acceleration is the time to go from idle thrust to 95% thrust. The response time for a deceleration is the time to go from idle thrust to 20% thrust. The target thrust response time was chosen to match t he response time of the engine w ith the baseline controller. The T SU is a metric developed by the author to quantify operability margin . The metric is a single value that quantifies the operability margin for a given transient . The metric is defined below: 𝑃𝑅 − 𝑃𝑅 𝑆𝑆 𝑇𝑆𝑈 = 𝑚𝑎𝑥 ( ) × 100% ( 3 ) 𝑃𝑅 − 𝑃𝑅 𝑠𝑡𝑎𝑙𝑙 𝑆𝑆 PR is the pressure ratio, PR is the pressure SS ratio at the same corrected flow rate along the steady - state operating line , and PR is the stall pressure ratio at the same corrected flow rate along the stall line. Each variable is a vector that varies throughout the transient. The T SU metric quantif ies what portion of the compressor operability stack is used during the transient. By quantifying the operability margin with a single value for the entire transient using map data that consider s the stall line and steady - state operating line, this metric giv es a decent summary of the compressor operability without some of the nuances of stall margin. Figure 2 i llustrates these terms on a compressor map . W on the c Figure 2 . Illustration of TSU x - axis is the corrected flow rate at the inlet of the compressor. It is noteworthy that the fitness function could be modified to utilize other operability metrics , such as minimum stall margin, or combinations of var ious operability metrics.

IV. Transient Optimization The transient maneuvers evaluated will be full power range bursts and chops with a new undegraded engine . A burst is characterized by a rapid increase in power/thrust and a chop is characterized by a rapid decrease in power/thrust. Typically , a hot Bodie re - acceleration (hot re - slam) is considered the worst - case scenario [ 9 ]. In such a case, the heating of the metal engine components cause the steady - state gas path characteristics to change such that the operating line is shifted closer to the stall line. Given t hat the model used in this study did not readily capture engine he at soak effects , standard burst and chops were utilized.

The results of the acceleration optimization at SLS conditions are shown for the HPC in Fig. 3 compared to results with the baseline acceleration schedule from Ref. [ 27 ]. Fig. 4 shows similar results for the LPC from the deceleration optimization. Referring to Fig. 3 , t he thrust response time with the optimized fuel flow input is the same as with the bas eline schedule , but the T SU is reduced significantly from 3 8.4 % to 20.5% , indicating an operability improvement along with observation of the flatter running line on the compressor map in Fig. 3 c . Referring to Fig. 4 , t he response time is slightly faster for decelerations with the optimized schedule and the T SU wa s reduced from 9. 1% to 7.4%. The optimized fuel flow input profile is also observed to re sult in a better trajectory for the LPC running line as it begins to decelerate.

The optimization was conducted for accelerations and decelerations at other flight conditions . Included altitude (Alt), and Mach Number (MN) combinations were: 5000 ft at Mach 0.2, 15000 ft at Mach 0.3, 20000 ft at Mach 0.6, Figure 3 . Optimized acceleration results at SLS conditions . HPC data shown in (c).

Figure 4 . Optimized deceleration results at SLS conditions . LPC data shown in (c) 30000 ft at Mach 0.7, and 35000 ft at Mach 0 .8. It was observed that the fuel flow input profile was very similar for all cases . Figure 5 shows the normalized fuel flow input , where w and t are defined below .

f,norm norm 𝑤 − 𝑤 𝑓 𝑓 , 𝑚𝑖𝑛 𝑤 = ( 4 ) 𝑓 , 𝑛𝑜𝑟𝑚 𝑤 − 𝑤 𝑓 , 𝑚𝑎𝑥 𝑓 , 𝑚𝑖𝑛 𝑡 𝑡 = ( 5 ) 𝑛𝑜𝑟𝑚 𝑡 𝑟 In Eq. ( 5 ) t i s the time vector during the transient and t is the thrust response time . The acceleration fuel flow input r profile tends to increase gradually before increasing more rapidly and sharply taper ing to the maximum fuel flow rate value. The deceleration fuel flow input profile tends to decease relatively sharply in a nearly linear fashion and then change to a less aggressive trajectory that is nearly linear as the minimum fuel flow rate value is ap proached. This forms a “kink” in the profile that is associated with the inflection in the LPC running line (see Fig. 4 c) , which bends to run parallel to the stall line as the variable bleed valve begins to open to manage LPC operability .

Figure 5 . Normalized fuel flow command profiles for burst (a) and chop (b) I t is hypothesized that the fuel flow input profile could be generali zed across flight conditions and still achieve near optimal results. Alternatively, the fuel flow input profile could be a function of operating conditions that is derived from optimizations at select operating conditions , such as those listed above . The benefit of either of these choices is a reduction in the workload to perform optimization. With an approximation of the optimal fuel flow input profile an approach similar to that of TTECTrA can be applied throughout the flight envelope to produce acceleration and deceleration schedules . With the form of the fuel flow input profile constant, an iterative solver can be used to stretch or compress the profile to achieve a desi red response time while respecting operability constraint s . For demonstration, the average normalized fuel flow rate command profile from the optimization s w as applied to the transient scenarios at each of the 6 operating conditions investigated prior. A n iterative solver was used to achieve a similar thrust response time as the baseline controller .

Figure 6 show s the acceleration results on the HPC compressor map and Fig. 7 shows the deceleration results on the LPC compressor map . Fig. 8 shows a zoomed - in image of the results at 15,000 ft and Mach 0.3 to provide an example of how little variation is observed in the transient running lines. These plots compare the generalized profile to results with the optimized profile for each flight condition . Also present are results that implement acceleration/deceleration schedules that are derived from the generalized profile results. Following the logical and sequential steps laid out in the introduction, those results will be highlighted later in this sect ion . Table 1 and 2 quantif y the terms of t and TSU for acceleration s and decelerations respectively . These tables also include results with the r derived schedules, which will be discussed later. Note that some of the “optimized” results did not fully converge upon the optimal solution (but are close) , as is evidenced by some lower response times , particula rly for the deceleration study . This was later improved in the optimization strategy by introducing an iterative solver to achieve the desired thrust response time , rather than incorporating it as a constraint in the fitness function . A slower response wit h the same fuel flow rate profile should produce more favorable operability results. Overall, the results are very Figure 6 . HPC running lines during accelerations transients at various flight conditions similar and provid e support for the theory that the optimal fuel flow input profile can be generalized across various operating conditions . However, it is noted that for the deceleratio n, the TSU with the generalized fuel flow profile is slightly higher than t hat wit h the baseline controller at the SLS and 1 5000 ft Mach 0. 3 flight conditions . In general , the generalized input profile provide s s imilar results to the optimized profile but tends to be slightly less optimal in terms of TSU . Given the relatively small devi ation between the baseline transient running line and the steady - state operating line, the margin for improvement is very small and therefore very small deviations in the running line can make a significant enough change in TSU to explain this observat ion. Observation of the running lines in Fig. 7 provide assurance that the response with the generalized results is quite good , as it is nearly indistinguishable from the optimal solution . The HPC running line during accelerations naturally has much more deviation from the steady - state operating line an d in all case s the generalized profile provides a significantly lower TSU than the response with the baseline schedule . The TSU result s and the running lines plotted in Fig. 6 demonstrate similar performance of the optimal and generalized fuel flow input profile s.

Once the fuel flow rate optimization simulation results are obtained, they need to be converted into acceleration and deceleration control schedules that are implementable in practice. To convert the generalized profile results into acceleration and decele ration limit schedules , data from the simulations are utilized. An RU limit approach is considered here , but the same idea is applicable to an Ṅ limit approach. Derived RU schedule s are constructed as a function of corrected fan speed, N . To prevent t he acceleration and deceleration limit schedules from limiting the c,Fan fuel flow at the start of the transients , the RU limits for acceleration s were increased by 3.5 % to 5.5% and reduce d by 1.5% for decelerations . The resulting schedules are show n in Fig. 9 . Fig ure 6 and 7 show the compressor running lines c ompared with the optimal and generalized optimal results . It is evident that the results are nearly identical , demonstrating that the limit schedules are working as intended . Table 1 and 2 provide the response time and operability metrics , which help to illustrate the s imilarity in performance of the engine with the schedule and with an optimized fuel flow input profile. Implementation of the schedule does result in some deviation of the fuel flow input from the data used to produce the schedule and this results in a slight increase in the TSU , which may also be affected by slight changes in the thrust response time. Still , the schedule appears to provide a near optimal solution and certainly an improvement over the baseline schedule .

Figure 7 . LPC running lines during decelerations transients at various flight conditions Figure 8 . Zoomed in plots of the transient running lines at 15,000 ft and Mach 0.3. (a) shows the HPC data for an acceleration and (b) shows the LPC data for a deceleration.

The next set of simulations demonstrate the impact of the acce leration limit schedule on peak temperature s , which can impact the life span and maintenance costs of engine components. These simulations consider a n acceleration transient conducted at SLS conditions for a full power rapid burst take - off and a de - rated ta ke - off in which the aircraft only needs ~80% of the maximum thrust . The first two rows of Table 3 compare the peak turbine inlet temperature , T 4 ,peak , and T 4 overshoot for the baseline control schedule and a schedule optimized for operability . While the new optimized schedule i mprove s transient operability, peak T is higher . To strike a better balance between operability and peak T , the optimization can be redone with a modified fitness function that penalizes T overshoot. An 4 4 optimization was conducted with the fitness function in Eq. ( 6 ).

3 10 ( 6 ) 𝑓 = + 𝑇𝑆𝑈 ( 𝑇 − 𝑇 ) / 𝑇 + 1 4 , 𝑝 𝑒 𝑎𝑘 4 , 𝑆𝑆 4 , 𝑆𝑆 T is the final steady - state turbine inlet temperature after the acceleration. The algorithm was also modified with a 4 ,SS root solver to stretch or contract the fuel flow input profile to achieve the desired response time , and therefore the response time term such as the one used in Eq. ( 2 ) becomes unnecessary . Using this fitness function , a new optimized Table 1 . Table of t r and TSU for accelerations Flight Baseline Controller Optimized w Input Generalized w Input Derived RU Schedule f f Condition t , s TSU , % t , s TSU , % t , s TSU , % t , s TSU , % r r r r SLS 4.98 38.4 4.96 20.5 4.98 23.0 4.93 25.8 5000ft, 5.31 38.7 5.31 24.3 5.31 25.1 5.34 27.3 Mach 0.2 15000ft, 5.50 39.6 5.50 33.5 5.50 33.8 5.56 35.4 Mach 0.3 20000 ft, 6.01 39.5 6.01 33.0 6.01 33.0 6.07 35.1 Mach 0.6 30000 ft, 8.56 46.7 8.53 32.8 8.56 33.9 8.50 37.4 Mach 0.7 35000 ft, 10.17 49.5 10.15 31.6 10.17 39.9 10.09 42.7 Mach 0.8 Table 2 . Table of t and TSU for decelerations r Flight Baseline Controller Optimized w Input Generalized w Input Derived RU Schedule f f Condition t , s TSU , % t , s TSU , % t , s TSU , % t , s TSU , % r r r r SLS 11.61 8.41 10.86 6.67 11.61 8.85 11.10 9.24 5000ft, 11.13 10.08 11.01 7.56 11.13 8.23 10.68 8.38 Mach 0.2 15000ft, 11.02 12.74 10.71 9.99 11.02 15.10 10.63 15.33 Mach 0.3 20000 ft, 10.27 15.18 10.17 11.38 10.27 12.61 9.99 13.52 Mach 0.6 30000 ft, 10.27 19.56 10.15 15.75 10.27 16.47 10.06 17.17 Mach 0.7 35000 ft, 10.11 23.97 10.11 21.69 10.11 21.58 9.97 22.06 Mach 0.8 Figure 9 . Acceleration (a) and deceleration (b) derived schedules Table 3 . Comparison of transient limit schedules on the basis of TSU and peak T Transient Limit Logic Full Power Burst Derated Take - Off Burst TSU, % T °R T Overshoot, % T °R T Overshoot, % 4,peak , 4 4,peak , 4 Baseline Schedule 38.4 3172 0.1 2951 3.0 Schedule Optimized for 25.8 3181 0.4 2989 4.3 Operability Schedule Optimize d with 29.76 3182 0.4 2972 3.7 consideration of Operability and T Figure 10 . Comparison of optimized fuel flow inputs (a) and acceleration schedules (b) fuel flow input profile shown in Fig. 10 a was obtained for a derated takeoff and a new RU limit schedule was extracted.

The schedule is plotted in Fig. 10 b with reference to the p reviously optimized schedule . Figure 11 shows the HPC running lines during the acceleration transient . It is evident in these results that the new optimization leads the transient with more fuel flow input to sacrifice some operability s uch that the maximum fuel flow rate can be approached more gradually to reduce overshoot in T . The last row of Table 3 includes the TSU and peak T res u lts for the new optimization . During the more common derated takeoff burst scenario , the peak T is reduced by 17°R when compared with the prior optimized schedule . It can also be noted that the while the peak T is still higher than that achieved with the baseline acceleration limit schedule, the transient operability margin is significa ntly better with a TSU of 29.76 % vs.

38. 4 % .

The conclusion of these analyses is : (1) optimized transient limit logic can have a significant impact on engi ne operation and influence system design , (2) a generalized or simplified fuel flow profile could be substituted in the control design process to simplify and expedite control design w hile achieving similar results to the optim al solution, and (3) the optimization objectives can be modified to a chieve different goals or to strike a balance Figure 11 . Comparison of the transient running lines of the between competing goals.

two optimized acceleration schedules V. Engine Life span Optimization Since the controller is developed to handle engine deterioration, the transient limit logic is overly conservative until the engine is fully deteriorated. This means that the engine will tend to respond slower than it is able. While this may not be a concern for commercial aircraft engines, there are circumstances where better responsiveness is desirable , particularly during emergency scenarios where thrust is needed quickly . M ilitary engine applications would benefit from the ability to improve thrust responsiveness over the lifespan of the engine more so than commercial engines .

This goal is in the same spirit of the works referenced prior including Ref. [ 12 ], [ 13 ], [ 15 ], and [ 18 ]. In a similar spirit, t he example presented in this paper will consider minimizing thrust responsiveness under operability constraints.

However, it c an be noted that the goals of the lifespan optimization problem formulation could be modified to achieve other goals more relevan t to commercial engines . For instance, the objective could be to balance factor s th at contribute to engine deterioration , such as peak temperatures and pressures, with operability improvements that enable engine design benefits .

MBEC enables the use of cl osed loop control on unmeasured parameters such as thrust and stall margin. MBEC could also eliminate the need to create extensive schedules for managing acceleration and deceleration. However, the schedule approach considered here is more traditional , and field - tested . Utilizing traditional methods could be argued as being advantageous in the certification process . It is thought that small and gradual updates to the schedule over time , and the ability to revert to the original generic and conservative schedule are favorable qualities with regard to certification and could be view ed as advantageous when compared with advanced MBEC approaches that heavily rel y on a model . A n optimization s cheme with a digital twin will enable the schedule to be reassessed as the engine ages or after changes occur ( e.g., performed maintenance and sensed shifts in performance ) . Coupling this with engine sensor data and a reinforcement learning (RL) strategy c ould offer a trusted means of maintaining a good balance between transient performance and operability as the engine ages. The RL would leverage the optimization results obtained using the digital twin but would not trust those results . The results would o nly be used to guide its decisions as it cautiously explores the solution space . The RL algorithm will limit changes to the schedules and penalize any observations of unexpected reduc tions in operability margin that could be the result of error between the model and physical syste m . In theory, it could also inform the updating of the digital twin. Figure 12 shows the proposed architecture.

Si nce engine deterioration occurs gradually, the optimization could occur periodically and may even be conducted off - line with only the results of the optimization being supplied to the control system. The RL algorithm will attempt to adjust the schedule tow ard the optimal schedule but will limit how much the schedule can change . It will require feedback from the physical system and evaluation of the transient response characteristics prior to allowing further updates. If operability is observed to degrade too much , then the schedule will be shifted back toward the original conservative schedule.

To demonstrate th e concept, the genetic algorithm will be employed at four different times during the life span of the engine : new engine (NEW) , mid - life engine (MID) , three quarters of life engine (3Q), and end - of - life engine Figure 12 . Schematic of the proposed transient control architecture and the iterative process to update it (EOL) . This will be simulated by modifying the health parameters for the turbomachinery components. The degradation states of the engine have been sparsely selected to demonstrate the concept . F or the sake of demonstration, all the examples focus on sea level static operation. An RL algorithm is then used to modify the schedule with guidance from the optimization results. The rest of this section is broken into three sub - section s : (1) the RL algorithm , (2) the results of a demonstration , and (3) discussion of some related topic s .

A. Reinforcement Learning Algorithm A Q - learning RL algorithm [ 32 ] is utilized to update the schedule. Q - learning uses a quality function, Q(s,a) , that describes the value or quality of being in a particular state, s , and taking a particular action, a. In this case, the state is the enforced limit schedule, and the action is the decision for how to modify it in the future. For demonstration, a discrete set of schedules are initialized and the RL algorithm will have the option to shift the schedule toward t he “optimal” transient schedule, retain the same schedule, or shift the schedule toward the original conservative schedule.

The quality of the state and action combination is based upon accumulated rewards, R , which are a function of the responsiveness and operability of the system. The reward will be the sum of response time and operability rewards ( R and R ).

r o 𝑅 = 𝑅 + 𝑅 𝑟 𝑜 1 𝑖𝑓 𝑡 − 𝑡 < − 𝑋 𝑟 𝑟 , 𝑜𝑙𝑑 0 𝑖𝑓 𝛥𝑃𝑅 ≥ 0 3 𝑖𝑓 𝛥𝑃𝑅 < 0 & ( 𝛥𝑃𝑅 − 𝛥𝑃𝑅 ) > 0 𝑅 = { − 0 . 1 𝑖𝑓 | 𝑡 − 𝑡 | < 𝑋 , 𝑅 = { ( 7 ) 𝑜𝑙𝑑 𝑟 𝑟 𝑟 , 𝑜𝑙𝑑 𝑜 − 5 𝑖𝑓 𝛥𝑃𝑅 < 0 & ( 𝛥𝑃𝑅 − 𝛥𝑃𝑅 ) ≤ 0 − 3 𝑖𝑓 𝑡 − 𝑡 > 𝑋 𝑜𝑙𝑑 𝑟 𝑟 , 𝑜𝑙𝑑 𝛥𝑃𝑅 = 𝑚𝑖𝑛 ( 𝑃𝑅 − 𝑃𝑅 ) 𝑚𝑎𝑥 The parameter t is the response time. PR is the pressure ratio and PR is the maximum allowa ble pressure ratio r max given as a function of corrected speed N . PR effectively defines a “do not exceed” line on the compressor map.

c max The subscript “old” refers to quantities resulting from the prior action while variables without the subscript refers the current state and action pair. X is a threshold value below which the change in thrust response time is considered negligible. In the presented example X = 0 , which takes advantage of the idealistic simulati on conditions under which the RL is applied . However, it should be noted that variations and sources of uncertainty will be present in real world applications. For that reason, X would need to be non - zero. The idea is simple : r eward reductions in response time , p enalize increases in response time , and penalize reductions in operability beyond the prescribed limit . The second R o condition shows that if the operability limit is in violation (i.e., 𝛥𝑃𝑅 < 0 ) , an improvement i n operability (i.e., ( 𝛥𝑃𝑅 − 𝛥𝑃𝑅 ) > 0 ) is rewarded . To encourage the search for better solutions, a slight penalty is incurred by 𝑜𝑙𝑑 staying at the same schedule or achieving the same thrust response time . The relative magnitude of the value s chosen in Eq. ( 7 ) should be considered to achieve efficien t learni ng .

It is acknowledged that quantify ing compressor operability with measurement data remains an open question.

Ideally, this would be done without any reliance on a model. Ref. [ 12 ] proposes an approach to adapting acceleration schedules based on the combust or discharge gas temperature or an estimate of it. In any case, it is outside of the focused intent of this paper. For demonstration , the maximum pressure ratio constraint fills this role . The compressor pressure ratio will be measured via sensors , as will the corrected HPC speed . While the measurement is feasible, the practical viability of the approach is unknown and beyond the scope of this paper .

In RL terminology, a policy is enacted by an agent. The agent gathers information about the environment and the impact of its actions and uses that information to update the quality function and the policy used to determine future action s . The quality function will be initialized and later updated as the agent applies actions and observes the rewards of those actions. Equation ( 8 ) is used to update the quality function at state s with action a .

( ) ( ) ( ) ( ) 𝑄 𝑠 , 𝑎 = 1 − 𝛼 𝑄 𝑠 , 𝑎 + 𝛼 ( 𝑅 + 𝛾 max 𝑄 ( 𝑠 ′ , 𝑎 ′ ) ) ( 8 ) 𝑜𝑙𝑑 𝑎 ′ Q is the quality function value from the previous update , R is the reward for the state and action pair, α is the learning old rate, and γ is the discount factor. The learning rate is a value between 0 and 1 with higher values encouraging faster adoption of new data. Since the measured feedback data from the actual engine is assumed to be more accurate tha n the digital twin, the learning rat e will be set to a relatively high value of 0.9 . The discount factor is a value between 0 and 1 with lower values favoring immediate rewards over long - term rewards. In th e demonstration presented in the next sub - section , γ was set to 0. 7 . The term multiplied by the discount factor is an estimate of the optimal future value.

The quality function value for each state and action combination can be initialized using information from the genetic algorithm optimization.

A design consideration w ith any RL application is the trade - off between exploration and exploitation. Exploration refers to the tendency of the policy to explore new state and action pairs in search for a better solution, and exploitation is the tendency of the policy to use know n information to maximize reward. Assuming the optimization is decent, initial quality approximations are relatively trustworthy and therefore the RL algorithm can favor exploitation. The epsilon greedy strategy [ 32 ] is employed for choosing which action to take . The parameter ε will be set at a value between 0 and 1. A random number generator will output a number between 0 and 1 and if the number is less than ε then the exploration method will be implemented. Otherwise, the exploitation method will be implemented. The exploration method will randomly choose which action to take while the exploitation option will choose the action with highest quality value. For the demonstration presented in the next sub - section, ε was chosen to be 0.05.

The nature of this application allows for some simplifications due to the limited number of action options and the ability to assume trends between action and impact . For instance, a more aggressive acceleration schedule will lead to a faster response time and reduced operability while a less aggressive schedule will lead to a slower response time and improved operability. When the quality function is updated, there a re instances when the quality adjustment for other state and action pairs can be inferred. For example, if the schedule was shifted to be more aggressive and the operability was reduced such that R was significantly reduced, it can be assumed that any acti on to make the schedule more aggressive will make the issue worse and will result in more negative rewards. Therefore, the quality function can be updated to discourage such an action.

B. Demonstration Optimizations were conducted for the NEW , MID, 3Q, and EOL engine. The optimization for the EOL utilized the fitness function provided by Eq. ( 2 ). These results were used to create the conservative schedule at whic h the limit controller will be initialized and will default to if the transient operability were to become a concern. The EOL results were also used to define a “ do not exceed ” HPC PR as a function of HPC c orrected speed. This limit was set slightly above the EOL transient running line. Optimizations for the 3Q, MID , and NEW engine sought to minimize the response time while not exceeding the maximum PR limit. An iterative root solver was used to just meet the PR limit and the optimizer used the fitness function defined below: 𝑓 = ( 9 ) 𝑡 𝑟 Results from the optimizations were used to create an RU schedule for each degradation level. The schedules are max plotted in Fig. 13 a while the transient running lines with the optimized fuel flow input are plotted in Fig. 13 b along with a representation of the maximum HPC PR limit. It is noted that the normalized fuel flow input profiles have similar shape but do vary significant ly w hen compared to each other in Fig. 14 . A newer engine can lead more Figure 13 . Schedules derived from the optimization results (a) and transient running lines with the optimized fuel flow inputs (b) aggressively with fuel flow input . However, these profiles appear to collapse to a similar normalized fuel flow input profile if they are stretched or contracted. To demonstrate this Fig. 14 also show s the NEW, MID, and 3Q profiles stretched to better match the EOL profile. The profiles are stretched by factors of 1.2 and 1.15 , and 1.1 respectively. T herefore , it is noted that near optimal results are expected by fixing a normalized fuel flow input profile which can be stretched or contracted to just meet the desired operability constraint . This approach would avoid the need to perform additional optimizations.

T he RL algorithm was applied assuming the digital twin is an exact match to the physical system. Figure 15 shows the possible acceleration schedules as the controller attempts to adjust from the original conservative schedule to a schedule Figure 14 . Normalized fuel flow command that minimizes response time for a new engine . Figure 16 a shows the progression of the acceleration schedule , and the profiles from the optimizations response time after each transient. Figure 16 b shows the running lines for the first, last , and most severe transients during the training period . Since the implementation of the schedule does not perfectly match the optimal results used to derive it, the operability limit is hit prior to reaching the most aggressive schedule (schedule 10) and instead settles on schedule 7, which just meets the opera bility limit. The thrust response time was reduced from 3.82s to 3.37s , a n 11.8% reduction .

Next, the health state of the engine was set to a mid - life state to simulate aging of the engine without updating the optimal sched ule solution . This scenario illustrates how the RL algorithm will adapt the schedule to assure adequate operability margin . This scenario could also be viewed as representing a mismatch in the actual system and the digital twin , for which the digital twin is optimistic about the engine’s health state . The results are shown in Fig. 17 . The RL agent Figure 15 . Acceleration schedule options for shifted from usin g schedule 7 to schedule 6 to mee t the the RL agent operability constraint while increasing the thrust response Figure 16 . Training of the RL agent from an initialized conservative acceleration schedule to a more aggressive schedule that minimiz es response time. (a) shows the schedule selection for each sequential transient and (b) shows the transient running lines time from 3.68 s to 3.76 s. These results demonstrate the ability of the RL algorithm to learn that the existing schedule is too aggressive and needs to be shift ed toward the more conserva tive original schedule .

The agent can learn and shift the schedule ind efinitely as the engine ages , but some additional benefit could be possible if the “optimal” schedule were updated , and the schedule options were refine d . To demonstrate this , the engine health state was changed to 3Q and the option of acceleration schedule s was refined to exist betwe en the optimal 3Q schedule and the conservative EOL schedule. The Q values were reset , and the schedule was reinitialized on the conservative side of the prior used schedule . T he RL agent was observed to shift toward the optimal schedule and settle around the schedule that just meets the imposed operability limit . For comparison, another set of Figure 17 . Actions of the RL agent when the engine simulations were conducted to observe results that help state is upda ted to a MID engine would occur without making these updat es . This resulted in shifting from the original schedule 6 to the original schedule 5 . It was observed that refinement in the schedule options resulted in a thrust response time reduction of only 0.02 s , which suggests tha t performing optimizations and deriving new optimal schedules over the lifespan of the engine is un necessary .

The results demonstrate the ability of the RL algorithm to minimize acceleration time while respecting operability limits. The RL was also able to slow the response down and prevent the schedule from becom ing more aggressive when operability measures indicated an undesirable decrease in operability margin due to engine degradation .

C. Discussion Results have suggested that a near optimal normalized fuel flow input profile can be used throughout the engine life span . Thus , the need to use the genetic algorithm or some other optimization scheme throughout the lifespan of the engine can be reduced or eliminated . However, i f the genetic algorithm were employed as the engine ages , optimizer settings could be modified to strike a good balance between exploration and exploitation. For instance, at the beginning of the life of the engine when the controller is representative of the fleet of engines rather than the specific engine , or after maintenance and sensed anomalies, exploration is more favorable. Exploitation is more favorable as the engine slowly degrades .

A benefit to using the RU limit approach over the Ṅ limiter approach is that it does not require the design of a closed loop controller. However, u nlike an Ṅ limit schedule , an RU limit schedule will not produce a consistent response time as the engine ages . One benefit of updating the RU limit schedule is that it could be used to maintain similar thrust response times for engines on the same aircraft, thus minimizing undesirable asymmetric thrust. This could be done by modifying the acceleration schedule of the faster engine(s) to match the responsiveness of the slowest engine. The acceleration schedule of the slowest engine could be optimized for responsiveness while applying operability constraint s , while the operability margin of the faster engine(s) is optimized using thrust responsiveness as a constrai nt .

To implement the RL method presented here in any practical sense , additional factors will need to be considered.

Among them will be strategies for considering variation s in operation and secondary effects such as power extraction and heat soak. If the RL were to remain active while the engine is operating in the field, the agent would need to be able to recognize when a transient is occurring and how to characterize it , using sensor fe edback to guide the learning process . A variety of factors could make each transient unique including the manner in which the throttle position is moved, the flight condition, the thermal state of the engine components, etc. These factors could become part of the state and accounted for in the quality function , thus allowing the schedule to adapt to these conditions . An alternative could be to build some conservativeness into the operability limit to ensure worst case transient scenarios are accommodated. A nother approach could be to put the engine through a “training” period at various intervals with in its life span so that the agent can adjust the schedule under controlled conditions.

VI. Summary T o optimize engine performance, transient operability must be considered. The proper combination of responsiveness, operability, and efficiency should be sought . T o do this, optimization techniques can be applied. A genetic algorithm has been employed to de termine the optimal fuel flow input for extreme transient scenarios , and the data has been used to create transient limit schedules. Results have demonstrated significant improvements in operability margin over the original transient limit schedules. An ex ample of this includes a 31% reduction in the amount of operability stack utilized during a snap acceleration at sea level static conditions. Studies have demonstrated , with some success , how the form of the fuel flow input profile can be generalized acros s various flight conditions to achieve near optimal results. Furthermore, it has been demonstrated how other objectives, such as the reduction of peak turbine inlet temperature, can be considered in the optimization to strike a balance between competing go als. Finally, a method has been proposed and demonstrated f or updating engine acceleration limit logic to minimize response time under operability constraints. T he method employs reinforcement learning techniques t hat leverage a digital twin to modify the acceleration schedule while protecting the engine by limiting the changes in the schedule and utilizing sensor feedback to assess the quality of each schedule modification and each action taken to adjust the schedu le . A simple application illustrated the ability of the approach to (1) efficiently find the optimal acceleration schedule that minimized thrust response time while respecting operability constraints , and (2) to adapt to changes in engine operation due to component aging . The example illustrated a reduction in thrust response time of ~11.8% compared to a schedule designed conservatively for an end - of - life engine. Results from the example also suggest that it is unnecessary to update the optimal schedule as the engine ages, thus reducing the workload for implementation. Topics for future investigation could include attempts to address the various challenges identified for practical implement ation of the reinforced learning approach for updating t ransient limit schedules . In addition, the approach could be modified to achieve goals more aligned with commercial engine applications.

Acknowledgments The author would like to acknowledge the Transformation al Tools and Technologies (T TT) project under the NASA Aeronautics Research Mission Directorate (ARMD ) that has supported this work.

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