Gas Turbine Model Identification Based on Online Sequential Regularization Extreme Learning Machine with a Forgetting Factor

Abstract

Due to the advantages of high convergence accuracy, fast training speed, and good generalization performance, the extreme learning machine is widely used in model identification. However, a gas turbine is a complex nonlinear system, and its sampling data are often time-sensitive and have measurement noise. This article proposes an online sequential regularization extreme learning machine algorithm based on the forgetting factor (FOS_RELM) to improve gas turbine identification performance. The proposed FOS_RELM not only retains the advantages of the extreme learning machine algorithm but also enhances the learning effect by rapidly discarding obsolete data during the learning process and improves the anti-interference performance by using the regularization principle. A detailed performance comparison of the FOS_RELM with the extreme learning machine algorithm and regularized extreme learning machine algorithm is carried out in the model identification of a gas turbine. The results show that the FOS_RELM has higher accuracy and better robustness than the extreme learning machine algorithm and regularized extreme learning machine algorithm. All in all, the proposed algorithm provides a candidate technique for modeling actual gas turbine units.

1. Introduction

Gas turbine (GT) modeling is necessary to investigate and improve gas turbine performance [1]. Without an accurate mathematical model, the dynamic characteristics of the unit cannot be learned before the GT is installed, and the design of the control system cannot be carried out. The modeling of a gas turbine can be used to predict engine performance [2], fault diagnosis [3], and control strategy investigation [4]. An accurate mathematical model must genuinely reflect the gas turbine’s steady-state and dynamic response characteristics.

The methods of gas turbine modeling are divided into two categories: one of which is to build a component-level model based on the internal operation mechanism, energy and momentum conservation, and other physical laws of the engine. The other one is to identify the mathematical model of the engine based on the actual data during operation. Due to the consideration of physical laws, component-level modeling can explain the operating rules of gas turbines. Still, the internal complex mathematical derivation process also limits the scope of application of component-level modeling. The model identification method does not require coupling the complex internal aerodynamic thermal mechanism of an actual unit. It characterizes the operating characteristics as a mapping relationship between input and output parameters to create a black box model describing the system’s dynamic properties. The method has the effect of model dimensionality reduction compared to the component-level model and therefore has better real-time performance. Asgari et al. [5] developed a NARX-ANN algorithm for the offline identification of low-power gas turbines. The results showed that the NARX-ANN model outperformed the physics-based model. Talaat et al. [6] published an identification method for predicting a gas turbine’s performance degradation using a feedforward ANN algorithm, but this method is not applicable to control system design. Jurado F. [7] developed a Hammerstein model for a micro gas turbine, and the results showed that the proposed Hammerstein model could accurately describe the system’s dynamic characteristics. In this article, we focus on the application of model identification in the field of micro gas turbines.

Single hidden layer feedforward networks (SLFNs) can approximate any continuous function defined on a compact set and have been widely used in model identification. Traditional SLFNs training methods are usually based on gradients, such as backpropagation (BP) [8] and neuron-by-neuron algorithms [9]. However, because of the need to adjust all parameters of SLFNs during training, these algorithms expose the drawback of slow convergence when identifying objects in high-dimensional spaces. In addition, gradient-based algorithms are more accessible to converge to local extremes due to the overdependence on parameter initialization and the complexity of the feature space. To solve the above problems, Huang et al. [10] proposed a new training method for SLFNs called the extreme learning machine (ELM). Unlike gradient-based learning methods, an ELM randomly assigns the network’s input weights and the hidden layer’s biases, and the nonlinear activation function of the hidden layer guarantees the system’s nonlinearity. Therefore, the ELM algorithm must only train the weights between the hidden and output layers. This improvement reduces the computational burden of weight adjustment caused by the widely used gradient descent method and ensures the convergence speed of an ELM. At the same time, random hidden nodes guarantee the global convergence of the algorithm [11].

Due to its special training effect, the ELM has been widely used in time series analysis [12], control [13], system modeling [14] and prediction [15], fault detection and diagnosis [16], and so on. Although the ELM has been successful in practical applications, its unique learning approach, which brings many advantages, inevitably causes disadvantages as well. By only considering empirical risk minimization, an ELM may provide weaker stability, generalization performance, and sparsity results. Random initialization of the hidden layer node parameters makes the ELM converge faster but also leads to fluctuations in the identification effect. The performance of an ELM is highly dependent on the number of hidden layer nodes. Incorrect selection of the number of hidden layer nodes may lead to overfitting or underfitting [17]. Output weights are another parameter related to the performance of an ELM. It is easier to direct to overfitting because an ELM seeks the output weight values by minimizing the training error [11]. Deng et al. [18] pointed out that the ELM algorithm is based on the empirical risk minimization (ERM) principle and tends to overfitting. According to statistical learning theory, an ELM with good generalization performance should consider structural risk minimization (SRM) instead of ERM, which requires controlling the model complexity of the ELM.

In 1963, Tikhonov proposed a new method for solving the ill-posed problem known as regularization. To avoid the occurrence of overfitting, the regularization principle is often used to improve the generalization ability of an ELM. The regularization principle reduces the complexity of ELM networks by applying the norm of the output weights to the objective function, which avoids the occurrence of overfitting. Zhao et al. [14] proposed a robust regularized ELM to solve an unsatisfactory modeling problem of the original ELM in the presence of outliers and noisy environments. Huang et al. [19] introduced the L₂ norm penalty term into the ELM algorithm, and the results show that the model minimizes the empirical risk.

Initially, ELM algorithms were designed for offline learning. However, offline learning does not address the time-varying nature presented by the gas turbine. The online sequential extreme learning machine (OS_ELM) proposed by Liang et al. [20] can address the above problem precisely. The OS_ELM acquires samples in the form of data blocks and retrains the entire dataset containing the old samples to purchase the output weights of the ELM when new samples arrive. In general, with the arrival of new samples, the old sample data contribute less and less to the model, but the traditional OS_ELM uses the same weights for the whole dataset. To reduce the impact of old samples on model training, an extreme learning machine with a forgetting mechanism algorithm was proposed in Ref. [21]. This algorithm improves the learning effect by quickly discarding the outdated data so that the weights of the old sample data are set to zero. In addition, Wang et al. [22] introduced the sliding window concept into the OS_ELM algorithm, which moves forward with time and gradually forgets the old sample dates.

The actual operating parameters of a gas turbine have two significant characteristics: On the one hand, due to the influence of measurement errors, operation errors, and changes in operating conditions, the acquired sample data usually contain noisy interference signals, and the presence of noise signals is more likely to lead to the occurrence of overfitting. On the other hand, the characteristics of a gas turbine are varied by environmental conditions, performance degradation, and other factors. Studies have shown that gas turbine performance is directly related to ambient temperature [23]. Suppose the ambient temperature is used as a state variable for identification. In that case, a large number of data containing all possible temperature ranges is required, and often the acquisition of sample data is difficult.

In this paper, we combine the advantages of the regularization principle and online learning mechanism to propose a regularized online sequential extreme learning machine based on an adaptive forgetting factor (FOS_RELM). Our main contributions include the following:

• We improve the ELM algorithm using the Tikhonov regularization principle to reduce the complexity of the model and thus improve the algorithm’s performance in handling noisy data.
• An ELM is a batch learning algorithm while data acquisition for gas turbines is a continuous process, and it is impossible to obtain the complete data set at one time. When some new data arrive, batch learning has to repeat the training with old and new data, so it takes a lot of time. This paper improves the ELM algorithm by using an online learning method to process the data sequentially, which saves training time.
• A forgetting factor adaptive update strategy is designed to forget the old sample data selectively. This forgetting mechanism can eliminate the adverse effects of old data which do not match with current characteristics of model training and solve the problem of low training accuracy caused by the time-varying nature of gas turbines.

This article is organized as follows: Section 2 describes the basic structure of an ELM and regularized ELM (RELM). Section 3 introduces the principle of our proposed FOS_RELM algorithm. In Section 4, a comprehensive demonstration of the FOS_RELM algorithm is presented. Section 5 summarizes this paper’s work, and the application prospects of the FOS_RELM algorithm are discussed.

2. The Traditional Algorithms

This section briefly introduces the principles of the traditional ELM algorithm and RELM algorithm.

The extreme learning machine is a method for training single hidden layer feedforward networks (SLFNs). As shown in Figure 1, SLFNs consist of three parts: the input layer, the hidden layer, and the output layer.

With a training sample $S={\left\{\left(x_i,t_i\right)\right\}}_{i=1}^N$, where $x_i=\left[x_{i1},x_{i2},\mathrm{\dots }{x}_{im}\right]\in R^m$ is the ith input vector, and $t_i=\left[t_{i1},t_{i2},\mathrm{\dots }{t}_{in}\right]\in R^n$ is the ith target vector. SLFNs with L hidden nodes can be described as Equation (1).

$$\widehat{f}(x)={\displaystyle \sum _{i=1}^L{\beta }_i}G\left(a_i,b_i,x\right)\begin{array}{cc}& a_i,x\in R^m\end{array},b_i\in R$$

(1)

where $x\in R^m$ is the input parameter of the SLFNs, ${\beta }_i$ is the output weight coefficient connecting the hidden layer and the output layer, $G(\cdot )$ is the nonlinear activation function, and $a_i,b_i$ are the randomly generated node parameters of the hidden layer.

The ELM algorithm reduces the training parameters by randomly initializing the parameters of hidden layer nodes so that the network between the input and hidden layers becomes a fixed nonlinear transformation. After the fixed nonlinear transformation of the hidden layer, the input parameters are mapped to a new linear space named the ELM feature space [24]. The ELM trains the network by minimizing the estimated and actual values error. Thus, the objective loss function of the model is shown as follows:

$$\underset{\beta }{\min }:\Vert H\mathsf{\beta }-T\Vert$$

(2)

where

$$H=\left[\begin{array}{c}\begin{array}{cc}\begin{array}{l}G\left(a_1,b_1,x_1\right)\\ ⋮\end{array}& \begin{array}{cc}\cdots & \begin{array}{l}G\left(a_L,b_L,x_1\right)\\ ⋮\end{array}\end{array}\end{array}\\ \begin{array}{cc}G\left(a_1,b_1,x_N\right)& \begin{array}{cc}\cdots & G\left(a_L,b_L,x_N\right)\end{array}\end{array}\end{array}\right]$$

(3)

is the hidden nodes output matrix, $\beta ={\left[{\beta }_1,{\beta }_2\mathrm{\dots },{\beta }_L\right]}^T$, and $T={\left[t_1,t_2\mathrm{\dots },t_N\right]}^T$. Combining Equations (1) and (2), the optimization problem of ELM can be expressed as Equation (4).

$$\begin{array}{l}\underset{\beta }{\min }\left\{J_{\mathrm{ELM}}=\frac{1}{2}{\Vert E\Vert }_F^2\right\}\\ T=H\mathsf{\beta }+E\end{array}$$

(4)

where $E={\left[e_1,e_2\mathrm{\dots },e_N\right]}^T$ is the modeling error matrix, $e_i$ is the modeling error of the ith sample, and ${\Vert ·\Vert }_F$ represents the Frobenius norm.

By solving Equation (4) using the least squares method, the expression for the output weight factor $\beta$ is obtained as follows:

$$\beta =\{\begin{array}{c}{\left(H^{T}H\right)}^{-1}{H}^{T}T\begin{array}{cc}& N\ge L\end{array}\\ H^T{\left(H{H}^T\right)}^{-1}T\begin{array}{cc}& N<L\end{array}\end{array}$$

(5)

Equation (3) minimizes the identification error on the training set as the objective function and trains the parameters of the network with the least squares principle. In this approach, only the empirical risk minimization is considered, and it is easy to overlearn the features of the training set when performing model identification, which results in overfitting. Deng et al. [18] used the regularization condition to improve the extreme learning machine, forming a regularized extreme learning machine (RELM) that integrates SRM and ERM. The objective function of RELM can be expressed as Equation (6).

$$\begin{array}{l}\underset{\beta }{\min }f\left(\beta \right)=\frac{1}{2}{\Vert \beta \Vert }_{}^2+\frac{C}{2}\Vert H\mathsf{\beta }-T\Vert \\ \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& =\end{array}\frac{1}{2}{\beta }^T\beta +\frac{C}{2}{\left(H\mathsf{\beta }-T\right)}^T\left(H\mathsf{\beta }-T\right)\end{array}$$

(6)

where C > 0 is the regularization factor for balancing the output weight matrix and the modeling error. Solving Equation (6) by setting the partial derivative of the function to zero gives the following expression:

$$\frac{\partial f\left(\beta \right)}{\partial \beta }=\beta -C{H}^T\left(T-H\mathsf{\beta }\right)=0\Rightarrow \beta =\{\begin{array}{c}{\left(H^{T}H+\frac{I}{C}\right)}^{-1}{H}^{T}T\begin{array}{cc}& N\ge L\end{array}\\ H^T{\left(H{H}^T+\frac{I}{C}\right)}^{-1}T\begin{array}{cc}& N<L\end{array}\end{array}$$

(7)

where I is the unit matrix. It can be seen from Equation (7) that the RELM degenerates to an ELM when $C\to \infty$. Hence, the ELM algorithm can be considered a particular case of the RELM algorithm. By adding the examination of the output weight coefficient $\beta$ to the objective function of the RELM algorithm, we can finally obtain a model with a small value of $\beta$. It has been shown in [25] that the smaller the output weight $\beta$, the better the generalization performance of the network.

3. The Proposed Algorithm

The traditional ELM and gradient-based discrimination algorithms are offline discrimination methods. Offline identification methods can achieve satisfactory modeling results when sufficient sample excitation is obtained, and the system characteristics keep unchanged with time. However, a gas turbine is a complex power system with nonlinearity and high time lag. During actual operation, characteristics of a gas turbine will be changed by the external environment and performance degradation. For advanced control strategies, such as predictive and adaptive control, real-time acquisition of the object’s model is also required. Traditional identification methods can only establish a single mapping relationship, which cannot solve the time-varying problem of the gas turbine. Fortunately, the OS_ELM algorithm proposed by Liang et al. [20] can solve the above issues well. In this paper, we combine the advantages of the OS_ELM and the regularization principle to propose an FOS_RELM algorithm that applies to gas turbine model identification.

As shown in Figure 2, the training process for the FOS_RELM is mainly divided into a prior knowledge learning phase and an online updating phase. In the prior knowledge learning phase, the FOS_RELM algorithm trains the model according to the RELM algorithm. The online update phase updates the output weight coefficients of the model using new sample dates. The detailed steps of the FOS_RELM are described below.

4.

Prior knowledge learning phase:

The initialized training set $S_0=\left\{\left(x_i,t_i\right)|x_i\in R^m,t_i\in R^n,i=1,2,3\cdots N_0\right\}$ is formed by taking N₀ samples from the sample set. Initialized training samples should satisfy N₀ > L. Consider regularization, and train the initialized output layer weight coefficients by Equation (8).

$${\beta }_0={\left(H_{{}_0}^T{H}_0+\frac{I}{C}\right)}^{-1}{H}_{{}_0}^T{T}_0$$

(8)

Let $P_0=H_{{}_0}^T{H}_0+\frac{I}{C}$ to obtain ${\beta }_0=P_0{}^{-1}{H}_{{}_0}^T{T}_0$.

5.

Online updating phase:

When a new sample $\left(x_{N_0+1},t_{N_0+1}\right)$ is available, the implied layer matrix $h_1$ is expressed as Equation (9).

$$h_1=\left[\begin{array}{cc}G\left(a_1,b_1,x_{N_0+1}\right)& \begin{array}{cc}\cdots & G\left(a_L,b_L,x_{N_0+1}\right)\end{array}\end{array}\right]$$

(9)

Therefore, the output weights can be calculated as follows:

$$\begin{array}{l}{\beta }_1={\left({\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]}^T\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]+\frac{I}{C}\right)}^{-1}{\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]}^T\left[\begin{array}{l}{T}_0\\ t_1\end{array}\right]\\ \begin{array}{cc}& =\end{array}{\left(H_{{}_0}^T{H}_0+h_{{}_1}^T{h}_1+\frac{I}{C}\right)}^{-1}\left(H_{{}_0}^T{T}_0+h_{{}_1}^T{t}_1\right)\end{array}$$

(10)

Let $P_1=H_{{}_0}^T{H}_0+h_{{}_1}^T{h}_1+\frac{I}{C}\Rightarrow P_1=P_0+h_{{}_1}^T{h}_1$.

From ${\beta }_0=P_0{}^{-1}{H}_{{}_0}^T{T}_0\iff P_0{\beta }_0=H_{{}_0}^T{T}_0$, the expression for ${\beta }_1$ can be rewritten as Equation (11).

$$\begin{array}{l}{\beta }_1={\left({\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]}^T\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]+\frac{I}{C}\right)}^{-1}{\left[\begin{array}{l}{H}_0\\ h_1\end{array}\right]}^T\left[\begin{array}{l}{T}_0\\ t_1\end{array}\right]\\ \begin{array}{cc}& =\end{array}{P}_1^{-1}\left(\left(P_1-h_{{}_1}^T{h}_1\right){\beta }_0+h_{{}_1}^T{t}_1\right)\\ \begin{array}{cc}& =\end{array}{\beta }_0+P_1^{-1}{h}_{{}_1}^T\left(t_1-h_1{\beta }_0\right)\end{array}$$

(11)

From this, it can be deduced that the updated formula for the output weight coefficient $\beta$ is shown in Equation (12).

$$\{\begin{array}{l}{P}_0=H_{{}_0}^T{H}_0+\frac{I}{C}\\ \begin{array}{cc}& ⋮\end{array}\\ P_{k+1}=P_k+h_{k+1}^T{h}_{k+1}\\ {\beta }_{k+1}={\beta }_k+P_{k+1}^{-1}{h}_{{}_{k+1}}^T\left(t_{k+1}-h_{k+1}{\beta }_k\right)\end{array}$$

(12)

As seen in Equation (12), the updating process of the output weight coefficient $\beta$ records the information of the whole sample. The data of the old sample will affect model identification accuracy when the features of the gas turbine are changed. Zhao et al. [21] modified the structure of the OS_ELM with the forgetting mechanism and defined the concept of the time window to distinguish the old and new observations. Observations in the time window are used to update the network, and the old sample data outside the time window are forgotten. However, this approach to bypassing the sample data cannot accurately reflect how much the training samples contribute to the network. In addition, it will decrease the utilization of sample data or affect the recognition accuracy of the network when the time window is not set correctly. The training error is the most accurate parameter to describe the contribution of data samples. If a small training error can still be achieved for the new sample data, it is proven that the original network can be well adapted to the latest data [24]. In this paper, we use the forgetting factor λ to adjust the weight values of the old and new samples. A strategy for adaptively changing the forgetting factor according to the training error $e$ was designed, as shown in Figure 3. The training error is defined as $e=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(t_i-{\widehat{t}}_i\right)}^2}}$. Defining a standard training error $e_t$. The required adjustment strategy should be set with $e_t$ as the critical point for forgetting the old sample date. The correction function of the forgetting factor λ is shown in Equation (13). When $e<e_t$, it is considered that the model has achieved a high identification effect at this time so that λ = 1. When $e\ge e_t$, it is proven that the model identified by the old sample data has failed to describe the characteristics of the new sample data. At this time, λ should tend to the set minimum forgetting factor ${\lambda }_{\min }$. Figure 3 is the schematic diagram of the correction function.

$$\lambda =\{\begin{array}{c}\begin{array}{cc}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& 1\end{array}& \end{array}& \end{array}& \end{array}& \end{array}& \end{array}& \end{array}& \end{array}0<e\le e_t\end{array}\\ \begin{array}{cc}\frac{2(1-{\lambda }_{\min })}{\pi }\left[\frac{\pi }{2}-\mathrm{atan}\left(e-e_t\right)\right]+{\lambda }_{\min }& e_t<e\end{array}\end{array}$$

(13)

where $\mathrm{atan}(\cdot )$ is the arctangent function. The updated formula of the output weight coefficient $\beta$ is shown in Equation (14).

$$\{\begin{array}{l}{P}_0=H_{{}_0}^T{H}_0+\frac{I}{C}\\ \begin{array}{cc}& ⋮\end{array}\\ P_{k+1}=\lambda P_k+h_{k+1}^T{h}_{k+1}\\ {\beta }_{k+1}={\beta }_k+P_{k+1}^{-1}{h}_{{}_{k+1}}^T\left(t_{k+1}-h_{k+1}{\beta }_k\right)\end{array}\left(0<\lambda \le 1\right)$$

(14)

The forgetting factor λ determines the proportion of old and new samples in the online identification process, as seen from Equation (14). It indicates that all sample data are considered in the identification process when λ = 1. When λ < 1, the old sample data are gradually forgotten as the iterations increase.

The pseudocode of the proposed FOS_RELM algorithm is shown below (Algorithm 1).

Algorithm 1: FOS_RELM

Input:${\left\{\left(x_i,t_i\right)\right\}}_{i=1}^N$, N0$,L,C,e_t$$,{\lambda }_{\min }$ Output:$\beta$ Random initialization hidden layer coefficients $a_i,b_i$. Select initialize training dataset $S_0=\left\{\left(x_i,t_i\right)|x_i\in R^m,t_i\in R^n,i=1,2,3\cdots N_0\right\}$. Calculate hidden layer output matrix H0 by Equation (2). Calculate initialization output layer weight ${\beta }_0$ by Equation (8). k = 1. while$\left(x_{N_0+k},t_{N_0+k}\right)$∈${\left\{\left(x_i,t_i\right)\right\}}_{i=1}^N$do 5. Calculate the hidden layer matrix $h_k$ with the new sample $\left(x_{N_0+k},t_{N_0+k}\right)$ by $h_k=\left[\begin{array}{cc}G\left(a_1,b_1,x_{N_0+k}\right)& \begin{array}{cc}\cdots & G\left(a_L,b_L,x_{N_0+k}\right)\end{array}\end{array}\right]$. 6. Calculate the objective function estimate ${\widehat{t}}_{N_0+k}=h_{N_0+k}{\beta }_{k-1}$ and Training error e. 7. Update the value of λ through Equation (13). 8. The updated value of the output weight coefficient ${\beta }_k$ can be obtained by Equation (14). 9. k = k + 1. end

4. Results and Discussion

4.1. Performance Evaluation of Simulation

In this section, the component-level mechanism model of the gas turbine is used to obtain sample data to verify the rationality and feasibility of the proposed FOS_RELM algorithm.

4.1.1. Input Signal Selection

During model training, the choice of the input excitation signal will directly affect the quality of the model [26]. The pseudorandom binary signal (PRBS) is a periodic signal with white noise characteristics, and it is usually used as an input signal in the model identification of linear systems. However, a gas turbine is a complex nonlinear system, and a PRBS does not cover all its relevant frequencies and attractive spaces [27]. Nelles et al. [28] improved the PRBS by assigning different amplitudes in the range A_min to A_max to each constant signal part to form the amplitude tunable pseudorandom binary signal (APRBS). Figure 4a shows that APRBS for A_min = 0 to A_max = 1. It has been demonstrated that an APRBS is the best excitation signal to obtain information about the process behavior of nonlinear systems [29]. It is worth noting that the minimum dwell time should be at least equal to the stability time of the system so that the system output has time to approach the new set point. Otherwise, only wrong output amplitudes will be generated. On the other hand, a very long minimum dwell time will lead to quasistatic excitation. Therefore, the minimum dwell time was selected to be 5 s in this paper.

In the actual operation of a gas turbine, we usually change the operating state of the gas turbine by controlling the load. We use the gas turbine power signal command to obtain the sample data as the input signal for the mechanism model simulation. The training samples were obtained using the input excitation, as shown in Figure 4a. To avoid the influence of the sample data on the identification effect, we used the data set mutually exclusive with the training set as the testing set. As shown in Figure 4b, we selected the combustion engine load command from no-load to 0.4 operating conditions to obtain the testing set. The sampling time was 0.01 s.

4.1.2. Network Architecture Design

To fully retain the system’s dynamic characteristics, the identified model should contain current and historical information on the essential state parameters. In this paper, the present and historical information on fuel, turbine output speed $N_t$, exhaust temperature $T_e$, and power $P_t$ were selected as input signals. The current moment information of $N_t$, $T_e$, and $P_t$ was used as the output signal to establish the structure of the identification model. All the state parameters are normalized. The model can be described as Equation (15).

$$\begin{array}{l}T=f\left(X\right)\\ \{\begin{array}{l}X=\left[\begin{array}{l}{W}_f\left(k\right)\dots W_f\left(k-m_1\right),N_t\left(k-1\right)\dots N_t\left(k-m_2\right)\\ T_e\left(k-1\right)\dots T_e\left(k-m_3-1\right),P_t\left(k-1\right)\dots P_t\left(k-m_4-1\right)\end{array}\right]\\ T=\left[N_t\left(k\right),T_e\left(k\right),P_t\left(k\right)\right]\end{array}\end{array}$$

(15)

To ensure the model’s accuracy, we need as much information about the input parameters as possible. However, excessive input parameters will increase the structural complexity of the network and quickly lead to the overfitting phenomenon. After debugging, we took $m_1=m_2=m_3=m_4=2$ in this paper. The sigmoid function $G\left(x\right)=1/\left(1+\exp \left(-x\right)\right)$ was chosen as the activation function. The number of nodes in the hidden layer was taken as $L=100$.

It can be seen in Equations (7) and (12), the value of the regularization coefficient C will affect the model’s learning accuracy. To understand the influence law of C on model identification, we compared the identification accuracy of the RELM algorithm and traditional ELM network when C takes different values. Parameter C took values in the set $\left\{2^{-20},2^{-19}\cdots 2^{24}\right\}$, which has 45 elements. To avoid the influence of accidental factors, each algorithm was run 40 times independently, and the final results were judged by the root mean square error (RMSE). The comparison results are shown in Figure 5.

$$\mathrm{RMSE}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(t_i-{\widehat{t}}_i\right)}^2}}$$

(16)

As shown in Figure 5, when the regularization parameter C is smaller, the performance of the RELM identification network is worse. When C ∈ [2¹⁰ 2²⁴], the RELM’s training and testing error of the output speed $N_t$ and exhaust temperature $T_e$ are smaller than the ELM network, and both the RELM and ELM have the same identification accuracy of the power $P_t$. When C = 2¹⁹, the RELM has the highest identification accuracy for the output speed signal. From Figure 5b, we can see that the RELM has the smallest training error for the exhaust temperature signal when C = 2²¹ and the minimum testing error when C = 2¹⁹. Based on the comprehensive analysis of the results in Figure 5, it is concluded that the RELM performs best at C = 2¹⁹. Therefore, parameter C was taken as C = 2¹⁹ in the subsequent calculations and extended to the FOS_RELM algorithm. The parameters of the FOS_RELM algorithm are set as shown in

Table 1. FOS_RELM algorithm parameter settings.

FOS_RELM
L	100
$G(\cdot )$	sigmoid
C	219
$e_t$	0.01
${\lambda }_{\min }$	0.98

.

The model identification results of the ELM, RELM, and FOS_RELM algorithms at C = 2¹⁹ are recorded in

Table 2. Model identification effects of ELM, RELM, and FOS_RELM algorithms.

		ELM	RELM	FOS_RELM
Training RMSE	$N_t$	1.30 × 10−4	9.46 × 10−6	6.60 × 10−6
	$T_e$	1.12 × 10−4	2.78 × 10−5	2.67 × 10−5
	$P_t$	0.0226	0.0226	0.0226
Testing RMSE	$N_t$	1.29 × 10−4	1.16 × 10−5	1.01 × 10−5
	$T_e$	1.12 × 10−4	1.55 × 10−5	1.46 × 10−5
	$P_t$	0.0115	0.0115	0.0115
Training time (s)		0.0809	0.0879	0.6609

. From Table 2, we can see that the FOS_RELM has the best identification effect, and the ELM has the worst impact. The ELM algorithm and RELM algorithms’ training times are similar since they have the same number of nodes in the hidden layer. The FOS_RELM requires iterative updates of the discriminated parameters, so the I’FOS_RELM has the longest training time.

4.1.3. Robustness Testing

The performance of a gas turbine is directly related to external environmental conditions [23]. When the temperature increases, the air density becomes less, and the characteristic line of the compressor is shifted, which changes the gas turbine’s thermal efficiency and power output. To verify the ability of the proposed FOS_RELM algorithm to cope with external environmental changes. The ambient temperature was changed from 22.6 °C to 17 °C to compare the identification effects of the FOS_RELM with the other algorithms.

The parameters of the FOS_RELM algorithm are chosen as follows: ${\lambda }_{\min }=0.98,e_t=0.01,C=2^{19}$.

The model identification effect is shown in Figure 6. After the change in ambient temperature, both the ELM and RELM algorithms failed to describe the system’s dynamic characteristics accurately. In contrast, the FOS_RELM algorithm could still follow the changes in the system state parameters accurately. From this result, we can see that this method of online updating network parameters adopted by the FOS_RELM algorithm has a significant advantage in identifying the nonlinear system with time-varying nature, such as a gas turbine.

Further, to verify the improvement effect of the adopted adaptive update forgetting factor and regularization parameters, the FOS_RELM, FOS_ELM, and OS_ELM algorithms were used to identify the state parameters of the gas turbine when the ambient temperature was reduced from 26 °C to 17 °C, respectively. The operating data of the gas turbine at 17 °C were used for verification. The results are shown in

Table 3. Identification effects of FOS_RELM, FOS_ELM, and OS_ELM algorithms.

Algorithm		OS_ELM	FOS_ELM	FOS_RELM
Testing RMSE	$N_t$	1.06 × 10−5	9.71 × 10−6	8.93 × 10−6
	$T_e$	2.18 × 10−4	1.17 × 10−4	1.19 × 10−4
	$P_t$	0.011	0.011	0.011
Training time (s)		25.82	25.52	26.08

. The proposed FOS_RELM algorithm has the highest recognition accuracy followed by the FOS_ELM algorithm, and the OS_ELM obtained the worst result.

4.2. Performance Evaluation of the Experiment

Section 4.1 compares the identification effects of the FOS_RELM algorithm with other algorithms at the simulation level. However, in actual operation, the placement of sensors and measurement errors can affect the data samples we obtain. The most obvious one is that noise interference signals are inevitably introduced in the measurement process. The noise signal coupled with the gas turbine’s characteristic data will affect the model identification’s effectiveness. The mechanical model does not capture these. Therefore, to evaluate the ability of the proposed FOS_RELM algorithm to resist noise signal interference, the measured data of the gas turbine was used as sample data to compare the identification effect of the FOS_RELM algorithm and the conventional OS_ELM algorithm.

The experimental micro gas turbine unit is shown in Figure 7. It contains a single-stage centrifugal compressor, a single-cylinder combustion chamber, a two-stage axial turbine, and a three-phase AC synchronous generator. Gas turbines support the combustion of various fuels [30,31]. For this article, we used diesel. The detailed parameters of the micro gas turbine are shown in

Table 4. Basic parameters of the micro gas turbine unit.

Rated Speed	Rated Output Power	Rated Power Generation Efficiency	Dimensions (Length × Width × Height)	Compressor Type	Turbine Type	Combustion Chamber Type	Weight
51,000 rpm	125 kW	15%	2950 × 2200 × 2450 mm3	Single-stage centrifugal type	Two-stage axial flow type	Single cylinder type	4.3 t

. The experimental platform utilizes SUPCON’s real-time monitoring software AdvanTrol-Pro(V2) to record the experimental data. The AdvanTrol-Pro software records sampled data from the sensors at a frequency of one datum per second. Due to the inhomogeneity of the gas turbine temperature field, we used three thermocouple sensors to collect the gas turbine temperature signals. The average of the three sensor signals was used as the final output signal. We used magneto-electrical speed sensors to measure the output speed of the turbine. The experimental steps of this paper are as follows:

Process: Firstly, check whether the fuel and lubricating oil systems are standard, and determine whether the unit startup conditions are met. Secondly, start the gas turbine. The unit can realize two kinds of triggering modes, manual and automatic, and we used the automatic mode to accelerate the gas turbine to the no-load state. Thirdly, learn the variable working condition of the gas turbine operation. The gas turbine adopts the constant speed control mode; at the moment of load switching, the speed changes due to the power mismatch of the gas turbine, and the control system stabilizes the speed signal by adjusting the fuel injection. Finally, after all data measurements are completed, switch the gas turbine to shutdown mode.

4.2.1. Performance Test of Noise Resistance

To verify the ability of the FOS_RELM algorithm to resist noise interference, we independently ran the gas turbine twice with the same external conditions. The sampled data were selected as the training and testing sets, respectively. It is worth noting that the selected sample data exclude the startup phase data. The training set contains 1830 data points, as shown in Figure 8. The testing set includes 400 data points, as shown in Figure 9. All data are normalized. It can be seen that the sample data contain random noise signals.

The identification algorithm has 100 hidden nodes, and the sigmoid function was chosen as the activation function. The main parameters of the FOS_RELM algorithm were determined by debugging as ${\lambda }_{\min }=0.98,e_t=0.01,C=2^{10}$.

Figure 10a–c depicts the effectiveness of the OS_ELM and FOS_RELM algorithms in identifying the gas turbine output speed, turbine exhaust temperature, and output power, respectively. In this paper, the sample data were obtained from the actual operation of the micro gas turbine. During the stable operation, the output speed of the gas turbine was the rated value, one after dimensionless. The speed of a gas turbine will fluctuate when the operating condition is changed. As shown in Figure 10a, the FOS_RELM can eliminate the interference of noise signals and accurately describe the dynamic characteristics of the gas turbine. However, the OS_ELM incorrectly treats the noise signal as an inherent characteristic of the gas turbine, which also appears as an overfitting phenomenon. As shown in Figure 10c, the stability of the OS_ELM algorithm is poor, and the OS_ELM does not accurately identify the output power when the operating condition of the gas turbine is changed. Compared with the OS_ELM algorithm, the FOS_RELM algorithm reduces the complexity of the model by using the regularization condition. From the perspective of numerical analysis, the FOS_RELM algorithm can obtain more stable results than the OS_ELM algorithm by reducing the value of the output weight coefficients.

4.2.2. Robustness Testing

The experimental unit shown in Figure 7 was run on 8 September 2021 (the ambient temperature was 28.5 °C) and 23 December 2021 (the ambient temperature was 14 °C) to obtain two operational data sets. The dynamic characteristics in the process of changing gas turbine operating conditions can affect the actions of the control system. If the identified parameters are not in line with the output parameters of the actual unit, it will mislead the controller to take wrong actions and lead to severe accidents. Therefore, we chose the operating data of the gas turbine variable condition process as the sample data for model identification. The model was first trained using the December sample data. Next, 800 data samples from the data collected in September were selected to update the model online. Finally, 202 samples from the data contained in September were used to test the model.

We compare the identification results of the OS_ELM and FOS_RELM in Figure 11. As shown in Figure 11, the FOS_RELM can still accurately describe the dynamic characteristics of the gas turbine when the environmental conditions changed. The speed of the gas turbine first dropped sharply and then slowly rose to the rated value during the ramp-up condition, as shown in Figure 11a. The FOS_RELM algorithm accurately tracks the speed variation law of the gas turbine. However, the OS_ELM algorithm was not accurate in estimating the variation law of the speed. From Figure 11c, it can be seen that the OS_ELM algorithm works poorly in identifying the power $P_t$. The reason is that the old sample data cannot represent the gas turbine after changing external environmental conditions. However, the OS_ELM continues to use the old sample data in the updating process. However, the adaptive forgetting factor strategy applied in the FOS_RELM algorithm provides more attention to new sample observations which can characterize the gas turbine properties. As new sample data becomes available, the old observations are gradually forgotten, and the identification model can keep following the state changes of the gas turbine.

5. Conclusions

Compared with the traditional gradient descent method, the extreme learning machine has apparent advantages in computational efficiency and convergence accuracy. Still, it cannot handle time-varying and measurement noise in the data. Hence, a FOS_RELM algorithm was proposed in this paper. On the one hand, the Tikhonov regularization principle was used to control the complexity of the model and reduce the interference of noise signals to the model. On the other hand, a forgetting factor adaptive adjustment strategy was designed to gradually discard obsolete data in learning and reduce their adverse effects on the network. The FOS_RELM algorithm also inherits the advantages of the extreme learning machine in terms of training time and convergence accuracy.

To illustrate the effectiveness and feasibility of the proposed algorithm, validation experiments were conducted using a gas turbine simulation dataset. The results show that the identification accuracy of the proposed FOS_RELM algorithm is higher than ELM and RELM algorithms. When the gas turbine characteristics change, the identification accuracy of ELM and RELM algorithms deteriorates while the FOS_RELM algorithm still has a better identification effect. More importantly, when it is used to model a natural micro gas turbine system, it shows good robustness and stability and can maintain good identification accuracy when the training data are affected by noise interference and time variability. The proposed algorithm has a broad application prospect which can be used for both onboard modeling and fault diagnosis of the gas turbine.