# Research on Initial Installed Power Loss of a Certain Type of Turbo-Shaft Engine Using Data Mining and Statistical Approach

Authors: Heng Wu; Benwei Li; Shufan Zhao; Xinyi Yang; Hanqiang Song
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The installed positions of three domestic turbo-shaft engines mounted on a certain type of ship-borne helicopter interfere with the intake air flow of the engines, resulting in a decline of engine performance after initial installation. Due to the difference of load and adjustment method under the bench and installed conditions, it is necessary to study the change in gas turbine power rather than output shaft power of the engine before and after installation to evaluate the engine initial installed power loss. In this paper, quantum-behaved particle swarm optimization (QPSO) is applied to optimize the calculation of gas turbine power at different steady states based on the component-level aerodynamic thermal model of gas generator. Then, extreme learning machine (ELM) is adopted for regressive identification of the established gas generator state assessment model based on data mining and the identification model is applied to engine installed condition. Finally, statistical analysis of engine initial installed gas turbine power loss at three installed positions is carried out, respectively. Results show that the values of engine initial installed gas turbine power loss at three installed positions all conform to the normal distribution, the mean values are 1.658%, 9.828%, and 5.089%, respectively, and a confidence interval with 95% confidence level of the mean values are (1.388%, 1.928%), (9.178%, 10.478%) and (4.308%, 5.870%), which can provide references for determining the power monitoring thresholds after engine installation.


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## Body

## 1. Introduction

The actual available power of turbo-shaft engine after installation is not only one of the main factors that determine the maximum flying speed, maximum climb rate and the use ceiling of helicopter, but also an important reference standard for setting health monitoring threshold of the engine [1, 2]. A certain type of ship-borne helicopter assembles with three domestic turbo-shaft engines. Due to the impact of installed positions and use environment, the actual available power of each engine after initial installation is lower than the power in engine bench test condition [3], and this power loss is analyzed. Usually, engine output shaft power is used to evaluate the engine performance. However, for a certain type of turbo-shaft engine, the output shaft is connected to a hydraulic dynamometer on engine bench test condition, and the hydraulic dynamometer does not have a feedback regulation. While, under installed condition, the engine output shaft is connected to the reducer and rotor, the rotor speed regulator adjusts the state of engine to keep the rotor speed constant. The differences in load and adjustment method make it impossible to determine the change value of engine output shaft power in the same state before and after installation to analyze the initial installed power loss caused by the disturbance of intake air flow. So it is necessary to evaluate the decline of doing work capacity of the engine after initial installation according to the change of gas turbine power [4, 5].Gas turbine power of the engine is usually not measurable. At present, calculations of gas turbine power for different types of engines at home and abroad basically adopt the methods of aerodynamic thermal modeling [6–13] and simulation modeling [14, 15], among which the typical ones are the following: Coban et al. [6] combined the aerodynamic performance calculation method with bench test data to evaluate the energy and dynamic characteristics of a military turbo-shaft engine. Zhu [7] applied a differential evolution algorithm to optimize the computational performance of the aerodynamic thermal model with the consideration of machining error and component performance degradation. Onder et al. [8] adopted numerical methods to analyze power generation and installed application of turbo-shaft engines from the perspective of energy and dynamics. Nkoi et al. [14] established simulation model for performance calculation of an original engine and the corresponding modified engine. Ghoreyshi et al. [15] proposed a method of computational fluid dynamics (CFD) flow field simulation for engine gas path components to calculate the engine output shaft power. All of the above studies require adequate engine design and measurable parameter data, but, under installed condition, few measurable parameters of a certain type of turbo-shaft engine limit the application of methods of aerodynamic thermal modeling and simulation modeling.In recent years, there have been a growing number of data-driven model identification studies [16–19]. Lu et al. [16] proposed a novel wiener model constructed by an optimized kernel extreme learning machine to identify the dynamic and static behavior of a gas turbine engine. Pan et al. [17] applied an artificial neural network for identifying the engine’s nonlinear auto regressive model with exogenous inputs. However, none of the above studies have taken gas turbine power as a parameter of identification model, coupled with the particularities of mission requirements; there is almost no research on initial installed gas turbine power loss of a certain type of turbo-shaft engine according to the literatures that have been reported.To address this problem, in this study, quantum-behaved particle swarm optimization (QPSO) was applied to optimize the calculation of gas turbine power at different steady states based on aerodynamic thermal model of gas generator, then the converted gas generator rotor speed was set as an input parameter, and the converted gas turbine outlet temperature and the converted gas turbine power were set as output parameters, to establish a gas generator state assessment model. In combination with the selected sample data, extreme learning machine (ELM) was adopted for regressive identification of the model, and then the identification model is applied to engine installed condition. Finally, statistical analysis of engine initial installed gas turbine power loss at three installed positions had been carried out, respectively.The rest of the paper is organized as follows. Section2 gives an introduction of calculation method of gas turbine power based on QPSO. Methodology for the study of initial installed gas turbine power loss is overviewed in Section 3 and then a specific research process is proposed. Section 4 presents the calculation and statistical analysis results. The conclusion is followed in Section 5.


## 2. Optimized Gas Turbine Power Calculation Method

### 2.1. Brief of QPSO

Particle Swarm Optimization (PSO) [20] was first proposed by Eberhart and Kennedy in the United States in 1995. It is a population-based evolutionary algorithm that simulates the bird flocking process and believes that information sharing among individuals in a population can provide evolutionary advantages, and cooperation as well as competition among individuals can solve the optimization problem. Based on PSO, QPSO [21] was proposed through the introduction of quantum mechanics principle. In quantum space, the state of particles is described by wave function, the Schrödinger equation is solved to obtain the probability density function of particles appearing at a certain point, and the particle search position is determined by the probability density function. The algorithm discards the particle velocity, so it is not only simple, but also has good stability as well as strong global search and optimization capabilities. The search equation for particle’s movement can be expressed as(1)xijk+1=Pijk±βkmbestjk-xijkln⁡1uijk(2)Pijk=φijkpbestijk+1-φijkgbestjk(3)mbestjk=1N∑i=1Npbestijk(4)βk=1-0.5T-kT+0.5where i=1,2,⋯,N, j=1,2,⋯,D, k=1,2,⋯,T, N is the population of particles, D is the particles’ dimension in solution space, T is the total number of iterations. pbesti represents the best position of the ith particle and gbest is the best position of the population. mbest denotes the mean best position defined as the mean of all the best positions of the population. Pij is the local attractor of ith particle to jth dimension. φ and u are the random number distributed uniformly in (0,1), respectively. xij is jth dimension in ith particle’s position. The plus and minus sign in (1) is determined according to the comparison between a random number in (0,1) and 0.5 before the position of each particle is updated. If the random number in (0,1) is less than 0.5, the plus sign is taken; otherwise the minus sign is taken. β is called contraction-expansion coefficient, which is the only control parameter of QPSO and adopts the linear decreasing strategy shown by (4).Relevant studies have proved that QPSO shows better convergence performance than some other algorithms such as PSO and genetic algorithm in solving some typical optimization problems [21].


### 2.2. Calculation Method of Gas Turbine Power Based on QPSO

In this study, we focus on a two-shaft turbo-shaft engine with a free turbine (for confidentiality reasons the engine type is omitted). A schematic diagram of studied turbo-shaft engine is displayed in Figure1, in which the gas generator shown in the dashed box is mainly composed of an intake port, a combined compressor, a combustion chamber, a gas turbine, and the front-end attachment transmissions including starter generator transmission, fuel regulator, and oil pump transmission. The power emitted by gas turbine drives the operation of compressor and front-end attachment transmissions. The numbers in the figure stand for the inlet or outlet of different components. For example, “3” stands for the outlet of combined compressor or the inlet of combustion chamber and “51” stands for the outlet of gas turbine.Figure 1
Schematic of studied turbo-shaft engine. (A) Intake port; (B) combined compressor; (C) combustion chamber; (D) gas turbine; (E) starter generator transmission; (F) fuel regulator and oil pump transmission; (G) free turbine; (H) reducer; (I) rotor or dynamometer; (J) tail nozzle.When engine is operating under steady states, the common working conditions on aerodynamics and rotor dynamics must be followed between components. Combining the component-level aerodynamic thermal model [7] of gas generator with the collected engine bench test parameter data, including atmospheric temperature t0, atmospheric pressure p0, fuel consumption of combustion chamber Wf, intake air flow W0, total pressure of compressor outlet p3, and gas generator rotor speed ng, the parameter values of each component’s inlet and outlet section can be achieved through calculation, especially gas turbine outlet temperature T5 and gas turbine power Pgt.Considering that this type of engine was modified from the prototype, only compressor component characteristics were tested, while gas turbine was adjusted only according to the performance change of the engine under bench test condition, and gas turbine component characteristics were not tested individually after the adjustment, coupled with the error in engine manufacturing and assembly process, the component-level aerodynamic thermal model of gas generator needs to be modified to obtain an optimized, more accurate value ofPgt. Since sensitivity of the target performance parameters to the selected component characteristic parameters to be modified has the most direct influence on the model correction accuracy, and the compressor working state and performance can be determined with high accuracy based on measured data and the component characteristics obtained from compressor characteristic test, change of gas turbine component characteristics directly determines the performance change of gas generator. Therefore, the critical parameters that affect the calculation accuracy of gas generator performance are gas turbine pressure-drop ratio πgt and gas turbine efficiency ηgt. In this study, the model simplification and the inaccuracy of component characteristics all boil down to the modification of gas turbine component characteristic parameters, and take πgt and ηgt as parameters to be optimized individually. Power balance of gas generator and error between the calculated and measured values of gas turbine outlet temperature make up the target equation which means that(5)F=αPgt-Pc+Pf+βT5-T5swhere Pc indicates the power absorbed by the compressor and Pf represents sum of the power consumed by front-end attachment transmissions, its value in engine design point is 18 kW and minor adjustments to it are performed to get the values of Pf in other steady states according to ng and characteristics of parts in front-end attachments. T5 and T5s are calculated and measured values of gas turbine outlet temperature, respectively. In order to simplify the calculation, the weighting coefficients α and β are all set to 1.The combination ofπgt and ηgt which makes the objective function value minimum is the solution obtained by optimization algorithm.In this paper, QPSO was invited to optimize the calculation of gas turbine power at different steady states. QPSO first generates a population of particles, and the number of particles usually takes 30-50. Each particleXi=[πgt,i,ηgt,i] in the population denotes a potential solution consisting of the component characteristic parameters to be optimized. All elements of a particle are initialized randomly into continuous values between 0 and 1. Since each element in the particle has different meaning and different range, it is necessary to convert them into their real values before being input to the aerodynamic thermal model for calculation.Forπgt, it is given by(6)zi1=zπmax-zπminXi1+zπminwhere zπmax=4 and zπmin=1 are the upper and lower boundary values of gas turbine pressure-drop ratio πgt, respectively.Forηgt, it is given by(7)zi2=zηmax-zηminXi2+zηminwhere zηmax=0.89 and zηmin=0.82 are the upper and lower boundary values of gas turbine efficiency ηgt, respectively.Then, input the convertedπgt and ηgt as well as the measured steady state data of ng, Wf, t0, p0, W0, and p3 in engine bench test to the component-level aerodynamic thermal model of gas generator, and T5, Pgt, and Pc can be achieved through calculation. In combination with the data Pf and T5s measured at the same steady state, QPSO evaluates the advantages and disadvantages of given parameters to be optimized according to the fitness function value which is calculated by the objective function shown in (5), and the goal of QPSO is to minimize the fitness function value. When the iterative optimization of the algorithm meets the set precision requirement, that is F(Xi)<σ=0.001, or reaches the set maximum number of iterations, stop the iteration and output the optimal solution gbest. Otherwise, update each particle’s best previous position pbesti and the population’s global best position gbest, calculate the local attractor Pi and mean best position mbest of each particle using (2) and (3), and then update particle’ new position according to (1), finally repeat the process of calculation on fitness function value and iterative optimization. After the optimized calculation, Pgt of a steady state can be obtained.


## 2.1. Brief of QPSO

Particle Swarm Optimization (PSO) [20] was first proposed by Eberhart and Kennedy in the United States in 1995. It is a population-based evolutionary algorithm that simulates the bird flocking process and believes that information sharing among individuals in a population can provide evolutionary advantages, and cooperation as well as competition among individuals can solve the optimization problem. Based on PSO, QPSO [21] was proposed through the introduction of quantum mechanics principle. In quantum space, the state of particles is described by wave function, the Schrödinger equation is solved to obtain the probability density function of particles appearing at a certain point, and the particle search position is determined by the probability density function. The algorithm discards the particle velocity, so it is not only simple, but also has good stability as well as strong global search and optimization capabilities. The search equation for particle’s movement can be expressed as(1)xijk+1=Pijk±βkmbestjk-xijkln⁡1uijk(2)Pijk=φijkpbestijk+1-φijkgbestjk(3)mbestjk=1N∑i=1Npbestijk(4)βk=1-0.5T-kT+0.5where i=1,2,⋯,N, j=1,2,⋯,D, k=1,2,⋯,T, N is the population of particles, D is the particles’ dimension in solution space, T is the total number of iterations. pbesti represents the best position of the ith particle and gbest is the best position of the population. mbest denotes the mean best position defined as the mean of all the best positions of the population. Pij is the local attractor of ith particle to jth dimension. φ and u are the random number distributed uniformly in (0,1), respectively. xij is jth dimension in ith particle’s position. The plus and minus sign in (1) is determined according to the comparison between a random number in (0,1) and 0.5 before the position of each particle is updated. If the random number in (0,1) is less than 0.5, the plus sign is taken; otherwise the minus sign is taken. β is called contraction-expansion coefficient, which is the only control parameter of QPSO and adopts the linear decreasing strategy shown by (4).Relevant studies have proved that QPSO shows better convergence performance than some other algorithms such as PSO and genetic algorithm in solving some typical optimization problems [21].


## 2.2. Calculation Method of Gas Turbine Power Based on QPSO

In this study, we focus on a two-shaft turbo-shaft engine with a free turbine (for confidentiality reasons the engine type is omitted). A schematic diagram of studied turbo-shaft engine is displayed in Figure1, in which the gas generator shown in the dashed box is mainly composed of an intake port, a combined compressor, a combustion chamber, a gas turbine, and the front-end attachment transmissions including starter generator transmission, fuel regulator, and oil pump transmission. The power emitted by gas turbine drives the operation of compressor and front-end attachment transmissions. The numbers in the figure stand for the inlet or outlet of different components. For example, “3” stands for the outlet of combined compressor or the inlet of combustion chamber and “51” stands for the outlet of gas turbine.Figure 1
Schematic of studied turbo-shaft engine. (A) Intake port; (B) combined compressor; (C) combustion chamber; (D) gas turbine; (E) starter generator transmission; (F) fuel regulator and oil pump transmission; (G) free turbine; (H) reducer; (I) rotor or dynamometer; (J) tail nozzle.When engine is operating under steady states, the common working conditions on aerodynamics and rotor dynamics must be followed between components. Combining the component-level aerodynamic thermal model [7] of gas generator with the collected engine bench test parameter data, including atmospheric temperature t0, atmospheric pressure p0, fuel consumption of combustion chamber Wf, intake air flow W0, total pressure of compressor outlet p3, and gas generator rotor speed ng, the parameter values of each component’s inlet and outlet section can be achieved through calculation, especially gas turbine outlet temperature T5 and gas turbine power Pgt.Considering that this type of engine was modified from the prototype, only compressor component characteristics were tested, while gas turbine was adjusted only according to the performance change of the engine under bench test condition, and gas turbine component characteristics were not tested individually after the adjustment, coupled with the error in engine manufacturing and assembly process, the component-level aerodynamic thermal model of gas generator needs to be modified to obtain an optimized, more accurate value ofPgt. Since sensitivity of the target performance parameters to the selected component characteristic parameters to be modified has the most direct influence on the model correction accuracy, and the compressor working state and performance can be determined with high accuracy based on measured data and the component characteristics obtained from compressor characteristic test, change of gas turbine component characteristics directly determines the performance change of gas generator. Therefore, the critical parameters that affect the calculation accuracy of gas generator performance are gas turbine pressure-drop ratio πgt and gas turbine efficiency ηgt. In this study, the model simplification and the inaccuracy of component characteristics all boil down to the modification of gas turbine component characteristic parameters, and take πgt and ηgt as parameters to be optimized individually. Power balance of gas generator and error between the calculated and measured values of gas turbine outlet temperature make up the target equation which means that(5)F=αPgt-Pc+Pf+βT5-T5swhere Pc indicates the power absorbed by the compressor and Pf represents sum of the power consumed by front-end attachment transmissions, its value in engine design point is 18 kW and minor adjustments to it are performed to get the values of Pf in other steady states according to ng and characteristics of parts in front-end attachments. T5 and T5s are calculated and measured values of gas turbine outlet temperature, respectively. In order to simplify the calculation, the weighting coefficients α and β are all set to 1.The combination ofπgt and ηgt which makes the objective function value minimum is the solution obtained by optimization algorithm.In this paper, QPSO was invited to optimize the calculation of gas turbine power at different steady states. QPSO first generates a population of particles, and the number of particles usually takes 30-50. Each particleXi=[πgt,i,ηgt,i] in the population denotes a potential solution consisting of the component characteristic parameters to be optimized. All elements of a particle are initialized randomly into continuous values between 0 and 1. Since each element in the particle has different meaning and different range, it is necessary to convert them into their real values before being input to the aerodynamic thermal model for calculation.Forπgt, it is given by(6)zi1=zπmax-zπminXi1+zπminwhere zπmax=4 and zπmin=1 are the upper and lower boundary values of gas turbine pressure-drop ratio πgt, respectively.Forηgt, it is given by(7)zi2=zηmax-zηminXi2+zηminwhere zηmax=0.89 and zηmin=0.82 are the upper and lower boundary values of gas turbine efficiency ηgt, respectively.Then, input the convertedπgt and ηgt as well as the measured steady state data of ng, Wf, t0, p0, W0, and p3 in engine bench test to the component-level aerodynamic thermal model of gas generator, and T5, Pgt, and Pc can be achieved through calculation. In combination with the data Pf and T5s measured at the same steady state, QPSO evaluates the advantages and disadvantages of given parameters to be optimized according to the fitness function value which is calculated by the objective function shown in (5), and the goal of QPSO is to minimize the fitness function value. When the iterative optimization of the algorithm meets the set precision requirement, that is F(Xi)<σ=0.001, or reaches the set maximum number of iterations, stop the iteration and output the optimal solution gbest. Otherwise, update each particle’s best previous position pbesti and the population’s global best position gbest, calculate the local attractor Pi and mean best position mbest of each particle using (2) and (3), and then update particle’ new position according to (1), finally repeat the process of calculation on fitness function value and iterative optimization. After the optimized calculation, Pgt of a steady state can be obtained.


## 3. Methodology for the Study of Initial Installed Gas Turbine Power Loss

### 3.1. Brief of ELM

ELM is an excellent feed-forward neural network algorithm with single hidden layer. It only needs to set the input weight and the number of hidden layer nodes to generate a unique optimal solution so that its learning efficiency increases dramatically. Only one ELM can realize multi-input multi-output model identification, the complexity of the algorithm is low; meanwhile the identification accuracy of the model is high [22, 23].ForN arbitrary distinct samples xi,yi, where xi=xi1,xi2,…,xinT∈Rn is a n×1 dimensional vector and yi=yi1,yi2,…,yimT∈Rm is a m×1 dimensional target vector, standard SLFN with N~ hidden neurons and activation function g(x) can approximate these N samples with high accuracy which means that(8)∑i=1N~αigiwi·xj+bi=yj,j=1,2,…,NEquation (8) can be simplified as Hα=Y, where H is the hidden layer output matrix and α is the weight vector connecting the hidden layer nodes and the output nodes. w is the weight vector connecting the input nodes and the hidden layer nodes. b denotes the bias of hidden layer nodes. Y is the matrix of desired output. The determination of the output weights is to find the least square solution with minimum norm to the linear system Hα=Y, which can be expressed as(9)α~=H+Y=HTH-1HTYwhere H+ is the Moore-Penrose generalized inverse of matrix H.


### 3.2. Identification of Gas Generator State Assessment Model

Gas turbine power and gas turbine outlet temperature are the principal parameters for evaluating engine gas generator performance. The former parameter characterizes the doing work capacity of engine gas generator, and the latter one determines the usage time of engine’s different states as well as the life of components. Therefore, gas turbine power and gas turbine outlet temperature are set as output parameters of the gas generator state assessment model. When the engine gradually rises from the ground idle state to the maximum state, parameters related to state change of the engine all increase accordingly. Therefore, only gas generator rotor speed is selected as input parameter of the gas generator state assessment model to characterize the different steady states of the engine. In addition, during the actual operation, state parameters and performance parameters of the engine are affected by different engine operating environment. In order to facilitate installed application of the model in the later period, the input and output parameters of the model are all converted to standard atmospheric condition(P0=101325Pa,T0=288.15K), as is listed in Table 1.Table 1
Input and output parameters of the model.
Parameters Unit type n g , c r/min Input P g t , c kW Output T 5 s , c °C Output10 turbo-shaft engines of the same type are selected, and optimized calculations are carried out for 5 typical steady states of each engine based on engine bench acceptance test data, respectively. The theoretical converted gas generator rotor speed of the 5 typical steady states are 25000 r/min, 30000 r/min, 31500 r/min, 32400 r/min and 33400 r/min, data selections of different steady states are according to the constant throttle position or fuel consumption basically, as well as the stability of gas generator rotor speed. All the selected data are within a certain range of the corresponding theoretical converted speed of each typical steady state. After the optimized calculations, convertng, Pgt and T5s of each steady state to standard atmospheric condition. Then, a total of 50 sample data points make up the training and validation sample sets of the gas generator state assessment model, and ELM is invited to perform regressive identification of the model.ELM can realize the multi-input multi-output model identification, in this application,y∈Rm denotes the target vector composed by model output parameters Pgt,c(k) and T5s,c(k), x∈Rn indicates the vector composed by model input parameter ng,c(k), the core idea of ELM for identifying the gas generator state assessment model is as follows: for unknown nonlinear function y=f(x), along with N arbitrary distinct samples xi,yi, seek a function f~ that minimizes the root mean square error between the network output vector y~=f~(x) of ELM and the target vector y.Network structure of ELM is displayed in Figure2, when ELM performs function approximation, the established SLFNs with N~ hidden neurons and sigmoid activation function g(x) can approximate the N samples with high accuracy which means that(10)∑i=1N~αigiwi·xj+bi=yj,j=1,2,…,NFigure 2
Network structure of ELM.Its matrix representation is(11)Hα=y(12)Hw1,⋯,wN~,b1,⋯,bN~,x1,⋯,xN=gw1·x1+b1⋯gwN~·x1+bN~⋯⋯⋯gw1·xN+b1⋯gwN~·xN+bN~N×N~(13)α=α1T⋮αN~TN~×m,y=y1T⋮yNTN×mWheng(x) is infinitely differentiable, not all the network parameters need to be adjusted, the input weights w and bias b of the hidden layer can be randomly selected during training. Then the determination of the output weights is to find the least square solution with minimum norm to the linear system (11) which can be expressed as(14)α~=H+y=HTH-1HTyFinally, the output of ELM is(15)y~=Hα~=f~xAfter identification and validation, apply the identified gas generator state assessment model to engine installed condition.


### 3.3. Specific Research Process on Initial Installed Power Loss

As the takeoff state of the helicopter is the critical state for monitoring, so the study mainly focuses on the corresponding engine installed power loss at three installed positions. Flight data of the ground takeoff state under the condition that the engine initial installed flight time is 50 hours are selected as the test data. The data extraction and processing method of the takeoff state is as follows: Determine the flight state information based on flight data file, keep the pitch angle and the tilt angle stable, the landing gear wheel-carrier signal is turned off (being 1), and the radio altitude is greater than 0, and make sure the above conditions are satisfied; after that, determine the “pitch position” at the maximum rate of change (first peak) position, 10 seconds delay from this position, and the data that is continuously stable for 5 seconds are selected; remove the maximum and minimum values for each parameter; the remaining data are averaged as the data under the takeoff state. In addition, since the rotor speed NR1 is measured in the actual flight condition and the output shaft torque is presented as a percentage of the limit torque value (2108.4N·m), the actual engine output shaft power Przj needs to be calculated through the following formulas:(16)nr=NR1×28.57(17)C=M×2108.4(18)Przj=nr×C9550where nr is the engine output shaft speed, 28.57 is gear ratio of the main reducer, M represents the percentage of torque limit value, and C is the actual torque value. For ease of comparison, the selected flight data also need to be converted to the standard atmospheric condition.Input converted gas generator rotor speedng,czj in the test data to the model to obtain the corresponding output parameters: converted gas turbine power Pgt,czj and converted gas turbine outlet temperature T5s,czj. For the same engine, data of steady state on engine bench acceptance test condition are selected according to the condition that converted engine output shaft power Pr,czj at the takeoff state under engine installed condition is equal to converted engine output shaft power Pr,ctj on engine bench acceptance test condition, and input converted gas generator rotor speed ng,ctj from the selected data to the model to obtain the corresponding output parameters: converted gas turbine power Pgt,ctj and converted gas turbine outlet temperature T5s,ctj as well. Through calculation and comparative analysis, the engine initial installed gas turbine power loss Pss is achieved.(19)Pss=Pgt,czj-Pgt,ctjPgt,czj×100%In summary, the specific research process of initial installed power loss of a certain type of turbo-shaft engine based on QPSO and ELM can be illustrated as in Figure3.Figure 3
Specific research process of initial installed power loss using QPSO and ELM.Step 1 (initializing).
First, a population of candidate solutions is generated, and all elements in the particleXi=[πgt,i,ηgt,i] are initialized randomly within the range of (0,1).Step 2 (calculation of gas generator performance).
The real values of elements in each particle, along with the selected data of steady states in engine bench test, are inputted to the aerodynamic thermal model of gas generator to get the values ofT5, Pgt and Pc.Step 3 (calculation of fitness function value).
In combination with the dataT5s and Pf of the same steady state, the fitness function value is obtained using (5). If F(Xi)<σ=0.001 (or the maximum number of iterations is reached), output the optimal solution and calculate Pgt; otherwise, QPSO generates a new population of particles and returns to Step 2 to repeat the calculation process. After the optimized calculation, Pgt of a steady state can be obtained.Step 4 (identification of gas generator state assessment model).
10 turbo-shaft engines of the same type are selected, and optimized calculations are carried out for 5 typical steady states of each engine, respectively. After data conversion, a total of 50 sample data points make up the sample sets to identify the gas generator state assessment model using ELM.Step 5 (apply the identified gas generator state assessment model to engine installed condition).
A total of 30 engines of the same type from 10 helicopters are selected as research objects, and flight data of the ground takeoff state under the condition that each engine’s initial installed flight time is 50 hours are selected as the test data. After data extraction and processing, input converted gas generator rotor speedng,c0zj⋯ng,c30zj in the test data to the model to obtain the corresponding outputs: converted gas turbine power Pgt,c0zj⋯Pgt,c30zj and converted gas turbine outlet temperature T5s,c0zj⋯T5s,c30zj. For each engine, data of steady state on engine bench acceptance test condition are selected according to the condition that converted engine output shaft power at the ground takeoff state under engine installed condition is equal to converted engine output shaft power on engine bench acceptance test condition, that is Pr,c0zj=Pr,c0tj⋯Pr,c30zj=Pr,c30tj. And input ng,c0tj⋯ng,c30tj from the selected engine bench acceptance test data to the model to obtain the corresponding outputs: Pgt,c0tj⋯Pgt,c30tj and T5s,c0tj⋯T5s,c30tj as well.Step 6 (statistical analysis ofPss at three installed positions).
In the light of (19), initial installed gas turbine power loss of the selected 30 engines Pss,0⋯Pss,30 is achieved through calculation, and Pss of 10 engines from each installed position are statistically analyzed.


## 3.1. Brief of ELM

ELM is an excellent feed-forward neural network algorithm with single hidden layer. It only needs to set the input weight and the number of hidden layer nodes to generate a unique optimal solution so that its learning efficiency increases dramatically. Only one ELM can realize multi-input multi-output model identification, the complexity of the algorithm is low; meanwhile the identification accuracy of the model is high [22, 23].ForN arbitrary distinct samples xi,yi, where xi=xi1,xi2,…,xinT∈Rn is a n×1 dimensional vector and yi=yi1,yi2,…,yimT∈Rm is a m×1 dimensional target vector, standard SLFN with N~ hidden neurons and activation function g(x) can approximate these N samples with high accuracy which means that(8)∑i=1N~αigiwi·xj+bi=yj,j=1,2,…,NEquation (8) can be simplified as Hα=Y, where H is the hidden layer output matrix and α is the weight vector connecting the hidden layer nodes and the output nodes. w is the weight vector connecting the input nodes and the hidden layer nodes. b denotes the bias of hidden layer nodes. Y is the matrix of desired output. The determination of the output weights is to find the least square solution with minimum norm to the linear system Hα=Y, which can be expressed as(9)α~=H+Y=HTH-1HTYwhere H+ is the Moore-Penrose generalized inverse of matrix H.


## 3.2. Identification of Gas Generator State Assessment Model

Gas turbine power and gas turbine outlet temperature are the principal parameters for evaluating engine gas generator performance. The former parameter characterizes the doing work capacity of engine gas generator, and the latter one determines the usage time of engine’s different states as well as the life of components. Therefore, gas turbine power and gas turbine outlet temperature are set as output parameters of the gas generator state assessment model. When the engine gradually rises from the ground idle state to the maximum state, parameters related to state change of the engine all increase accordingly. Therefore, only gas generator rotor speed is selected as input parameter of the gas generator state assessment model to characterize the different steady states of the engine. In addition, during the actual operation, state parameters and performance parameters of the engine are affected by different engine operating environment. In order to facilitate installed application of the model in the later period, the input and output parameters of the model are all converted to standard atmospheric condition(P0=101325Pa,T0=288.15K), as is listed in Table 1.Table 1
Input and output parameters of the model.
Parameters Unit type n g , c r/min Input P g t , c kW Output T 5 s , c °C Output10 turbo-shaft engines of the same type are selected, and optimized calculations are carried out for 5 typical steady states of each engine based on engine bench acceptance test data, respectively. The theoretical converted gas generator rotor speed of the 5 typical steady states are 25000 r/min, 30000 r/min, 31500 r/min, 32400 r/min and 33400 r/min, data selections of different steady states are according to the constant throttle position or fuel consumption basically, as well as the stability of gas generator rotor speed. All the selected data are within a certain range of the corresponding theoretical converted speed of each typical steady state. After the optimized calculations, convertng, Pgt and T5s of each steady state to standard atmospheric condition. Then, a total of 50 sample data points make up the training and validation sample sets of the gas generator state assessment model, and ELM is invited to perform regressive identification of the model.ELM can realize the multi-input multi-output model identification, in this application,y∈Rm denotes the target vector composed by model output parameters Pgt,c(k) and T5s,c(k), x∈Rn indicates the vector composed by model input parameter ng,c(k), the core idea of ELM for identifying the gas generator state assessment model is as follows: for unknown nonlinear function y=f(x), along with N arbitrary distinct samples xi,yi, seek a function f~ that minimizes the root mean square error between the network output vector y~=f~(x) of ELM and the target vector y.Network structure of ELM is displayed in Figure2, when ELM performs function approximation, the established SLFNs with N~ hidden neurons and sigmoid activation function g(x) can approximate the N samples with high accuracy which means that(10)∑i=1N~αigiwi·xj+bi=yj,j=1,2,…,NFigure 2
Network structure of ELM.Its matrix representation is(11)Hα=y(12)Hw1,⋯,wN~,b1,⋯,bN~,x1,⋯,xN=gw1·x1+b1⋯gwN~·x1+bN~⋯⋯⋯gw1·xN+b1⋯gwN~·xN+bN~N×N~(13)α=α1T⋮αN~TN~×m,y=y1T⋮yNTN×mWheng(x) is infinitely differentiable, not all the network parameters need to be adjusted, the input weights w and bias b of the hidden layer can be randomly selected during training. Then the determination of the output weights is to find the least square solution with minimum norm to the linear system (11) which can be expressed as(14)α~=H+y=HTH-1HTyFinally, the output of ELM is(15)y~=Hα~=f~xAfter identification and validation, apply the identified gas generator state assessment model to engine installed condition.


## 3.3. Specific Research Process on Initial Installed Power Loss

As the takeoff state of the helicopter is the critical state for monitoring, so the study mainly focuses on the corresponding engine installed power loss at three installed positions. Flight data of the ground takeoff state under the condition that the engine initial installed flight time is 50 hours are selected as the test data. The data extraction and processing method of the takeoff state is as follows: Determine the flight state information based on flight data file, keep the pitch angle and the tilt angle stable, the landing gear wheel-carrier signal is turned off (being 1), and the radio altitude is greater than 0, and make sure the above conditions are satisfied; after that, determine the “pitch position” at the maximum rate of change (first peak) position, 10 seconds delay from this position, and the data that is continuously stable for 5 seconds are selected; remove the maximum and minimum values for each parameter; the remaining data are averaged as the data under the takeoff state. In addition, since the rotor speed NR1 is measured in the actual flight condition and the output shaft torque is presented as a percentage of the limit torque value (2108.4N·m), the actual engine output shaft power Przj needs to be calculated through the following formulas:(16)nr=NR1×28.57(17)C=M×2108.4(18)Przj=nr×C9550where nr is the engine output shaft speed, 28.57 is gear ratio of the main reducer, M represents the percentage of torque limit value, and C is the actual torque value. For ease of comparison, the selected flight data also need to be converted to the standard atmospheric condition.Input converted gas generator rotor speedng,czj in the test data to the model to obtain the corresponding output parameters: converted gas turbine power Pgt,czj and converted gas turbine outlet temperature T5s,czj. For the same engine, data of steady state on engine bench acceptance test condition are selected according to the condition that converted engine output shaft power Pr,czj at the takeoff state under engine installed condition is equal to converted engine output shaft power Pr,ctj on engine bench acceptance test condition, and input converted gas generator rotor speed ng,ctj from the selected data to the model to obtain the corresponding output parameters: converted gas turbine power Pgt,ctj and converted gas turbine outlet temperature T5s,ctj as well. Through calculation and comparative analysis, the engine initial installed gas turbine power loss Pss is achieved.(19)Pss=Pgt,czj-Pgt,ctjPgt,czj×100%In summary, the specific research process of initial installed power loss of a certain type of turbo-shaft engine based on QPSO and ELM can be illustrated as in Figure3.Figure 3
Specific research process of initial installed power loss using QPSO and ELM.Step 1 (initializing).
First, a population of candidate solutions is generated, and all elements in the particleXi=[πgt,i,ηgt,i] are initialized randomly within the range of (0,1).Step 2 (calculation of gas generator performance).
The real values of elements in each particle, along with the selected data of steady states in engine bench test, are inputted to the aerodynamic thermal model of gas generator to get the values ofT5, Pgt and Pc.Step 3 (calculation of fitness function value).
In combination with the dataT5s and Pf of the same steady state, the fitness function value is obtained using (5). If F(Xi)<σ=0.001 (or the maximum number of iterations is reached), output the optimal solution and calculate Pgt; otherwise, QPSO generates a new population of particles and returns to Step 2 to repeat the calculation process. After the optimized calculation, Pgt of a steady state can be obtained.Step 4 (identification of gas generator state assessment model).
10 turbo-shaft engines of the same type are selected, and optimized calculations are carried out for 5 typical steady states of each engine, respectively. After data conversion, a total of 50 sample data points make up the sample sets to identify the gas generator state assessment model using ELM.Step 5 (apply the identified gas generator state assessment model to engine installed condition).
A total of 30 engines of the same type from 10 helicopters are selected as research objects, and flight data of the ground takeoff state under the condition that each engine’s initial installed flight time is 50 hours are selected as the test data. After data extraction and processing, input converted gas generator rotor speedng,c0zj⋯ng,c30zj in the test data to the model to obtain the corresponding outputs: converted gas turbine power Pgt,c0zj⋯Pgt,c30zj and converted gas turbine outlet temperature T5s,c0zj⋯T5s,c30zj. For each engine, data of steady state on engine bench acceptance test condition are selected according to the condition that converted engine output shaft power at the ground takeoff state under engine installed condition is equal to converted engine output shaft power on engine bench acceptance test condition, that is Pr,c0zj=Pr,c0tj⋯Pr,c30zj=Pr,c30tj. And input ng,c0tj⋯ng,c30tj from the selected engine bench acceptance test data to the model to obtain the corresponding outputs: Pgt,c0tj⋯Pgt,c30tj and T5s,c0tj⋯T5s,c30tj as well.Step 6 (statistical analysis ofPss at three installed positions).
In the light of (19), initial installed gas turbine power loss of the selected 30 engines Pss,0⋯Pss,30 is achieved through calculation, and Pss of 10 engines from each installed position are statistically analyzed.


## 4. Calculation Results and Statistical Analysis

### 4.1. Calculation Results of Gas Turbine Power at Selected Steady States

According to the above specific research steps, set the basic parameters of QPSO, which mainly include the number of particles in population is 30 and the maximum number of iterations is 30. Then, QPSO can be adopted for the optimized calculation ofPgt and T5 of each steady state from 10 turbo-shaft engines, among them, bench acceptance test data of 5 typical steady states and the corresponding performance calculated results of three engines installed in a helicopter are listed in Table 2, and change of fitness function value of a certain steady state in the optimization process is demonstrated in Figure 4.Table 2
Data of typical steady states and the corresponding performance calculated results.
Engine n g/(r/min) W f/(L/h) T 0/(°C) p 0/(Pa) W 0/(kg/s) π c/(-) P r/(kW) T 5 s/(°C) T 5/(°C) P g t/(kW) 1 26819.7 270.38 11.4 102533 5.038 3.825 385.9 491.7 485.6 860.6 30046.7 402.745 11.8 102539 5.788 5.024 777.5 585.5 591.7 1216 31032.4 451.931 11.2 102519 6.164 5.466 931.4 618.5 623.4 1400.8 32519.9 556.208 11.2 102503 6.677 6.12 1231.4 709.7 704 1631.5 33025.3 592.828 11.1 102519 6.824 6.325 1330 735.9 728.9 1719.1  2 25024.1 211.953 21.8 101660 4.294 3.146 229.4 472.3 466.5 659.6 30021.8 374.736 21.5 101650 5.556 4.776 686.5 587.7 588.6 1174.4 31712.4 469.258 21.2 101650 6.143 5.473 956.9 664.3 657.7 1448.4 32924.1 551.137 21.3 101636 6.532 5.988 1198.8 727.2 722.7 1641.2 33405.1 584.424 21.2 101629 6.679 6.205 1290.1 752.5 745.5 1704.6  3 25055.8 224.408 9.6 102250 4.505 3.311 253.1 452.3 448.3 691.9 30030.6 398.017 9.5 102257 5.796 5.033 767.3 598.5 593.5 1214.9 31731.4 496.431 9.2 102257 6.397 5.756 1071.1 689.9 686.7 1480.2 32398.5 539.857 9.1 102271 6.623 6.053 1204.5 728.5 722.6 1608.7 33234.8 586.745 9.0 102277 6.859 6.403 1357.8 771.6 766.1 1732.8Figure 4
Change of fitness function value.It can be seen from Table2 that the maximum absolute deviation between the actual measured value and the optimized calculation value of gas turbine outlet temperature for each steady state of the three turbo-shaft engines is 7°C, the average absolute deviation is 5.22°C, and the maximum relative deviation is 1.241%. Taking the influence of data acquisition accuracy and other factors into account, the optimized calculation results are very close to the actual values. In combination with the better convergence effect of the fitness function in the optimization process which is demonstrated in Figure 4, the accuracy of gas turbine power obtained by optimized calculation at each steady state is further verified.


### 4.2. Calculation Results of Initial Installed Gas Turbine Power Loss

Training and validation sample sets of gas generator state assessment model are formed by converting the data ofng, Pgt and T5s from each steady state to standard atmospheric condition, and then ELM is applied to carry out regressive identification of the model. Due to the large difference in the range of values of different parameters, the data of sample sets need to be normalized, and restore the data of each parameter after the training and validation. Apply the identified gas generator state assessment model to engine installed condition. According to the flight data extraction and processing method of ground takeoff state under the condition that the engine initial installed flight time is 50 hours, the extracted flight data of the same helicopter and data processing results are listed in Table 3.Table 3
Extracted flight data of the ground takeoff state and data processing results.
Torque 1 /(%) Torque 2 /(%) Torque 3 /(%) Rotor speed /(r/min) Engine speed 1 /(r/min) Engine speed 2 /(r/min) Engine speed 3 /(r/min) Pitch /(°) Atmospheric static temperature /(°C) Extracted flight data 62.808 50.891 60.659 206.6 31036 30747 31429 15.1 23.2 65.739 52.356 64.469 206.1 31297 30971 31416 15 23.2 66.227 52.794 65.446 207.5 31395 30977 31534 14.7 23.2 63.101 51.282 63.101 207.4 31068 30747 31405 14.5 23.2 60.269 47.57 59.878 207.4 30843 30590 31401 14.9 23.2 Data processing results 63.883 51.510 62.743 206.2 31134 30822 31417 14.9 23.2Convert the data processing results to standard atmospheric condition. For the same engine, data of steady state on engine bench acceptance test condition are selected according toPrzj=Prtj. Then input ng,czj and ng,ctj to the identified gas generator state assessment model; the corresponding outputs Pgt,czj, T5s,czj and Pgt,ctj, T5s,ctj can be achieved, respectively. Pss of each engine is calculated as well. The detailed results are listed in Table 4.Table 4
Performance calculation results on engine installed and bench condition.
Engine n g , c z j/(r/min) P g t , c z j/(kW) T 5 s , c z j/(°C) n g , c t j/(r/min) P g t , c t j/(kW) T 5 s , c t j/(°C) P s s/(%) 1 30643 1289.5 604.784 30516 1267.2 601.086 1.73 2 30336 1260 594.017 29485 1126.6 561.85 10.59 3 30921 1302.2 645.608 30484 1239.1 623.292 4.85From Table4, it is found that when the engine output shaft emits the same power before and after installation, Pgt,czj is greater than Pgt,ctj, which means that the engine needs to consume more energy after installation; in other words, energy loss is generated. Pss for three installed positions of a certain helicopter are 1.73%, 10.59%, and 4.85%, respectively.For the ground takeoff state, the critical factor forPss is the impact of engines’ installed positions on the intake air flow of each engine. Figure 5 displays the layout of the sample helicopter power plant; it is seen that the air inlets of engine 1 and engine 3 are located on both sides of the fuselage, while the air inlet of engine 2 is located at the upper right of the fuselage and the exhaust system turns to exhaust backwards with the direction of horizontal oblique 30 degrees. From the perspective of relative installed position of the three engines, engine 2 is located behind engine 1 and engine 3, and its exhaust system is in front of the intake system, making the intake airflow vulnerable to the influence of the other two engines, its own exhaust flow field, and the main rotor wake. As a result, the engine performance is affected, so Pss of engine 2 is relatively largest. The doing work capacity of engine 3 is also affected by the dual action of main rotor wake as well as the intake and exhaust flow of engine 2, while engine 1 is less affected by the other two engines, so the performance is relatively stable.Figure 5
Layout of the sample helicopter power plant.


### 4.3. Statistical Analysis for the Calculation Results in Three Installed Positions

For better quantitative analysis ofPss in three installed positions, a total of 30 engines of the same type from 10 helicopters are selected as the research objects. Initial installed flight data and the corresponding bench test data of each engine are extracted and processed. In combination with the identified gas generator state assessment model, Pss of each engine is obtained through calculation. Then the selected 30 engines are classified in accordance with their installed positions, and statistical analyses are carried out for Pss of 10 engines from each installed position. The serial number of engines in three installed positions are 1-10, 11-20, and 21-30, the performance calculation results of each engine are listed in Tables 5, 6, and 7, respectively. Pgt of different engines in bench and installed condition as well as Pss of each engine are intuitively demonstrated in Figures 6 and 7, respectively.Table 5
Performance calculation results of 10 engines in installed position 1.
The serial number of engines 01 02 03 04 05 06 07 08 09 10 P g t , c z j(kW) 1289.5 1268.8 1212.6 1237.4 1251.2 1257.2 1315.7 1228.5 1214.3 1276.1 P g t , c t j(kW) 1267.2 1245.3 1194.1 1209.4 1228.9 1240.2 1288.9 1205.3 1200.1 1263.6 P s s(%) 1.73 1.85 1.53 2.26 1.78 1.35 2.04 1.89 1.17 0.98Table 6
Performance calculation results of 10 engines in installed position 2.
The serial number of engines 11 12 13 14 15 16 17 18 19 20 P g t , c z j(kW) 1260 1135.8 1202 1217.7 1226.1 1148.6 1258.7 1216.2 1201.6 1198.5 P g t , c t j(kW) 1126.6 1029.4 1064.3 1091.3 1115.6 1052.4 1123.4 1096.1 1094.2 1084.3 P s s(%) 10.59 9.37 11.46 10.38 9.01 8.38 10.75 9.87 8.94 9.53Table 7
Performance calculation results of 10 engines in installed position 3.
The serial number of engines 21 22 23 24 25 26 27 28 29 30 P g t , c z j(kW) 1302.2 1342.6 1212.6 1331.5 1243.8 1218.5 1290 1257.3 1223.7 1264.6 P g t , c t j(kW) 1239.1 1263.2 1159.9 1239 1195.8 1176.2 1215.8 1205.4 1159.5 1184 P s s(%) 4.85 5.91 4.34 6.95 3.86 3.47 5.76 4.13 5.25 6.37Figure 6
P g t of different engines on bench and installed conditions.Figure 7
P s s of each engine.The Shapiro-Wilk W test is a normality test that has been designated as a national standard. It was proposed by Shapiro and Wilk in 1965 and requires a sample size of 3 to 50. This test method can be applied to examine whether a batch of observations or a batch of random numbers come from the same normal distribution [24]. The test question is as follows: H0, the population obeys normal distribution; H1, the population disobeys normal distribution.Statistical analyses are performed onPss of 10 engines from each installed position. Take Pss of 10 engines in installed position 2; for example, the specific implementation steps are as follows.( 1 ) Arrange the sample values in nondecreasing order.(20)8.38≤8.94≤9.01≤9.37≤9.53≤9.87≤10.38≤10.59≤10.75≤11.46( 2 ) Calculate the value of statistic W using (21). In the formula, ak(W) is read from schedule 10 in literature [24]. For ease of calculation, the list of values is illustrated as in Table 8.(21)W=∑k=1n/2akWXn+1-k-Xk2∑k=1nXk-X-2Table 8
The list of values for calculation.
k X ( k ) X ( 11 - k ) X ( 11 - k ) - X ( k ) a k ( W ) 1 8.38 11.46 3.08 0.5739 2 8.94 10.75 1.81 0.3291 3 9.01 10.59 1.58 0.2141 4 9.37 10.38 1.01 0.1224 5 9.53 9.87 0.34 0.0399( 3 ) Make a judgment: if W<Wα, reject H0; otherwise accept H0. Among them, Wα is read from schedule 11 in literature [24] for a given confidence level α and sample size n.It can be seen from Table8 that ∑k=110(X(k)-X-)2=8.254, ∑k=15ak(W)X(11-k)-X(k)=2.7426 and W=2.74262/8.254=0.9113. For α=0.05, it is found that W0.05=0.842. Owing to W>Wα, so accept H0, and it is considered that Pss at installed position 2 obeys the normal distribution.Assuming that the populationX~N(μ,σ2), μ and σ2are mean and variance of the normal population, respectively. Pss of 10 engines in installed position 2 are the samples from the population X. X- is the sample mean and S2 is the sample variance. It concludes that X- is an unbiased estimate of μ, and, on the basis of interval estimation theorem of sample mean from a normal population, a confidence interval with 95% confidence level of μ can be expressed as X-±(S/n)tα/2(n-1). According to the sample data, it is calculated that X-=9.828, S=0.8254=0.9085, n=10 and t0.025(9)=2.2622, then the confidence interval of the mean value of Pss at installed position 2 is obtained as 9.828±0.9085/10×2.2622, namely (9.178%, 10.478%).Similarly, the W-normality test and the confidence interval estimation forPss at installed position 1 and installed position 3 are carried out, respectively.For installed position 1,∑k=110(X(k)-X-)2=1.428, ∑k=15ak(W)X(11-k)-X(k)=1.1777, W=1.17772/1.428=0.9711, W>W0.05=0.842, so it is considered that Pss at installed position 1 obeys the normal distribution. According to the sample data, it is calculated that X-=1.658,S=0.1428=0.3779, n=10 and t0.025(9)=2.2622, then a confidence interval with 95% confidence level of the mean value of Pss at installed position 1 is obtained as 1.658±0.3779/10×2.2622, namely, (1.388%, 1.928%).For installed position 3,∑k=110(X(k)-X-)2=11.924, ∑k=15ak(W)X(11-k)-X(k)=3.3941, W=3.39412/11.924=0.9661, W>W0.05=0.842, so it is considered that Pss at installed position 3 obeys the normal distribution. According to the sample data, it is calculated that X-=5.089,S=1.1924=1.092, n=10 and t0.025(9)=2.2622, then a confidence interval with 95% confidence level of the mean value of Pss at installed position 3 is obtained as 5.089±1.092/10×2.2622, namely, (4.308%, 5.870%).In summary, Table9 lists the statistical analysis results of Pss at three installed positions. The boxplots of Pss at three installed positions is demonstrated in Figure 8.Table 9
Statistical analysis results ofPss at three installed positions.
Installed position Performance evaluation index m a x δ % m e d i u m δ % m i n δ % m e a n δ % s t d δ % 95% confidence interval 1 2.26 1.755 0.98 1.658 0.1428 (1.388%, 1.928%) 2 11.46 9.7 8.38 9.828 0.8254 (9.178%, 10.478%) 3 6.95 5.05 3.47 5.089 1.1924 (4.308%, 5.870%)Figure 8
Boxplots ofPss at three installed positions.It is concluded from Table9 and Figure 8 that the mean value of Pss at three installed positions of a certain type of ship-borne helicopter are 1.658%, 9.828%, and 5.089%, respectively. On the basis of sample estimation, a confidence interval with 95% confidence level of the mean value of Pss at three installed positions are (1.388%, 1.928%), (9.178%, 10.478%), and (4.308%, 5.870%), which are important references for determining the power monitoring thresholds after engine installation.In addition, statistical calculations have found that gas turbine outlet temperature of the engines rise in all three installed positions due to the increased energy consumption caused byPss. Although the maximum temperature limit is still not exceeded, the usage time of the engine in large states and the life of components have been affected to a certain extent. With the increase of engine using time, the engine performance will decline, and the value of gas turbine outlet temperature will further increase. It should be monitored with emphasis.


## 4.1. Calculation Results of Gas Turbine Power at Selected Steady States

According to the above specific research steps, set the basic parameters of QPSO, which mainly include the number of particles in population is 30 and the maximum number of iterations is 30. Then, QPSO can be adopted for the optimized calculation ofPgt and T5 of each steady state from 10 turbo-shaft engines, among them, bench acceptance test data of 5 typical steady states and the corresponding performance calculated results of three engines installed in a helicopter are listed in Table 2, and change of fitness function value of a certain steady state in the optimization process is demonstrated in Figure 4.Table 2
Data of typical steady states and the corresponding performance calculated results.
Engine n g/(r/min) W f/(L/h) T 0/(°C) p 0/(Pa) W 0/(kg/s) π c/(-) P r/(kW) T 5 s/(°C) T 5/(°C) P g t/(kW) 1 26819.7 270.38 11.4 102533 5.038 3.825 385.9 491.7 485.6 860.6 30046.7 402.745 11.8 102539 5.788 5.024 777.5 585.5 591.7 1216 31032.4 451.931 11.2 102519 6.164 5.466 931.4 618.5 623.4 1400.8 32519.9 556.208 11.2 102503 6.677 6.12 1231.4 709.7 704 1631.5 33025.3 592.828 11.1 102519 6.824 6.325 1330 735.9 728.9 1719.1  2 25024.1 211.953 21.8 101660 4.294 3.146 229.4 472.3 466.5 659.6 30021.8 374.736 21.5 101650 5.556 4.776 686.5 587.7 588.6 1174.4 31712.4 469.258 21.2 101650 6.143 5.473 956.9 664.3 657.7 1448.4 32924.1 551.137 21.3 101636 6.532 5.988 1198.8 727.2 722.7 1641.2 33405.1 584.424 21.2 101629 6.679 6.205 1290.1 752.5 745.5 1704.6  3 25055.8 224.408 9.6 102250 4.505 3.311 253.1 452.3 448.3 691.9 30030.6 398.017 9.5 102257 5.796 5.033 767.3 598.5 593.5 1214.9 31731.4 496.431 9.2 102257 6.397 5.756 1071.1 689.9 686.7 1480.2 32398.5 539.857 9.1 102271 6.623 6.053 1204.5 728.5 722.6 1608.7 33234.8 586.745 9.0 102277 6.859 6.403 1357.8 771.6 766.1 1732.8Figure 4
Change of fitness function value.It can be seen from Table2 that the maximum absolute deviation between the actual measured value and the optimized calculation value of gas turbine outlet temperature for each steady state of the three turbo-shaft engines is 7°C, the average absolute deviation is 5.22°C, and the maximum relative deviation is 1.241%. Taking the influence of data acquisition accuracy and other factors into account, the optimized calculation results are very close to the actual values. In combination with the better convergence effect of the fitness function in the optimization process which is demonstrated in Figure 4, the accuracy of gas turbine power obtained by optimized calculation at each steady state is further verified.


## 4.2. Calculation Results of Initial Installed Gas Turbine Power Loss

Training and validation sample sets of gas generator state assessment model are formed by converting the data ofng, Pgt and T5s from each steady state to standard atmospheric condition, and then ELM is applied to carry out regressive identification of the model. Due to the large difference in the range of values of different parameters, the data of sample sets need to be normalized, and restore the data of each parameter after the training and validation. Apply the identified gas generator state assessment model to engine installed condition. According to the flight data extraction and processing method of ground takeoff state under the condition that the engine initial installed flight time is 50 hours, the extracted flight data of the same helicopter and data processing results are listed in Table 3.Table 3
Extracted flight data of the ground takeoff state and data processing results.
Torque 1 /(%) Torque 2 /(%) Torque 3 /(%) Rotor speed /(r/min) Engine speed 1 /(r/min) Engine speed 2 /(r/min) Engine speed 3 /(r/min) Pitch /(°) Atmospheric static temperature /(°C) Extracted flight data 62.808 50.891 60.659 206.6 31036 30747 31429 15.1 23.2 65.739 52.356 64.469 206.1 31297 30971 31416 15 23.2 66.227 52.794 65.446 207.5 31395 30977 31534 14.7 23.2 63.101 51.282 63.101 207.4 31068 30747 31405 14.5 23.2 60.269 47.57 59.878 207.4 30843 30590 31401 14.9 23.2 Data processing results 63.883 51.510 62.743 206.2 31134 30822 31417 14.9 23.2Convert the data processing results to standard atmospheric condition. For the same engine, data of steady state on engine bench acceptance test condition are selected according toPrzj=Prtj. Then input ng,czj and ng,ctj to the identified gas generator state assessment model; the corresponding outputs Pgt,czj, T5s,czj and Pgt,ctj, T5s,ctj can be achieved, respectively. Pss of each engine is calculated as well. The detailed results are listed in Table 4.Table 4
Performance calculation results on engine installed and bench condition.
Engine n g , c z j/(r/min) P g t , c z j/(kW) T 5 s , c z j/(°C) n g , c t j/(r/min) P g t , c t j/(kW) T 5 s , c t j/(°C) P s s/(%) 1 30643 1289.5 604.784 30516 1267.2 601.086 1.73 2 30336 1260 594.017 29485 1126.6 561.85 10.59 3 30921 1302.2 645.608 30484 1239.1 623.292 4.85From Table4, it is found that when the engine output shaft emits the same power before and after installation, Pgt,czj is greater than Pgt,ctj, which means that the engine needs to consume more energy after installation; in other words, energy loss is generated. Pss for three installed positions of a certain helicopter are 1.73%, 10.59%, and 4.85%, respectively.For the ground takeoff state, the critical factor forPss is the impact of engines’ installed positions on the intake air flow of each engine. Figure 5 displays the layout of the sample helicopter power plant; it is seen that the air inlets of engine 1 and engine 3 are located on both sides of the fuselage, while the air inlet of engine 2 is located at the upper right of the fuselage and the exhaust system turns to exhaust backwards with the direction of horizontal oblique 30 degrees. From the perspective of relative installed position of the three engines, engine 2 is located behind engine 1 and engine 3, and its exhaust system is in front of the intake system, making the intake airflow vulnerable to the influence of the other two engines, its own exhaust flow field, and the main rotor wake. As a result, the engine performance is affected, so Pss of engine 2 is relatively largest. The doing work capacity of engine 3 is also affected by the dual action of main rotor wake as well as the intake and exhaust flow of engine 2, while engine 1 is less affected by the other two engines, so the performance is relatively stable.Figure 5
Layout of the sample helicopter power plant.


## 4.3. Statistical Analysis for the Calculation Results in Three Installed Positions

For better quantitative analysis ofPss in three installed positions, a total of 30 engines of the same type from 10 helicopters are selected as the research objects. Initial installed flight data and the corresponding bench test data of each engine are extracted and processed. In combination with the identified gas generator state assessment model, Pss of each engine is obtained through calculation. Then the selected 30 engines are classified in accordance with their installed positions, and statistical analyses are carried out for Pss of 10 engines from each installed position. The serial number of engines in three installed positions are 1-10, 11-20, and 21-30, the performance calculation results of each engine are listed in Tables 5, 6, and 7, respectively. Pgt of different engines in bench and installed condition as well as Pss of each engine are intuitively demonstrated in Figures 6 and 7, respectively.Table 5
Performance calculation results of 10 engines in installed position 1.
The serial number of engines 01 02 03 04 05 06 07 08 09 10 P g t , c z j(kW) 1289.5 1268.8 1212.6 1237.4 1251.2 1257.2 1315.7 1228.5 1214.3 1276.1 P g t , c t j(kW) 1267.2 1245.3 1194.1 1209.4 1228.9 1240.2 1288.9 1205.3 1200.1 1263.6 P s s(%) 1.73 1.85 1.53 2.26 1.78 1.35 2.04 1.89 1.17 0.98Table 6
Performance calculation results of 10 engines in installed position 2.
The serial number of engines 11 12 13 14 15 16 17 18 19 20 P g t , c z j(kW) 1260 1135.8 1202 1217.7 1226.1 1148.6 1258.7 1216.2 1201.6 1198.5 P g t , c t j(kW) 1126.6 1029.4 1064.3 1091.3 1115.6 1052.4 1123.4 1096.1 1094.2 1084.3 P s s(%) 10.59 9.37 11.46 10.38 9.01 8.38 10.75 9.87 8.94 9.53Table 7
Performance calculation results of 10 engines in installed position 3.
The serial number of engines 21 22 23 24 25 26 27 28 29 30 P g t , c z j(kW) 1302.2 1342.6 1212.6 1331.5 1243.8 1218.5 1290 1257.3 1223.7 1264.6 P g t , c t j(kW) 1239.1 1263.2 1159.9 1239 1195.8 1176.2 1215.8 1205.4 1159.5 1184 P s s(%) 4.85 5.91 4.34 6.95 3.86 3.47 5.76 4.13 5.25 6.37Figure 6
P g t of different engines on bench and installed conditions.Figure 7
P s s of each engine.The Shapiro-Wilk W test is a normality test that has been designated as a national standard. It was proposed by Shapiro and Wilk in 1965 and requires a sample size of 3 to 50. This test method can be applied to examine whether a batch of observations or a batch of random numbers come from the same normal distribution [24]. The test question is as follows: H0, the population obeys normal distribution; H1, the population disobeys normal distribution.Statistical analyses are performed onPss of 10 engines from each installed position. Take Pss of 10 engines in installed position 2; for example, the specific implementation steps are as follows.( 1 ) Arrange the sample values in nondecreasing order.(20)8.38≤8.94≤9.01≤9.37≤9.53≤9.87≤10.38≤10.59≤10.75≤11.46( 2 ) Calculate the value of statistic W using (21). In the formula, ak(W) is read from schedule 10 in literature [24]. For ease of calculation, the list of values is illustrated as in Table 8.(21)W=∑k=1n/2akWXn+1-k-Xk2∑k=1nXk-X-2Table 8
The list of values for calculation.
k X ( k ) X ( 11 - k ) X ( 11 - k ) - X ( k ) a k ( W ) 1 8.38 11.46 3.08 0.5739 2 8.94 10.75 1.81 0.3291 3 9.01 10.59 1.58 0.2141 4 9.37 10.38 1.01 0.1224 5 9.53 9.87 0.34 0.0399( 3 ) Make a judgment: if W<Wα, reject H0; otherwise accept H0. Among them, Wα is read from schedule 11 in literature [24] for a given confidence level α and sample size n.It can be seen from Table8 that ∑k=110(X(k)-X-)2=8.254, ∑k=15ak(W)X(11-k)-X(k)=2.7426 and W=2.74262/8.254=0.9113. For α=0.05, it is found that W0.05=0.842. Owing to W>Wα, so accept H0, and it is considered that Pss at installed position 2 obeys the normal distribution.Assuming that the populationX~N(μ,σ2), μ and σ2are mean and variance of the normal population, respectively. Pss of 10 engines in installed position 2 are the samples from the population X. X- is the sample mean and S2 is the sample variance. It concludes that X- is an unbiased estimate of μ, and, on the basis of interval estimation theorem of sample mean from a normal population, a confidence interval with 95% confidence level of μ can be expressed as X-±(S/n)tα/2(n-1). According to the sample data, it is calculated that X-=9.828, S=0.8254=0.9085, n=10 and t0.025(9)=2.2622, then the confidence interval of the mean value of Pss at installed position 2 is obtained as 9.828±0.9085/10×2.2622, namely (9.178%, 10.478%).Similarly, the W-normality test and the confidence interval estimation forPss at installed position 1 and installed position 3 are carried out, respectively.For installed position 1,∑k=110(X(k)-X-)2=1.428, ∑k=15ak(W)X(11-k)-X(k)=1.1777, W=1.17772/1.428=0.9711, W>W0.05=0.842, so it is considered that Pss at installed position 1 obeys the normal distribution. According to the sample data, it is calculated that X-=1.658,S=0.1428=0.3779, n=10 and t0.025(9)=2.2622, then a confidence interval with 95% confidence level of the mean value of Pss at installed position 1 is obtained as 1.658±0.3779/10×2.2622, namely, (1.388%, 1.928%).For installed position 3,∑k=110(X(k)-X-)2=11.924, ∑k=15ak(W)X(11-k)-X(k)=3.3941, W=3.39412/11.924=0.9661, W>W0.05=0.842, so it is considered that Pss at installed position 3 obeys the normal distribution. According to the sample data, it is calculated that X-=5.089,S=1.1924=1.092, n=10 and t0.025(9)=2.2622, then a confidence interval with 95% confidence level of the mean value of Pss at installed position 3 is obtained as 5.089±1.092/10×2.2622, namely, (4.308%, 5.870%).In summary, Table9 lists the statistical analysis results of Pss at three installed positions. The boxplots of Pss at three installed positions is demonstrated in Figure 8.Table 9
Statistical analysis results ofPss at three installed positions.
Installed position Performance evaluation index m a x δ % m e d i u m δ % m i n δ % m e a n δ % s t d δ % 95% confidence interval 1 2.26 1.755 0.98 1.658 0.1428 (1.388%, 1.928%) 2 11.46 9.7 8.38 9.828 0.8254 (9.178%, 10.478%) 3 6.95 5.05 3.47 5.089 1.1924 (4.308%, 5.870%)Figure 8
Boxplots ofPss at three installed positions.It is concluded from Table9 and Figure 8 that the mean value of Pss at three installed positions of a certain type of ship-borne helicopter are 1.658%, 9.828%, and 5.089%, respectively. On the basis of sample estimation, a confidence interval with 95% confidence level of the mean value of Pss at three installed positions are (1.388%, 1.928%), (9.178%, 10.478%), and (4.308%, 5.870%), which are important references for determining the power monitoring thresholds after engine installation.In addition, statistical calculations have found that gas turbine outlet temperature of the engines rise in all three installed positions due to the increased energy consumption caused byPss. Although the maximum temperature limit is still not exceeded, the usage time of the engine in large states and the life of components have been affected to a certain extent. With the increase of engine using time, the engine performance will decline, and the value of gas turbine outlet temperature will further increase. It should be monitored with emphasis.


## 5. Conclusions

In this paper, initial installed gas turbine power loss of a certain type of turbo-shaft engine has been studied using data mining and statistical approach, in which QPSO is employed to optimize the calculation of gas turbine power at different steady states based on component-level aerodynamic thermal model of gas generator, then ELM is adopted for regressive identification of the established gas generator state assessment model, and the identification model is applied to engine installed condition, and finally statistical analysis of engine initial installed gas turbine power loss at three installed positions are performed. The following conclusions can be drawn.( 1 ) The values of engine initial installed gas turbine power loss at three installed positions all conform to the normal distribution; the mean values are 1.658%, 9.828%, and 5.089% according to the sample estimation, and confidence intervals with 95% confidence level of the mean values are (1.388%, 1.928%), (9.178%, 10.478%), and (4.308%, 5.870%), respectively.( 2 ) The doing work capacity of engines after initial installation in three installed positions has declined to varying degree, among them, the initial installed gas turbine power loss of engine 2 is relatively largest, for the reason that engine 2 is located behind engine 1 and engine 3 and the exhaust system is in front of the intake system, making the intake airflow vulnerable to the influence of the other two engines, its own exhaust flow field and the main rotor wake so that the engine performance is affected. As for engine 3, it is also affected by the dual action of main rotor wake as well as the intake and exhaust flow of engine 2, while engine 1 is less affected by the other two engines and the performance is relatively stable.Some of the future research directions are (i) to get more accurate results based on more sample data, (ii) to analyze the effects of different flight states and flight environment on engine installed power loss, and (iii) to discuss the change of engine installed power loss under different engine using time.



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*Source: 9412350-2018-09-24.xml*