Identification of twin-shaft gas turbine based on hybrid decoupled state multiple model approach

Monitoring control of industrial systems is essential for the good productivity and safety of installations and operators, with better performance that must be guaranteed. This is often challenging due to the nonlinearities and dynamic complexities of these systems, adding operating constraints and instability. Hence, the multi-models constitute then an adapted tool for the modeling of the nonlinear systems to characterize their dynamic behaviors. Indeed, this work proposes the implementation of a hybrid identification approach of the operating variables of a gas turbine, thus making it possible to interconnect the various linear sub-models with decoupled states in order to generate the global output of their nonlinear model, from the exploitation in real time of the turbine's input/output data. However, the suggested decoupled-state multi-model approach offers an interesting alternative to the optimization procedure of the estimated turbine parameters. By using gradient and Gauss–Newton algorithms, improved by genetic algorithms combined with NSGA-II in hybrid form, in order to converge toward the best solutions with an optimal cost function, the obtained implementation results show that this approach allows the convergence of the estimated turbine variables and describes its behavior in real time, with the guaranteed efficiency of the proposed decoupled state multi-model method.


i
Index for the ith linear local model j Index for the jth output of the gas turbine f i The dynamic behavior of the ith local linear models ŷi k ð Þ The output of the ith local linear models The measured output of the considered nonlinear system ŷ k ð Þ Estimated output of the multiple model N The number of measurements h The parameter vector to be found A i ; B; C i;j and D i;j The matrices of state space model for the ith linear local model The dimension of h and the chromosome in GA h Iteration index of the search method The jth output of the multi-model The actual jth output of the gas turbine L Number of local linear models nðkÞ The decision variables vector The activation function The weighting functions x The space vector u The control vector c i Center of the Gaussian function used in the weighting functions r The dispersion used for the Gaussian function NGP Speeds of the high-pressure turbine NPT Speeds of the low-pressure turbine HP High-pressure turbine LP Low-pressure turbine J The cost function The augmented state space model The augmented output of the state space model NSGA-II Non-dominated Sorting Genetic Algorithm

Introduction
The governance of complex dynamic systems in monitoring requires reliable modeling with identification of the behavior of system variables.Nevertheless, the dynamic complexity of these processes with mathematical models is represented by nonlinear dynamic relations, which turned out, according to their mathematical complexity, to be difficult to exploit in a monitoring context (control or diagnosis).However, to describe the dynamic behaviors of nonlinear dynamic systems, the decomposition of the global model into linear sub-models makes it possible to describe the relations between the different state variables and to preserve their dynamics in linearized multi-model form.This modeling aims to find a linearized model around an equilibrium point, by a set of local models characterizing the behavior of the system in different operating zones.
Indeed, as gas turbines are overly complex systems with highly nonlinear dynamic behavior, the multi-model approach combined with linearization techniques is adopted in this work, to identify the nonlinear behavior of the gas turbine, with developments seeking a more complete and optimal presentation of the nonlinear model.
However, a gas turbine consists of a complex, expensive, and very accurate components that operates at high pressure and gas temperature.Gas turbines are one of the major sources of power generation in countries with natural gas resources and they are installed in many places around the world.Deterioration in turbine performance strongly affects its operation, and the loss of stability monitoring in some important industrial processes causes dramatic accidents that can lead to many serious problems, such as human accidents, loss of production or damage to equipment and property.As a result, requests received for simplified mathematical models for the gas turbine for further study are extremely challenging for experts.The aim of these studies was to investigate system stability, strategy development and contingency planning for system disruptions.
Indeed, the interest aroused by the methods and algorithms of decoupling in modeling in the form of multimodels was widely applied in the literature, for a large class of industrial systems.Hence, in 2022, Shuangshuang Chen et al. in Chen et al. (2022) proposed a two-gate decoupled algorithm based on numerical techniques for a multidimensional model applied to the fracture phenomenon, using gate approximation with specific contact interface conditions.That makes it possible to confirm the theoretical analysis of the problems of multi-fields with the effectiveness of the proposed method.As well as Jie Yan et al. in Yan et al. (2022) suggested a mechanism for decoupling model dimensions from time series, based on neural networks, by improving the predictive ability of neural networks.Jakub Kudela and Radomil Matousek in Kudela and Matousek (2022) made a review on surrogate models and their applications to approximate complex systems, where they confirmed that prediction, sensitivity analysis, uncertainty quantification, and optimization assisted by substitution are provided.Junxiang Yang et al. in Yang et al. (2022) realized a totally decoupled linear scheme applied to energy dissipation to elaborate the phase field fluid vesicle model.With this decoupling of models, it is possible to solve the linear elliptic equations separately.Also, Qiongwei Ye et al. in Ye et al. (2022) proposed a second-order numerical structure with decoupled states to determine the phase field model of two-phase flows, which is to solve the nonlinear equations in fully decoupled linear elliptic.
In addition, other works have confirmed the effectiveness of using the multi-model strategy, such as Zhao Liu et al. in Liu et al. 2022a realized a classic closed-loop control strategy with decoupled variable states of an air supply system, which helps to ensure the reliability and efficiency of operation.Also, Li D. H. and Zhu Z. J. in Li and Zhu (2022) have proposed a decoupled modeling technique for estimating the hardening of laminated composite plates in damage, with computer codes allowing the characterization of the effects of delamination and cracks of these composite laminated plates.Yudan Liu et al. in Liu et al. 2022b suggested the use of constraint management techniques for multi-input/multi-output (MIMO) systems using decoupled-state multi-models to establish an efficient computing solution, with respect to the robustness of this approach in the presence of disturbances and modeling uncertainties.
Indeed, the use of optimization algorithms based on artificial intelligence, such as genetic algorithms, has been highly successful because of their considerable number of applications with simplicity and efficiency.For example, Ahmed Zohair Djeddi et al. in Ahmed Zohair Djeddi (2022) and Choayb Djeddi et al.In (Djeddi et al. 2021) proposed to improve the protection system and availability of turbines, which is based on reliable models and mechanisms necessary for operation, to ensure the safety of facilities.This allows minimizing the downtime of these turbines and predicting and avoiding their malfunctions.Furthermore, Kyung-Min Lee and Chul-Won Park in Lee and Park (2022) applied genetic algorithms to fault detection of multifunctional digital substations in power grids, while ensuring the reliability of power grid systems.Hence, the application results confirmed the good speed of fault isolation with accurate decision-making in optimum time.Of course, other data mining techniques such as artificial neural networks have had success in application in several works, such as Guangna Zhang in Zhang ( 2022) proposed a simulation model for modeling dynamic systems based on artificial neural network techniques.Hence, the obtained results showed a good convergence of the proposed prediction model.Also, Vinit Gupta and Santosh Pawar in Gupta and Pawar (2022) suggested a structure based on neural networks with wavelets for the processing of teaching data with optimization for the processing of these voluminous data by eliminating redundant data.This allows having a high accuracy on the predicted data set.
Other studies have been done regarding the estimation of multi-models, such as Javier Arroyo et al. in Arroyo et al. (2020) identified multi-zone construction models for their use in predictive control, which improves the performance of building energy systems.Samah Ben Atia et al. in Samah Ben Atia (2018) proposed an algorithm for the real-time identification of nonlinear discrete systems, using a multi-model control structure over several operating zones, which makes it possible to guarantee better supervision of local predictors.Also, Seyed Saleh Mohseni et al. in Seyed Saleh Mohseni (2017) suggested an approach to reduce the order of nonlinear models, using a decoupled multi-model method with linearization per part of trajectory with a weighting of the multi-model controllers of the examined system.Furthermore, Guoliang Zhao et al. in Zhao et al. (2016) realized a sliding mode control strategy of a tensor product terminal using decoupled state models, using the transformation of these models, which was validated and tested on the pole cart system.Sriram Srinavasan et al. in Young Man Cho (2007) have done the modeling and identification based on input-output data of active magnetic bearing systems, compared by their theoretical behaviors, which allows improving their control systems.As well as optimization approaches have been widely applied in the technological field, such as Hakim Bagua et al. in Bagua et al. (2021), Kaoutar Senhaji et al. in Senhaji et al. (2020) Gonzalez (2020), Xue Jiang and Jin Na in Jiang and Na (2020).
Practically, nonlinear systems are characterized by several numbers of variables that are strongly coupled, with strong interaction between them.That obviously presents many problems in the modeling and analysis of these complex systems, should perhaps look for other representations in the form of linear models useful for the estimation of these nonlinear systems.However, Sidali Aissat et al. in Aissat et al. (2021); Aissat et al. 2022a) exploited decoupled state multi-models for the identification of the different operating variables of a twin-shaft turbine and applied to the detection of faults affecting this machine.Using newly developed modeling techniques and identification methods based on data collected from the input/ output system, for the representation of complex systems in the form of multi-models, are explored by Benyounes Abdelhafid et al. in Abdelhafid et al. (2016), where they studied the modeling of a gas turbine system based on fuzzy clustering from experimental data.Also, Nadji Hadroug et al. in Hadroug et al. (2017) conducted the modeling, analysis, and optimization of the dynamic and numerical modeling of a gas turbine.Furthermore, the parametric modeling of gas compression plants using real data has been studied by Yang Chen et al. in Chen et al. (2014) and the modeling of flow dynamics with a probabilistic method has been studied by Yu Zhang in Zhang et al. (2020).As well as the hierarchical modeling approach to characterize the dynamics of a gas turbine drive system and the gas turbine monitoring system based on the acquisition of real gas turbine vibration measurement data on site were studied by Guolian Hou et al. (Hou et al. 2020).However, the solutions provided to identify a nonlinear system, based on their experimental knowledge of its inputs/outputs, require a modeling phase to faithfully describe its behavior.What was implemented by Sidali Aissat et al. in Aissat et al. 2022b through several investigative tests on a turbine system, with developments on the multi-model identification approach with fuzzy decoupled states with recursive least square (RLS) linear optimization, which is to offer a satisfactory performance of the studied turbine.
This work presents a procedure for identifying the nonlinear dynamics of a two-shaft gas turbine, using a multi-model decoupled states approach, to obtain an efficient and equivalent linear approximation of their nonlinear model, based on a novel hybrid nonlinear optimization approach, to control the search area for the best sub-models for the examined turbine, with a minimal number of iterations of the used algorithms without divergence.Because the imposed optimization constraints are strongly related to the sensitivity of optimization algorithms, which is caused by the initial parameter vector to be determined, the parameter vector defines the direction of the search area used by the gradient which acts as a regulator.This algorithm sensitivity limit is the main motivation to apply a robust approach based on an analysis of variance to identify the influencing variables on the stability of the turbine sub-models.Therefore, the linearization of the turbine dynamic model via their input/output data around the equilibrium point will be proposed, to describe this nonlinear system, by a set of linear sub-models, and by an algorithm structure of optimization to continue turbine evaluation in real time.This makes it possible to provide solutions improving the efficiency of the decoupled-state multi-model approach to follow the evolution of the turbine operating point.Hence, the exploration of the multi-model approach with decoupled states to develop linear submodels from the operating data of the turbine studied makes it possible to improve the efficiency of the identification process for the estimation of their variables in several practical situations with precision and to get around the difficulties stated during the exploitation of these models.
Indeed, this work is organized as follows: The first introductory section offers a brief description of the emergence related to the use of artificial intelligence in the identification and multi-model estimation of systems, as well as their use in monitoring and governance in control and monitoring, besides the diagnosis of these industrial systems.Then, the studied gas turbine system is given in the second section, through the presentation of the different variables used in the proposed multi-model identification.Then, the third section of this work presents the proposed multi-model approaches, with the development of decoupled state models, specific to the studied turbine, as well as the choice of the used activation functions.Subsequently, the estimation of the multi-model parameters of the examined turbine is done in the fourth section, with reliable optimization of these parameters, while minimizing a cost function.The implementation of evolutionary genetic optimization algorithms is proposed in the fifth section in order to solve multi-objective optimization problems.Hence, the NSGA-II structure is adopted in this work to solve the problems of identification of the turbine variables under consideration.Similarly, the results of the application of genetic algorithms for the identification of twinshaft turbine variables in the form of hybrid decoupled state multiple models are detailed in the sixth section.And finally, conclusions are drawn from the whole work.

Gas turbine system
The gas turbine is an internal combustion turbine that converts the chemical energy of the fuel into mechanical energy.A typical gas turbine, or also a single-shaft turbine, consists mainly of three components: compressor, combustor and turbine as shown in Fig. 1.In this process, the compressor sucks in air and increases its pressure; this compressed air is then introduced into the combustion chamber, where heat is added by burning the fuel.The hot, high-pressure gases are then expanded in a turbine to extract useful power.Part of the turbine's power is absorbed by the compressor, providing power for the compression process via the shaft connecting the compressor and the turbine.
The operation of the gas turbine is dependent on these operating conditions such as weather conditions (dry, humid, hot climate).For the turbine under consideration, the turbine's remaining output power is used to drive a load, where ambient input conditions influence the output variables of this machine, while the actual data, the gas turbine inlet, and the outlet are shown in Figs.2.A, B and  C, where the expansion process of the twin-shaft gas turbine can be divided into two separate turbines.The first is used to drive the compressor, called the power turbine or NGP (the high-pressure turbine), and the low-pressure turbine (LP turbine), called the NPT turbine, is mechanically independent and drives the load, while the highpressure turbine, compressor and combustor are called the gas generator.

Multiple model approaches
The nonlinear system can be decomposed into a set of L L local linear or affine models, where f i presents the dynamic behavior of the i th i 2 L ð Þ local linear model at a specified range of operation area, as shown in Fig. 3. Indeed, the idea of the multi-model approach is based on this simple principle (Gregor and Lightbody 2008;Stelios 2006;Leith and Leithead 1999;Colette et al. 1999).The output of each sub-model ŷi k ð Þ, as it is given in Eq. ( 1), contributes by an activation function Þ, more or less to the approximation of the global behavior of the nonlinear system y k ð Þ which can be defined as follows (Aissat et al. 2021(Aissat et al. , 2022b;;Hadroug et al. 2017;Gregor and Lightbody 2008): where nðkÞ 2 < q is the decision variables vector and Þ is the activation function defined by Eq. ( 2), which determines the activation of the i th local model depending on the operation area of the system, this function indicates the more or less important contribution of the corresponding local model in the global model (multimodel); it must satisfy the following properties of convexity: Þ ensures a gradual transition from an actual model to the neighboring local models.This function has generally special shapes such as triangular, sigmoidal or Gaussian.
The main interests of the multiple-model approach are (Zhang 2022;Daneshfar and Bevrani 2012;Gubarev et al. 2019): -They are universal approximations; any nonlinear system can be approximated with an imposed precision by increasing the number of sub-models, -It can be used as an analysis tool for linear systems, -It is possible to link the multi-model to the physics of the nonlinear system in order to give meaning to the multiple models and more precisely to associate a particular behavior of the nonlinear system to a submodel.
In order to obtain a multiple model, three distinct methods can be used, by identification if the data of the inputs and outputs of the considered nonlinear system are available, by linearization around different operating points within a range in which the system can behave as a linear or affine system or by using a convex polytopic transformation if the analytical model of the considered system.

Decoupled state model
The state estimation of a nonlinear system based on the decoupled state linear multi-model approach is an adequate solution to follow the evolution of the nonlinear system behavior.Hence, this approach is addressed in several works carried out in the industrial literature, such as Junxiang Yang et al. in Yang et al. (2022), Gregorc ˇic Ǧregor and Gordon Lightbody in Gregor and Lightbody (2008), Katsanevakis Stelios in Stelios (2006), Leith Douglas and Leithead in Leith and Leithead (1999)  Indeed, a nonlinear system can be approximated by a decoupled state multi-model system, expressed as: where x 2 < n is the space vector, u 2 < m is the control vector or input vector, n is the index of activation functions l i , and h is the parameter vector to be found.
The construction of a multiple model from inputs and outputs requires the following steps (Aissat et al. 2021(Aissat et al. , 2022b))

Activation function
The weighting function as shown in Fig. 4.A is built as follows: where this function is centered in c i and r presents its dispersion which is the same for all the i th functions to all weighting functions.In order to respect the constraint of Eq. (2), the functions w i n ð Þ are normalized to obtain the required activation functions, as shown in Fig. 4.B: The decision variable n can be defined based on the measurable state variables or/and system input and output signal values, with the following condition (Aissat et al. 2021(Aissat et al. , 2022b): 4 Parameter estimations The aim is to identify and optimize the parameters of the local model in order to accurately reproduce the dynamic behavior of the system.The parametric estimation method is based on minimizing the function of the quadratic error presenting the difference between the estimated output of the multiple model ŷ k ð Þ and the measured output of the considered nonlinear system y k ð Þ, which is defined as follows (Aissat et al. 2021(Aissat et al. , 2022a(Aissat et al. , 2022b)): where N is the number of measurements, h is a vector containing the parameters of the local models; it is updated at each iteration as follows: where h is the iteration index of the search method, hðhÞ is the solution at iteration h, D represents the adjustment factor which allows to forcing the convergence of the solution, it is obtained based on iterative optimization algorithm, D h ð Þ is the research direction, for which the expression is dependent on the optimization method used.The main usual used optimization method, which are used for finding the best values of the adjustment factor and the research direction, can be summarized in the following section.

Gradient algorithms
The search direction is specified by the gradient G criterion as follows (Daneshfar and Bevrani 2012;Gubarev et al. 2019;Azlan Mohd Zain 2011;John 1992): oh is the sensitivity function.

Newton's algorithms
The direction and steps of the search are specified simultaneously by the equation based on second-order development: where H h ð Þ is the Hessian matrix of the criterion defined by: The main advantage of this method is that its ability of defining the direction of research and the speed of research at the same time.However, it has a main disadvantage, which is imposed by the hard calculation of the Hessian inverse matrix when passing from one iteration to the next iteration.

Gauss / Newton algorithms
By neglecting the second-order terms to simplify Newton's method, we obtain: This method allows obtaining a defined positive Hessian matrix; hence, it guarantees the convergence to a minimum.This algorithm, as given in Algorithm 1, is sensitive to the initial choice of the parameter vector which causes the convergence toward local minima when the size of the parameter space is significantly high.To avoid this problem, some improvements can be made to the initial identification procedure to ensure the convergence of the algorithm (Aissat et al. 2021; Claudia Gutie ´rrez Antonio 2010; Chica et al. 2011).Marquardt's algorithm is used to overcome problems related to the inversion of the H matrix when updating the parameters.Equation ( 8) is replaced by Eq. ( 13): where I is the appropriate dimension identity matrix and k a scalar (regularization parameter).

Genetic algorithms
The genetic algorithms are a research technique by imitation of observed processes and based on the principles of genetics and natural selection developed by John Holland in 1975, then described by his student David Goldberg in 1989, in 1975 De Jong showed the use of genetic algorithms as an optimization tool.The genetic algorithms develop a population of candidate solutions; each solution is usually coded as a binary string or real number, called a chromosome.The cost of each chromosome is then evaluated using a cost function after the chromosome has been decoded.Once the evaluation is complete, selection rules are established so that the best chromosomes undergo genetic operations such as crossing and mutation that mimic nature.The cost of the newly produced chromosomes is lower than the cost of the previous generation; they will replace the weaker chromosomes.And this process continues until the stop criteria are met.Real-coded genetic algorithms (RCGAs) implement real numbers in their candidate solutions rather than binary coding.Real or continuous coding uses representations that are easier to understand and useful in engineering applications (Yusoff 2011;Chen et al. 2010;Turkyilmaz et al. 2022; C ¸unkas ¸2010).

Cost function
The cost function is a mathematical function that generates an output from a set of input variables called chromosomes; in this work, the cost is the difference between the desired output and the actual output, which is represented by the global criterion: where ŷj k; h ð Þis the jth output of the multi-model, and y j k ð Þ is the actual jth output of the gas turbine, and is the number of measurements.This criterion favors a good characterization of the overall behavior of the nonlinear system by the multi-model, where the state space equations presenting the systems are defined as follows: where h is defined as the column vector of the multi-model parameters to be estimated which is partitioned into L blocks with h ¼ h 1 h 2 Á Á Á h L ½ T .And the matrices A i ; B; C i;j and D i;j of the multi-model are described.
Based on the equation described above using the realcoded genetic algorithms (RCGAs) given in Algorithm 2, all the outputs share the same A i and B matrices in all submodels, where p is the number of outputs of the present multiple models, which are the speeds of the high-pressure and the low-pressure turbines (NGP and NPT).
In order to reduce the complexity of the multiple models caused by the large number of parameters in the vector h, which lead to a deviation between the MM outputs and the measured outputs, the vector h is divided to two vectors such as: A vector contains the matrix A i¼1;L parameters A vector contains the parameters of C i¼1;L and D i¼1;L for each output: This parameter vector division will not only reduce the complexity of the multiple models but allow limiting the overall parameter search area for each output; however, this limitation can be used as constraints for all parameters or can be each parameter separately.
For the calculation of the gradient: While the derivative of oh requires the derivative of each sub-model depending on the number of the output, using Eq. ( 14) the following expression is obtained: The calculation of which is obtained as follows: To simplify the presentation of the sensitivity function, we use the following variables and parameters change: Equations ( 21) and ( 22) can be rewritten as follows: Furthermore, the following simplification can be used: The sensitivity function can be rewritten in the following simplified form: where The matrices A s ; B s ; C s and D s of Eq. ( 26) are defined as follows: ; • Chromosome The chromosome is a vector of variable values to be searched and optimized; the dimension of the chromosome has N var variables and necessarily has the same dimension of h.

• Initial Population
We define an initial population of candidate solutions, which are randomly generated.The number of individuals in the population is determined according to the chromosome.In some cases, the candidate solutions are seeded in the area of search space where the desired solution is likely to be found (Lee and Park 2022; Claudia Gutie ´rrez Antonio 2010).

• Natural Selection
An evaluation of each chromosome in the population is ranked from the lowest to the highest cost; the best chromosomes are kept for mating and the others are discarded and make way for new offspring.The selection process must take place at each iteration of the algorithm (Chen et al. 2010; C ¸unkas ¸2010).

• Pairing and Mating
To have random diversity in the population and prevent GA from converging too quickly or getting trapped in a local rather than a global minimum.A mutated variable is replaced by a new random variable.The rows and columns of the variables in the population to be mutated (John 1992; Yusoff 2011).

• Mutation
Mutations can cause a random diversity in the population by introducing traits not in the original parents and keep the GA from converging too fast or gets trapped in local minimum rather than a global minimum.Random numbers are chosen to select the row and columns of the variables in the population to be mutated.A mutated variable is replaced by a new random variable (Chica et al. 2011;John 1992).

• Termination
The GA is terminated when one of the following is true.
-The condition is reached, -Fixed number of generations or function evaluations is reached, -A solution with a desired fitness is found, -There is no change of cost indicating no improvement in candidate solutions.

Overview of NSGA-II
Evolutionary and stochastic genetic optimization algorithms based on the genetic operator as we said, with the task of solving multi-objective problems and giving optimization results satisfying all problems, based on the Pareto dominance concept, we can mention several multiobjective genetic algorithms.The main advantage of NSGA-II as one of the most widely used and efficient multi-objective algorithms is that the diversity preservation strategy used in NSGA-II does not require any parameters to be set (Aissat et al. 2021;Chica et al. 2011;John 1992).Therefore, we chose to solve our problem of estimating the state matrix between the two functions of NGP and NPT using this approach.And with three special features, a fast non-dominant sort approach, a fast overloaded distance estimation procedure and a simple overloaded comparison operator (Aissat et al. 2021(Aissat et al. , 2022b)).The NSGA-II can be roughly detailed as follows.
Step 1: Define the population based on the number of parameters in the L sub-models.
Step 2: Sort the population according to the nondomination between the NGP and NPT cost functions.
Step 3: Individuals are selected for the population based on rank and rounding distance.
Step 4: Combination of parent population and genetic operators actual GA coding, matching and mating, and mutation.
Step 5: Selection affects the offspring population and the population of the next generation.The cost function provided by Eq. ( 12) is used to select the new generation, which is then filled by each successive edge until the population size exceeds the existing population size.

Hybrid Genetics
The hybrid method refers to two or more research techniques combined to solve difficult problems.They diversify and intensify the research process to achieve a more robust solution (Aissat et al. 2022a;Turkyilmaz et al. 2022).Diversification refers to the ability to visit many different regions in the research space, while intensification refers to the ability to obtain high-quality solutions in a particular research space.Diversification is often referred to as global research or exploration.Intensification is referred to as local research, local refinement or exploitation.The main problem with hybrid methods is how to effectively balance the effects of diversification and intensification in the research process.The objective is to achieve a global optimum by obtaining quality solutions in a short time frame.In these hybrids, a genetic algorithm is integrated with parametric estimation to improve the performance of the multi-model approach in our application.
• MMA-GA Hybrid Relay This work has also implemented hybrid genetic algorithm with the multi-model approach, which is called genetic algorithm relay and multiple model approach (MMA-GA Relay).GA and MMA are combined in a two-step process.The first one, GA is launched for diversification to visit different regions of the search space, individuals in the population are given as initial solution in the search space of parametric estimation, the next step is parametric estimation using the capacity of intensification to obtain a solution in a specific space.The population generated only by the operators of genetic algorithms, and the cost of individuals in the population calculated after the application of parametric estimation for an estimation horizon, and there will be no exchange of information between the two algorithms, even if they are combined but operate separately.Figures 4 and 5 summarize the MMA-GA hybrid relay approach.Genetic algorithms and parametric estimation are combined in two steps also, the first step, GA is for diversification in order to reach different research spaces where individuals in the population are given as initial solution in the research spaces of the second step, and the parametric estimation is for intensification in order to find a better solution.
The difference between a teamwork and relay hybrid, as given by Algorithms 3, is the exchange of information between the two algorithms.The cost of each individual in the population calculated after parametric estimation, if a better solution or individuals are generated in the estimation horizon, the new ones in the population will replace the old individuals.The two algorithms will have an effect on the generated population; otherwise, the individuals generated by the GA will remain, and this will bring us back to hybrid relays.6 shows the comparison of the obtained results between hybrid relay and teamwork MMA-GA using 4 sub-models for NGP, and Fig. 7 shows the comparison of the obtained results between hybrid relay and teamwork MMA-GA using 4 sub-models for NPT.

Results and discussion
In this section, the parametric estimation methods, genetic algorithms, and hybrid teamwork are used to identify a model that can mimic the dynamics of the gas turbine.The cost function is derived from a global estimation criterion defined by Eq. ( 14) by computing the gradient and a second derivative to obtain the Hessian matrix.The form chosen of multi-model ( 15) is considered as a SIMO with fuel flow as input, HP Turbine Rotational Speed (NGP) and LP Turbine Rotational Speed (NPT) as outputs.Figure 8 shows the cost diminution using hybrid approaches of the NGP, and Fig. 9 shows the cost diminution using hybrid approaches of the NPT.Also, Fig. 10 shows the cost diminution using hybrid approaches of the NGPand Fig. 11 shows the cost diminution using hybrid approaches of the NPT.
The parameters used in this procedure with different function areas are 200 generations with 100 population, mutation rate of 0.15, and selection rate of 0.5.In order to present the results and to evaluate the performance of the proposed procedure, a different number of local models is used L ¼ 2; 6; the overall comparison of the obtained results is presented in Table 1.We start with an identification of the vector h NGP and h NPT , while the vector h A containing the parameters of A i is taken as a zero.The following figures show the decrease in the cost of NGP and NPT for different sub models.
Figure 12 shows the multi-objective optimization between the NGP and NPT for 3 and 4 sub-models, and Fig. 13 shows the multi-objective optimization between the NGP and NPT for 5 and 6 sub-models.The function cost decreases when the new generation appears, the search carried out by the two methods, GA for a global search with a population of 100 individual and 200 generations and the parametric estimation for a local search of each individual of the population in an estimation horizon.The next step is to find the vector h A that contains the parameters of the matrix A i , where the matrix is a function of the NGP and NPT vectors, and for this a multi-objective optimization is used.
The best results and new costs of the two outputs in our application are shown in Figs. 14 and 15, with the parameters obtained of the matrix A i in relation to the two  In order to validate the obtained result, another criterion is used, consisting in calculating the standard deviations given by the VAF (variance accounting for) criterion of variance accounting in the following formula: The results of the global criterion of identification and cost validation of the outputs of the multiple models according to the number of sub-models as well as the different parameters of the activation functions are presented in Table 2 and the obtained error from all sub-models of HP turbine is shown in Fig. 18 and the obtained error from all sub-models of LP turbine is shown in Fig. 19.As much we add the number of local models used in the identification, as much we get a better result, we can notice from Table 1 that L = 6 provide the best and lowest cost in the NGP identification cost, and for the NPT all the identification cost are close 193, and the worst result given by As we saw L = 6 provides the best approximation and we can see that the MOO NSGA-II does not add a lot to the identification cost validation cost, it makes the state space of each local model observable.
Figures 20 and 21 show the decrease in the costs of the two outputs for the different sub-models of the MA-GA Hybrid approach, where Fig. 20 shows the cost diminution for all sub-models of HP turbine rotational speed and Fig. 21 shows the cost diminution for all sub-models of LP turbine rotational speed.
We notice that L2 is the fastest method to join the lowest cost in relation to the number of local models, while L6 is the slowest method, to join the lowest cost in relation to the number of local models, and that because of the number of parameters to be searched for in the search area, for L2 we simply have 6 parameters to find, whereas for L6, 18 parameters to find, so the speed of the method has a direct relation with the number of parameters to find, and for the minimal cost obtained each time we add the number of local models, it is because of the real teamwork data of the gas turbine, and we clearly notice that the NGP and the NPT have several working points.

Conclusion
The representation of nonlinear systems by multi-models around the operating points is a strategic for the monitoring of these industrial systems, in order to characterize their nonlinear dynamic behaviors by local linear sub-models.This technique is implemented in this work for the linearization of the dynamic model of a gas turbine, via their input/output data around the equilibrium point.In order to describe this nonlinear system by a set of linear submodels, an optimization algorithm structure allows continuing the real-time turbine evaluation.This approach is based on reducing the complexity of the turbine system by decomposing its global nonlinear model into linear submodels, for each operating zone.Hence, the global dynamics of this rotating machine is then represented by considering the contribution of each sub-model with a weighting function associated with each operating zone.This makes it possible to provide solutions improving the efficiency of the decoupled-state multi-model approach to follow the evolution of the turbine operating point.However, the procedure for optimizing the estimated turbine  parameters was conducted using prediction algorithms (gradient and Gauss-Newton), improved using genetic algorithms combined with NSGA-II in a hybrid form.With a compromise between three constraints, namely the complexity, the number of parameters to be identified, and the accuracy of the approximation of the turbine parameters.This allows converging to the best multi-model parameters with an optimal cost function.Indeed, the obtained results in this work, through the validation tests carried out, show the ability of the multi-models with Fig. 18 Error obtained from all sub-models of HP turbine Fig. 19 Error obtained from all sub-models of LP turbine decoupled states to approach the behavior of investigated gas turbine and to characterize its operation, with the convergence of their variables estimated in real time using linear local models.Hence, the efficiency of the developed multi-model approach is ensured, during the identification of the turbine system variables.

Fig. 2
Fig. 2 Used real data of the examined gas turbine

Fig. 3
Fig. 3 Structure of the decoupled states multi-model identification

Fig. 4 A
Fig. 4 A Used Gaussian function B Activation function for L = 4

Fig. 5
Fig. 5 Flow chart of the hybrid Relay MMA-GA

Fig.
Fig. Cost diminution using hybrid approaches of the NGP

Fig. 10
Fig. 10 Cost diminution using hybrid approaches of the NGP

Fig. 12
Fig.12Multi-objective optimization between the NGP and NPT for 3 and 4 sub-models

Fig. 13
Fig.13Multi-objective optimization between the NGP and NPT for 5 and 6 sub-models

Fig. 15
Fig. 15 Model output for L = 3 and L = 4 with the real data of LP turbine

Table 1
Identification results

Table 2
Squared error with the standard deviation in percentage obtained by the identification of the multi-model of the output of the turbine for L = 6 with different types of algorithms