Digital twin model of gas turbine and its application in warning of performance fault

Minghui HU, Ya HE,*, Xinzhi LIN, Ziyuan LU, Zhinong JIANG,Bo MA

a Key Lab of Engine Health Monitoring-Control and Networking of Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China

b Beijing Key Laboratory of High-end Mechanical Equipment Health Monitoring and Self-Recovery, Beijing University of Chemical Technology, Beijing 100029, China

c Capital Aerospace Machinery Corporation Limited, Beijing 100076, China

d Hangli (Group) Industrial Co., Ltd, Chengdu 611936, China

KEYWORDS Gas turbine;Digital twin model;Gas-path fault warning;Deviation degree analysis;Performance simulation

Abstract The digital twin-driven performance model provides an attractive option for the warn gas-path faults of the gas turbines.However, three technical difficulties need to be solved: (1)low modeling precision caused by individual differences between gas turbines,(2)poor solution efficiency due to excessive iterations, and (3) the false alarm and missing alarm brought by the traditional fixed threshold method.This paper proposes a digital twin model-based early warning method for gas-path faults that breaks through the above obstacles from three aspects.Firstly, a novel performance modeling strategy is proposed to make the simulation effect close to the actual gas turbine by fusing the mechanism model and measurement data.Secondly, the idea of controlling the relative accuracy of model parameters is developed.The introduction of an error module to the existing model can greatly shorten the modeling cycle.The third solution focuses on the early warning based on the digital twin model, which self-learns the alarm threshold of the warning feature of gas-path parameters using the kernel density estimation.The proposed method is utilized to analyze actual measured data of LM2500+,and the results verify that the new-built digital model has higher accuracy and better efficiency.The comparisons show that the proposed method shows evident superiority in early warning of performance faults for gas turbines over other methods.

1.Introduction

The gas turbine has become the essential power unit in aviation, shipbuilding, and power industry1–2.However,long-term service easily results in performance faults such as fouling, erosion, corridors, abrasion, and damage.When the early fault evolves into more severe gas circuit failure, it is likely to be inseparable from enormous economic losses and potential safety hazards3–4.Therefore,it is urgent to accurately and timely warn performance faults to ensure the reliability and availability of gas turbines5–7.The performance model driven by the digital twin provides an effective solution for fault early warning of gas turbines8–10.

The performance model is the rudiment of the digital twin;In other words,before defining the term the digital twin,some scholars have studied the performance model and suggested applying it to fault warning11–12.The general idea is that the simulation results of the performance model are regarded as a reference for the gas path parameters,and the gas path state can be judged by comparing the measured data with the reference value.However,this method has not been popularized for the following reasons.(1) It is well known that the performance model is based on component maps.In fact,the manufacturer provides a set of general component maps for the same type of gas turbine, that is, the performance models of the same type share the same map13.However,due to different manufacturing, assembly, and service conditions, there are component performance differences among individuals of the same type, thus bringing obvious deviation between the simulation results of the model and the actual values.It is difficult to meet the high-precision requirements of the digital twin model.(2) Traditionally, higher simulation precision of the model is achieved by ensuring solution accuracy of each balance equation, which inevitably requires a large number of iterative calculations.However, this will seriously reduce the simulation efficiency,leading to excessive simulation and poor convergence, contrary to the real-time ability of the digital twin model.(3)Most works of literature focus on the establishment and optimization of gas turbine performance models14,but few are devoted to solving the interaction between model simulation parameters and the measured data and exploring excellent early fault features to achieve effective gas-path fault warning.

Many methods to improve the simulation accuracy are proposed,among which the most popularly used is the adjustment of component mapping that includes scaling method15, stagestacking method16, and regression-characteristic-fitting method17.The scaling method can obtain the approximate component characteristics of the equipment by multiplying the original curve value by a specific coefficient18.The stagestacking method mainly adjusts the component characteristic diagram through stage analysis19–20.The regression-featurefitting method aims to establish a new component feature map generation model, such as the elliptic function and trigonometric function21–23.The above method can improve simulation precision, but its premise is that the initial performance state of the model is consistent with that of the actual gas turbine, which limits its engineering application.Moreover, sufficient component data is unavailable in many cases,resulting in the actual design point performance of gas turbines with complex configurations that may not match the performance provided by the manufacturer.In addition, the above modeling only considers the main parts such as compressor,combustion chamber,and turbine,while ignoring the auxiliary parts such as fuel system.The differences in fuel composition significantly impact the gas turbine’s overall performance.Thus, there is still research work to be done in improving the performance simulation of the gas turbines.

The volume inertia substitution method plays an important role in improving simulation efficiency24–25.This method mainly involves the unsteady mass balance across the plenum,and establishes the differential equation of pressure and flow to replace the transient flow balance equation.The calculation is simplified by a differential equation instead of a balanced equation.Obviously, this technique avoids each iteration of the correct function in the balanced equation, so it reduces the calculation time of each integration step and improves the efficiency and real-time of the design point.However, the technique still has the problem of easily falling into the local optimum in the off-design point.The differential equation established by the inertial substitution method may also lead to non-convergence of simulation due to the excessive pursuit of the local exact solution.

Many scholars have explored the application of performance simulation to gas path fault detection.Urban firstly proposed the gas-path analysis (GPA) method based on a linear model26,which promotes the development of fault diagnosis technology for gas turbines.Since then, the nonlinear gas-path diagnosis method27and adaptive gas-path analysis method11have been developed successively.Although hundreds of documents have been related to the application of the GPA model in actual gas turbines since 2000, the simulation model to be embedded in the monitoring system has been in the testing stage,and has not been popularized and applied.Two main reasons are as follows.(1)Due to the variety of fault sources, degrees, and combinations, GPA and its improved version encounter many difficulties in actual fault diagnosis.(2) Most gas turbines are equipped with an online monitoring system when put into use to ensure safe and reliable operation.However, most of these systems achieve early warning using fixed threshold alarm,which has poor performance under variable operating conditions.Specifically,it is easy to lead to false alarms and missed alarms.Although the system can save data after failure for subsequent analysis,users prefer the system to provide early warning in case of early performance failure,because it can avoid more cumbersome maintenance and even fatal accidents.

The digital twin model was first defined by NASA in 201028, and Tao and Qi commented on its development in 201929.More specifically, some scholars have explored it to design, manufacturing, operation, and maintenance of machinery equipment, such as aero engines and gas turbines30–33.Among the application in gas turbines, one of the most important ideas is developed from performance simulation.For gas turbines and similar equipment, many frameworks of the digital twin have been proposed to monitor performance and detect failure33–34.Researchers not only focus on establishing an accurate digital temperature but also study gas turbine performance prediction and fault diagnosis based on the digital twin models35–37.Moreover, Panov and Cruz-Manzo38have explored the development of performance digital twin based on real-time embedded computing, which can be leveraged with Internet-of-Things (IoT) cloud platforms.However,the research on performance fault early warning driven by the digital twin model needs to be supplemented.

To tackle the above issues, a digital twin model-based warning method for performance faults of gas turbines was proposed.First, a high-precision performance model was established by fusing the mechanism model and measured data, specifically, the initial point of the model was adapted to make the simulation performance match the actual performance.Then, a relative precision control strategy is proposed to enhance the modeling efficiency.And the calculation of the model is reduced appropriately to avoid the troubles of over simulation and non-convergence.Thus, a digital twin model obtained will meet the demands of higher accuracy and lower time cost.Finally, based on the results of the model, a deviation degree is defined as the warning feature for performance faults of the gas turbines.The threshold is self-learned by kernel density estimation of the measured data.Therefore, the warning of performance fault can be realized by using the digital twin model of the gas turbine.

2.Theoretical background

2.1.Basic performance model

At present, modular modeling by volume inertia is the most generally used performance simulation method to obtain the basic simulation model of the gas turbines.According to the volume inertia method23,39, the gas turbine system can be divided into two modules.One is the plenum module, with a certain control volume (such as pipeline connection section and combustor).It considers the volume inertia,the imbalance of its inlet, and the outlet flow that caused the change in fluid pressure in the volume.The other is the compressor and turbine module, which is characterized by the clear model physical interface, and its flow characteristics are usually given in the form of characteristic maps or characteristic curves of the whole component.Moreover, the assistant modules, such as the rotor module and load module, can be combined to form the modular model of the whole system.The establishment of each module is briefly introduced as follows.

2.1.1.Plenum

The plenum module represents the flow connection part with a certain volume.In the dynamic calculation, the plenum module mainly considers the pressure change between the inflow and outflow of the volume:

where q represents mass flow rate,p represents pressure,T represents temperature, R represents the gas constant, V represents volume; and the subscripts‘‘out”and‘‘in”indicate outlet and inlet, respectively.

2.1.2.Combustor

The combustor mainly produces high-temperature and highpressure gas after the mixture of compressed air and fuel.According to the principle of mass conservation and energy conservation, the combustor module takes the volume inertia of the combustor into account and ignores the thermal inertia of the combustor.The equations are as follows:where cprepresents the specific heat at constant pressure, Hfrepresents heating value, ηbrepresents combustion efficiency;and the subscript f refer to the fuel.

2.1.3.Compressor module and turbine module

Both the compressor and turbine are important parts of a gas turbine with strong nonlinearity.The former consumes the power produced by high-temperature and high-pressure gas,and its working medium is air; the latter expands the gas and emits power,and the working medium is mixed gas.The solution of their characteristic parameters is similar.In the following section, this paper takes the compressor as an example to describe the solution process of the working characteristics.

where q represents mass flow rate;P represents power;and the subscripts c and t refer to the compressor and turbine,respectively.

The thermodynamic properties of the turbine are different from those of the compressor, but the thermodynamic parameters of the turbine can be obtained by a method similar to the compressor module mentioned above.Moreover, the turbine components can also be modeled using component maps.The precision of the design point in the component maps plays an essential role in the accuracy of the overall performance simulation.

2.1.4.Assistant module

In addition to the above basic components, some assistant modules such as the rotor module must be added to form a complete system model of a gas turbine.The rotor is a component that connects the compressor and the turbine.One end of the rotor is the turbine that sends out the power, which, in turn,drives the shaft to rotate.The other end is the compressor that consumes the power caused by the shaft, and the kinetic energy changes with speed.In the earlier stage,the rotor accelerates or decelerates due to the unbalanced power on the rotor.In the simulation model,the process from the initial simulation point to the calculation of target operating conditions can be regarded as dynamic.In the actual simulation, the rotor module is often over-simulated due to the pursuit of absolute power equality, which leads to a decrease in simulation speed or the non-convergence of simulation.According to the law of momentum, the differential equation of the rotational inertia of the rotor is as follows.

where N represents shaft rotational speed; I represents the moment of inertia;Perepresents power of shaft;Merepresents torque of shaft;and subscripts C and T refer to the compressor and turbine, respectively.

2.1.5.Thermal properties of gas

The properties of working fluid have a crucial influence on the performance of a gas turbine.In the process of simulation,these characteristics must be treated strictly.

Specific heat at constant pressure cp,specific heat at constant volume cv:This is the heat absorbed by raising the temperature of 1 kg of gas by 1°C at constant pressure or constant volume.For a gas turbine, gas flow is stable, so only cpcan be used directly.In the performance calculation,when the Mach number is less than 0.4,the total temperature can be used to calculate cp,which is negligible for the loss of accuracy because the dynamic temperature is small in proportion to the total temperature.

The gas constant R: The gas constant often appears in the pressure and temperature change formula.It is numerically equal to the difference between cpand cv.The gas constant of a gas is equal to the general gas constant divided by the molecular weight of the gas.In addition, the general gas constant is equal to 8314.3 J/(mol ?K) and is mainly affected by the fuel properties.When the fuel is gas, the gas constant changes significantly with a change in the fuel–air ratio; when the fuel is kerosene or other liquid, the gas constant can be considered constant.

Calculation method of gas constant: Many reports have detailed the basic gas properties.To facilitate the calculation,the polynomial fitting Eq.(9) (describing the relationship between the gas constant of natural gas as fuel and the change in the fuel–air ratio) is used in the simulation40–41.

where FR represents the fuel–air ratio.

2.2.Calculation and application of gas-path parameters deviation

After the basic performance model is established,the deviation of gas-path parameters can be defined using the simulation model and the measured data,as shown in Eq.(10).The deviation is mainly used for fault diagnosis and fault warning of gas turbines.

where EP is the deviation between the measured value and the simulated value of the gas-path parameter,Pmis the measured value of the gas-path parameter, and Psis the simulated value of the gas-path parameter, normal value.

The term normal value is considered to be the value of gaspath parameters when the gas turbine operates without faults under certain atmospheric conditions and control parameters.The normal value can be achieved in two ways:

(1) Experimental measurement; (2) Establishing a gas turbine performance model.As we all know,it takes a lot of manpower and time to measure the normal value of gas-path parameters under each atmospheric environment and fuel consumption.Therefore,it is a simple and effective way to get the normal value through performance simulation, which also requires high accuracy.

The deviation of gas path parameters can be applied to fault diagnosis and fault early warning.In the application of fault diagnosis,the deviation value of component performance parameters is iteratively solved by using the deviation value of measured parameters, and then the gas path fault is identified by using this value according to the corresponding diagnosis rules.In the application of fault warning, the deviation value of measured parameters is compared with the alarm threshold,so as to judge whether the components are unhealthy.The traditional alarm threshold is set based on the experience or experiment, which is relatively simple and ineffective.The threshold setting plays a decisive role in judging the fault state.Therefore, it is an urgent necessity to study the threshold and alarm rules for accurate early earning.

3.Proposed method

A digital twin-based performance fault warning method is proposed in this paper.The principle of the method is shown in Figs.1, 2, 3 and 4, and the basic steps are as follows:

3.1.Establishment of the digital twin model

3.1.1.Basic performance model

According to the modularization method,the gas turbine consists of a compressor, combustor, turbine, rotor, plenum, and other major component performance modules.The compressor module and the turbine module are composed of a characteristic part,a temperature part,and a power part,wherein the characteristic part is composed of individual characteristic maps, the temperature part is composed of Eqs.(4) and (5),and the power part is composed of Eqs.(6) and(7).The combustor module is composed of the pressure part and temperature part in Eqs.(2) and (3), respectively.The rotor module and the plenum module are composed of Eqs.(8) and (1),respectively.

3.1.2.High-precision performance model

An initial point optimization method is presented to improve the basic model’s accuracy, and a high-precision performance model can be obtained.

Fig.1 Establishment of virtual entity.

First,temperature(T),pressure(p),and the fuel flow(qf)in each section from the measured parameters under design conditions are selected as input parameters.The compressor inlet flow(qc)and the combustor outlet temperature(T3)are preset as initial values and generally set as design values.Then the fuel–air ratio (FR) is calculated by the measured fuel flow,and FRcalis calculated by the preset T3.Using Eq.(6),we iteratively calculate T3and FRcaluntil FR and FRcalare equal.Then, T3under the current condition can be obtained.

From the calculated T3, the power of a high-pressure turbine (PeT) and compressor (PeC) can be obtained.Then, we compare PeTand PeC;if the two are equal,we further calculate the efficiency(η)and flow rate(q)of the current operating conditions of each component;if they are not equal,we repeat the first step (reset the qc).

The formula for calculating the T3caof the combustor according to the FR iteration is as follows.

where T2represents combustor inlet temperature; T3represents combustor outlet temperature; FRmrepresents fuel–air ratio calculated by measured fuel flow; and FRcalrepresents fuel–air ratio calculated by the inlet and outlet temperature of the combustor, respectively.

According to this step, the initial point of the simulation is calculated and optimized, and the new initial point is introduced to the basic model to obtain the high-precision performance model.

3.1.3.Virtual entity: A real-time simulation model

To further obtain a highly efficient model, the concept of parameter relative accuracy control is proposed and applied to the high-precision performance model.The specific processes are as follows.

First, before calculating the power difference,PeTηmech-PeC= ΔPein Eq.(8),the gas turbine’s current operating condition should be identified, and then different error acceptance ranges [ΔPe] have to be set according to the high or low operating conditions.If the power difference is acceptable, set the power difference to zero; otherwise, output the original value of the power difference.Finally, the corrected power difference has to be output for subsequent calculations.

According to the above steps, we make it run in real-time based on the high-precision performance model.Then,the virtual entity of the digital twin model can be obtained,as shown in Fig.1.

3.1.4.Digital twin model

Fig.2 Framework of digital twin model.

After the virtual entity has been established, a digital twin model of the gas turbine can be obtained, as shown in Fig.2.The physical entity is the actual gas turbine, and we mainly focus on the measurable performance data.The virtual entity is the real-time and high-precision simulation model,and can simulate the performance data under normal status.Specifically, the physical entity provides some measured data,such as inlet pressure and temperature of the compressor, as the input parameters of the virtual entity.Then, the virtual entity holds that the gas turbine with no-fault and real-time simulated performance data according to input parameters.The virtual entity supplies simulated data to the physical entity as the baseline for comparative analysis.

The measured data and the simulated data compose the digital-twin data.It includes pressures,temperatures,and rotation speeds in each key unit such as compressor, turbine, and power turbine.Finally, the digital-twin data can be utilized to condition monitoring and fault warning of gas turbines,and its application effect continuously interactive feedback with the digital twin model.

3.2.Warning feature construction and threshold self-learning

To deal with the false alarm and missing alarm caused by the fixed threshold, a parameter deviation degree is defined as the warning feature,and the alarm threshold of this feature is selflearned using the measured data of the gas turbine.The specific steps are as follows.

3.2.1.Deviation degree of gas-path parameters

First,the actual operating conditions are simulated by the digital twin model.The simulated data obtained by the model is used as the reference value of the current conditions, and the maximum and minimum values of the reference value are calculated.Then, we calculate the deviation degree between the measured data and the reference value according to Eq.(12).

where r represents the deviation degree;Pmaxis the simulation maximum value of the gas-path parameter, and Pminis the simulation minimum value of the gas-path parameter.When the r value exceeds a given threshold,the parameter has serious deviation, and the system generates an alarm.The threshold setting was described in detail in Section 3.2.2.

Fig.3 Construction of warning feature and self-learning-alarm threshold based on KDE.

Herein,how to calculate the maximum and minimum value of the simulation is described in detail.The intake temperature and pressure of the model are changed to the upper and lower limits of the working ambient temperature and ambient pressure of the gas turbine.Then, the maximum value and minimum value of the simulation under the current operating condition can be obtained.

3.2.2.Threshold self-learning based on kernel density estimation(KDE)

For the r defined above,a reasonable alarm threshold has to be set so that the system can effectively warn performance fault.We utilize kernel density estimation and normal data to set the threshold.

First, we select the normal data of the gas turbine with a certain sample capacity and calculate the r value of each parameter.Then,we perform a kernel density estimation based on the r value samples of the normal data to get the kernel density function:

where K(x)is a kernel function,and it is non-negative,the integral is 1, and the mean is 0,which is consistent with the property of probability density.Commonly used kernel functions include Gaussian kernel, gamma kernel, triangular, and uniform kernel functions.Moreover, h is the bandwidth, and n is the sample size.

Fig.4 Flow chart of performance fault early warning method.

Finally, the data boundary value under the confidence 1-α of the selected sample can be obtained, and it is defined as the alarm threshold of the r value of the parameter:

where P(x)is the probability in the x range,and[r]is the data boundary under confidence 1-α, that is, the alarm threshold.

3.3.Warning of performance fault

The performance fault warning of gas turbines can be realized based on the digital twin model and the deviation degree of the gas path parameters.The specific processes are as follows.

First,the measured ambient temperature,pressure,and fuel flow are fed to the model, and the simulated value of each parameter is calculated through the digital twin model.After that, the simulated value and the measured value are substituted into Eq.(12) to calculate the r value of each parameter.Finally, the r-value of each parameter is compared with its alarm threshold.If the alarm threshold is exceeded,the performance of the gas turbine is considered to be in a fault state.As shown in Fig.3 and Fig.4.

4.Analysis of Case 1

To verify the effectiveness of the proposed method, the data under normal and fault states of a gas turbine are used for modeling and validation.The validation can be divided into two aspects: 1) the accuracy and efficiency of the digital twin model; 2) the effectiveness of the warning method.

4.1.Basic information of gas turbine

The gas turbine used in this study is LM2500+,a double-shaft gas turbine for a gas transmission pipeline42.The main components of the gas turbine are a high-pressure compressor(HPC),a combustor,a high-pressure turbine(HPT),and a power turbine, and the power turbine is connected with the load.The main design parameters of LM2500+ are shown in Table 143,and the rated operation parameters of LM2500+are shown in Table 2.

The gas turbine selected in this study was overhauled on November 26, 2017.Fouling was found in the blades of the HPT and the power turbine during the overhaul.Therefore,the data of the gas turbine after the last overhaul and before this overhaul are analyzed.The data of the four operating condition points from May 19 to May 25 are used to validate the accuracy and real-time ability of the digital twin model, and representative data are shown in Table 3.A total of 673 sets of data under normal conditions from June to November are used for alarm threshold self-learning.The data distribution interval is shown in Table 4.The data before the overhaul on November 24 is used to validate the fault warning method,as shown in Table 5.In the table, P1represents the inlet pressure of compressor;T1represents the inlet temperature of compressor; P2represents the outlet pressure of compressor; T2represents the outlet temperature of compressor;P4represents the outlet pressure of turbine; T4represents the outlet temperature of turbine;T5represents the outlet temperature of power turbine (see Fig.5).

4.2.Establishment of digital twin model

4.2.1.Basic performance model

According to the structure of LM2500+ and the steps in Section 3.1.1, we establish a basic performance model.The software platform used in this study is MATLAB / Simulink.The model consists of six modules: the compressor, combustor, high-pressure turbine, power turbine, plenum, and rotor module,as is shown in Fig.6.The input parameters are ambient temperature,ambient pressure,and fuel flow,and the output parameters are temperature, pressure, flow, efficiency,rotational speed, and the power-turbine output power of each component.

Compressor module: this component comprises a characteristic module, temperature module, and power module.The characteristic module uses compressor component maps, the temperature module is represented by Eq.(4), and the power module is represented by Eq.(6), as is shown in Fig.7.The input parameters of the compressor module are air temperature, pressure, outlet pressure, and rotational speed.The airtemperature and pressure are from the external input of the model,the outlet pressure is from the combustor,and the rotational speed is from the rotor module.Furthermore,the output parameters of the compressor module are the outlet flow,temperature,and power,in which the outlet flow and temperature are transmitted to the combustor,and the power is transmitted to the rotor module.

Table 1 LM2500 + main design parameters43.

Table 2 LM2500 + rated operation parameters.

High-pressure turbine module (HPT): it consists of the characteristic module, temperature module, and power module.The characteristic module uses the characteristic maps of the HPT components.The temperature module is represented by Eq.(5),and the power module is represented by Eq.(7),as is shown in Fig.8.The input parameters of the HPT module are inlet temperature, pressure, fuel–air ratio, outlet pressure,and rotational speed.The inlet temperature, pressure, and fuel–air ratio are from the combustor, the outlet pressure is from the plenum, and the rotational speed is from the rotor module.The output parameters of the HPT module are inlet flow,outlet flow,outlet temperature,and power.The inlet flow is transmitted to the combustor, the outlet flow is transmitted to the plenum, the outlet temperature is transmitted to the power turbine, and the power is transmitted to the rotor module.

Power turbine module: the power turbine and HPT have similar thermodynamic properties so that the power turbine module can be established in the same way.The module is mainly composed of the characteristic module, temperature module,and power module,in which the characteristic module uses the maps of power turbine components, the temperature module is represented by Eq.(5),and the power module is represented by Eq.(7).The structural diagram of the power turbine module is similar to that shown in Fig.8.The input parameters of the power turbine module are inlet pressure,outlet pressure, inlet temperature, and fuel–air ratio.The inlet pressure is from the plenum, the outlet pressure is from the outside of the model, the inlet temperature is from the HPT,and the fuel–air ratio is from the combustor.The output parameters of the power turbine module are inlet flow, outlet temperature, output power, and outlet flow.The inlet flow is transmitted to the plenum, and the outlet temperature,output power, and outlet flow are transmitted outside of the model.

Combustor module:this is mainly composed of the plenum and temperature rise part, wherein the plenum is represented by Eq.(2), and the temperature rise part is composed of Eq.(3),as is shown in Fig.9.The input parameters of the combustor module are inlet flow, inlet temperature, outlet flow, and fuel flow.The inlet flow and inlet temperature are from the compressor,the outlet flow is from the HPT,and the fuel flow is from outside of the model.The output parameters of thecombustor module are inlet pressure, outlet temperature, outlet pressure, and fuel–air ratio, in which the inlet pressure is transmitted to the compressor, the outlet temperature and pressure are transmitted to the HPT, and the fuel–air ratio is transmitted to HPT and power turbine.

Table 3 Data of four operating condition points from May 19 to May 25.

Table 4 Data distribution interval for alarm threshold self-learning.

Table 5 Data before this overhaul on November 24.

Fig.5 LM2500 + gas turbine42.

Pure plenum module and rotor module: the pure plenum module is represented by Eq.(1), as is shown in Fig.10.The input parameters of the pure-plenum module are inlet temperature,inlet flow,and outlet flow,in which the inlet temperature and inlet flow are from the HPT, and the outlet flow is from the power turbine.The output parameters of the pure plenum module are the outlet pressure, which is transmitted to the HPT and power turbine.The rotor module is composed of Eq.(8), as is shown in Fig.11.The input parameters of the rotor module are the compressor power and HPT power, in which the compressor power is from the compressor, and the power of HPT comes from the HPT.The output parameters of the rotor module are rotational speed, which is transmitted to the compressor and HPT.

The simulation results of the basic performance model are shown in Table 6.

To compare the measured and simulated parameters, we define an error, and its calculation formula is as follows.

Fig.6 LM2500 + performance model.

Fig.7 Compressor module.

The measured parameters are shown in Table 3.Then, the errors of the basic performance model can be obtained, as are shown in Table 7.

In the table,ER1_P2represents the error of P2in the basic performance model, ER1_P4, ER1_T2, ER1_T4and ER1_T5are the same.

From Table 7, we can see that the simulation error of the basic performance model is large, which cannot characterize the actual gas turbine.

4.2.2.Improvement of the simulation accuracy

To enhance the precision of the simulation,on the basis of the basic performance model, we optimize the initial simulation point according to Section 3.1.2, and the adaptation of each parameter is shown in Table 8.The selected parameters are involved in the simulation, representing the compressor flow,pressure ratio, and efficiency of each component.

Deviation value:this is the deviation of the selected parameter before and after adaptation in the initial state of the simulation.It can be defined as follows.

where dv represents the deviation value; Pais the adapted value of the gas-path parameter, and Pdis the design value of the gas-path parameter.

Fig.8 High-pressure turbine module.

Fig.9 Combustor module.

Fig.10 Plenum module.

A high-precision performance model can be obtained by optimizing the initial simulation point.The simulation results of the high-precision performance model are shown in Table 9,and errors of the high-precision performance model can be obtained, as are shown in Table 10.

In the table,ER2_P2represents the error of P2in the highprecision performance model, ER2_P4, ER2_T2, ER2_T4and ER2_T5are the same.

Then,the simulation results of the basic and high-precision models are compared.Fig.12(a)–12(d)show the error comparison between the simulation results of the basic model and the high-precision model under the design point condition, 0.85 condition, 0.8 condition, and 0.7 condition, respectively.The abscissa in the figure is the comparison parameter,the ordinate is the error between the simulation results and the measured data under normal operation.

Fig.11 Rotor module.

Table 6 Simulation results of the basic performance model.

Table 7 Errors of the basic performance model.

Table 8 Deviation value of the selected parameters.

The main comparison results are as follows:

(1) The simulation error of the basic performance model for each parameter is relatively large, and the error of the high-pressure turbine outlet pressure(P4)is even as high as 25%.

(2) The simulation errors of the high-precision model to various parameters are no more than ±4%.It can be seen from the comparison that the precision of the high-precision model significantly improves compared with the basic model.

4.2.3.Validation of the digital twin model

To further promote the real-time ability of the model,the error module is introduced into the rotor module of the highprecision performance model by using the step in Section 3.1.3,as is shown in Fig.13, so that the digital twin model can be obtained.

The convergence time of the model is used to evaluate the simulation efficiency of the digital twin model.We compare the convergence time of the high-precision model and the digital twin model under various operating conditions, as is shown in Fig.14.The abscissa in the figure is the selected operating condition,i.e.,the design point operating condition,0.85 operating condition, 0.8 operating condition, and 0.7 operating condition.The ordinate is the time required for convergence of the simulation model.

The main comparison results are as follows:

(1) Under each condition,the convergence time of the highprecision model is more than 0.9 s, and reached as high as 2.35 s, while the convergence time of the digital twin model is not more than 0.2 s.

(2) Comparing the time under the same operating condition,it can be seen that the convergence time of the digital twin model is less than one-tenth of that of the highprecision model, which shows that the real-time ability of the digital twin model is obviously better.

In fact,the minimum data acquisition interval of almost all gas turbine monitoring systems is 1 s,that is,the running time interval of the actual physical model.However,as can be seenfrom Fig.14, the running time of the virtual entity is far less than 1 s.Therefore, the digital twin model can meet the requirements of real-time monitoring and analysis, while the high-precision model cannot.

Table 9 Simulation results of the high-precision performance model.

Table 10 Errors of the high-precision performance model.

Fig.12 Error comparison of simulation results of two performance models.

The simulation results of the digital twin model are shown in Table 11, and the errors of this model can be obtained, as are shown in Table 12.

In the table,ER3_P2represents the error of P2in the digital twin model, ER3_P4, ER3_T2, ER3_T4and ER3_T5are the same.

To verify whether the digital twin model affects the simulation accuracy in improving the real-time performance,we compare the simulation accuracy of the two models under different operating conditions, as shown in Fig.15.The abscissa in the figure is the selected comparison parameter,the ordinate is the error between the model simulation results and the measured data under normal operation, and the calculation is shown in Eq.(15).

Fig.13 Rotor module after adding error module.

Fig.14 Comparison of simulation time.

As can be seen from Fig.15, the simulation results of the high-precision model and the digital twin model have similar errors, so we conclude that the influence of the error module on the accuracy is within an acceptable range.Meanwhile,the simulation accuracy of some parameters not only does not decrease, but has fewer errors than that of the highprecision model.In general, the simulation error of the digital twin model is less than ±1.5%, and in most cases, less than± 1%.The reason for this phenomenon is that the highprecision model over-simulated to pursue absolute balance,thereby reducing the simulation accuracy of some parameters.The error module avoids this situation as much as possible and improves the simulation accuracy.

In conclusion,the digital twin model can ensure high simulation accuracy, reduce the convergence time and improve the simulation efficiency,which can meet the demands of real-time monitoring and fault warning of gas turbines.

4.2.4.Comparison of the actual parameters and the modeling parameters

To further validate the simulation accuracy of the virtual entity of the digital twin model,we utilize the model to analyze five days of performance data of the gas turbine.Analysis results are shown in Figs.16–20.

In Figs.16–20,subgraph(a)has shown trends of the actual performance parameters and the virtual entity calculation parameters, and subgraph (b) has shown relative errors between the two obtained by Eq.(15).Based on these analysis results, we can conclude that:

(1) Whether the gas turbine operates under constant or variable working conditions,the parameters trend of the virtual entity is consistent with that of the actual performance values;

(2) The errors between the simulation values of the virtual entity and the actual performance values are very small,basically less than ±2%.

To summarize,the simulation accuracy of the virtual entity is high, so the virtual entity of the digital twin model can well characterize the physical entity.

Table 11 Simulation results of digital twin model.

Table 12 Errors of digital twin model.

Fig.15 Simulation accuracy comparison.

Fig.16 Analysis results of P2.

Besides, we also analyze the running time of the virtual entity.The minimum data acquisition interval is 1 s, and the simulation time of each group of data is less than 1 s, which indicates the virtual entity is more efficient than the physical entity.Therefore, the digital twin model can be used for realtime analysis and fault warning.

4.3.Self-learning of alarm threshold of r value

The alarm threshold of r has to be set before the warning method is carried out.The 673 sets of normal operating data mentioned above are brought into the model for simulation,and the r values of the corresponding parameters are calculated.Then,we utilize Eq.(13)to estimate the kernel density for the r value of each parameter.

Fig.17 Analysis results of P4.

Fig.18 Analysis results of T2.

Fig.19 Analysis results of T4.

Fig.20 Analysis results of T5.

In kernel density estimation, two aspects are crucial: one is the choice of the kernel function K(x), and the other is the choice of bandwidth h.The choice of bandwidth is more important and has a more significant impact on the final estimation result.The Gaussian kernel is selected as the kernel function, and its formula is as follows:

After the K(x) is determined, R(K), M(k), and f′′can be determined in the above formula to obtain the asymptotic optimal bandwidth hAMISE.

With the Gaussian kernel function, after obtaining its hAMISE, we apply Eq.(13) to estimate the kernel density of the sample.The results are shown in Fig.21.

The confidence of the sample data selected in this study was 99.7%,so according to the r-value KDE function of P2,P4,T2,T4, T5, the boundary value can be obtained using Eq.(14).That is, the area enclosed by the kernel density estimation curve and the coordinate axis on the left side of the boundary value in the above figure accounted for 99.7%of the total area.The boundary-value,i.e.,alarm threshold of each parameter is shown in Table 13.

4.4.Validation of warning method

This section uses the alarm threshold of r values, and selects the data of four working points from May 19 to May 25 and the data before overhaul on November 24 to verify the effectiveness of the proposed alarm method.

4.4.1.Analysis of data under normal state

According to Section 3.2.1, we substitute the measured data under normal state into the calculation.Then, the r value of each parameter under each operating condition can be obtained, as is shown in Fig.22.

Based on the above results,the r value of each parameter is lower than the alarm threshold value under each operating condition, so it should be judged that the performance of the gas turbine is healthy, which is consistent with the actual situation.

4.4.2.Analysis of fault data

Similarly, we calculated the r value of each parameter in Table 5, as is shown in Fig.23.

As Fig.23 shows, three parameters exceed their alarm threshold under each operating condition: P4, T4, and T5.Among them, the r value of T4exceeded 100% under the 1.0 condition.Therefore, an alarm of performance fault is generated by the proposed method.

Fig.21 KDE of r value of normal data.

Table 13 Alarm thresholds for r values based on KDE.

Fig.22 r value of each parameter under normal state.

The subsequent overhaul revealed that the HPT and power turbine of the gas turbine has a fouling phenomenon;in particular, the HPT blades have severe fouling.Two parameters related to compressor fouling, i.e., P2and T2, do not exceed the alarm threshold, while the other three parameters related to turbine fouling, i.e., P4, T4, and T5, exceed the threshold.This shows that the analysis results are consistent with the actual situation.

4.4.3.Comparison of false alarm rate and missing alarm rate

The case utilizes the normal and fault data of the design point condition, 0.85 condition, 0.8 condition and 0.7 condition,which involves five parameters,namely P2,T2,P4,T4,and T5.In order to more intuitively evaluate the warning effect of different methods, the BPM-based TA, HPPM-based TA,DTM-based TA, and the proposed methods are selected to compare and analyze the warning effectiveness.Herein, it can be seen from Figs.22 and 23 that the early warning accuracy of the proposed method is 100%, that is, the false alarm rate and missing alarm rate are 0.The BPM-based TA, the HPPM-based TA, and the DTM-based TA method refer to utilizing the simulation results of the basic model, the highprecision model, and the digital twin model respectively to set the threshold so as to realize early warning.For the three methods, we defined two quantitative indicators that include missed alarm (MA) and false alarm (FA) rate to quantify the alarming accuracy, as shown in Eqs.(21)–(22).

Based on the limited case data available, we calculate and obtain the MA rates, FA rates, and average accuracies of BPM-based TA, HPPM-based TA, DTM-based TA, and the proposed method,as shown in Table 14.The results show that the proposed method has the highest early warning accuracy of 100%,that is,the false alarm rate and missing alarm rate are 0.This means that this method has obvious superiority compared to the other three methods in the performance fault warning of gas turbines.Compared with BPM-based TA, HPPM-based TA, and DTM-based TA, it can be seen that in addition to the excellent performance model, the more important reason for the high early warning rate achieved by the proposed method is to construct an appropriate warning feature and self-learning threshold method.

Fig.23 r value of each parameter under fault state.

5.Analysis of Case 2

To further validate the effectiveness of the warning method,we analyze the normal and the fault data of another gas turbine.The gas turbine, in this case, is also LM2500+, and a digital twin model of this gas turbine can be established by the proposed method.

Table 14 Comparison of warning accuracy between four methods.

5.1.Alarm threshold self-learning of r value

In the same way,the alarm threshold of each parameter of this gas turbine can be obtained by kernel density estimation,which is shown in Table 15.

5.2.Validation of warning method

The gas turbine has been operating under 0.7 operating condition before the failure causes its shutdown.Therefore, we mainly analyze the performance parameters before and after the failure under an operating condition.The calculation results of r are shown in Fig.24.

From the above analysis results of this case, it can be concluded that:

(1) When the gas turbine is in a normal state,r values of all parameters are under their thresholds, which means the warning method has no problem of false alarm.

(2) When the gas turbine is in a fault state,the r value of P4,36.16% exceeds its threshold, which means that the warning method sends out the correct alarm.

After the gas turbine was shut down, it is found that blade tip exhaust edge fracture fault has occurred in the turbine through borescope, as is shown in Fig.25.The alarm parameter is the outlet pressure of the turbine P4, which can correspond to the fault phenomenon.Remarkably, the significantchange of the r value of P4has been enough to attract the attention of field staff.

Table 15 Alarm thresholds of another gas turbine.

Fig.24 r values of four parameters under normal and fault state.

Fig.25 Results of a borescope.

In Section 4 and Section 5,it is indicated that the proposed method can be utilized to realize a real-time and high-precision performance simulation, moreover, achieve effective warning of gas-path fault for gas turbine.

6.Conclusions

This paper presents a digital twin model of gas turbine and studies its application in warning of performance fault.To improve accuracy and efficiency, the initial simulation point is firstly adapted through the measured data so that the performance model is closer to an actual gas turbine.Then, an operation-condition-related error module is introduced into the rotor module among the model, which can significantly promote the real-time ability of the simulation, thus building the digital twin model with low error and high effectiveness.Finally, based on the above simulation, a deviation degree of gas-path parameters is defined as the warning feature, and the alarm threshold is self-learned of kernel density estimation of the measured data of gas turbine under normal state.

The proposed method is validated by two actual cases of LM2500 + gas turbines.The simulation results show that the errors between the simulated values of the digital model and the actual values are basically within ±2%,and each calculation time of the model is far less than 1 s.Moreover, the comparisons indicate that the proposed warning strategy can achieve accurate warning of performance faults for gas turbines, the false alarm and the missed alarm rate caused by the fixed threshold can be decreased with the aim of the digital twin model.

7.Discussions

This study presents an early warning scheme with a digital twin model that enables performance simulation,data sensing,processing, and mapping technologies.This is a good attempt of the digital twin in the health management of turbomachinery and realizes the accurate and timely early warning of gas path failures.There is great hope that it can contribute to the transformation of maintenance strategy for gas turbines.Predictive maintenance with digital twins will be a bright research direction.

Traditional gas turbine maintenance strategies mostly stay in routine maintenance and subsequent maintenance, which are carried out regularly according to the prior or historical knowledge of gas turbines.However, lack of accurate fault warning and diagnosis, the above strategies are conservative and insecure.This means that it not only increase the cost of labor, fuel and materials, but also make it difficult to ensure the reliable operation of the gas turbine.Therefore, by establishing a complete digital twin model, we can intelligently interact the state data collected in real-time during the operation of a gas turbine with the digital model,and then diagnose the health state and fault symptoms of the equipment for prognostic and health management.Therefore, the use of gas turbines will meet the requirements of high operation safety,low maintenance cost, and long availability.

Nevertheless, the simulation and application of the digital twin model for gas turbines still face some challenges as follows.(1) The performance model of a gas turbine depends on the design parameters such as the component characteristic curve, but it is difficult to obtain a complete curve.(2) In this paper, two cases have verified the accuracy of the digital twin model for LM2500+.However,if this method is used to multiple types of gas turbines,a lot of repeated work will undoubtedly be involved, and the capacity of the model base may be hard to self-converge.(3) This paper defines an early warning feature to represent the health state of the gas turbine.Section 4 and 5 have only verified its effectiveness on one type of gas turbine.In the future,it is hoped that this method can be applied to the real-time early warning engineering of various gas turbines or popularized to other turbomachinery.(4)Digital twin has great development potential in improving gas turbine from the design,operation,maintenance of life cycle.Therefore,the work of this paper still has a lot of research space, which still requires breakthroughs in crucial technical problems such as performance model migration, and data-driven analysis.

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

This study was co-supported by the National Postdoctoral Program for Innovative Talent (No.BX20180031).