# Rotor Transverse Slots’ Influence on Negative Sequence Loss and Heat Distribution Prediction of Large Generators Based on Improved Radial Basis Function Process Neural Network

Authors: Wu Guo; Jian Guo
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Owing to the advantages of scientific computation and the data feature support provided by artificial intelligence technology, the theoretical exploration and application research of new computing methods for large generators, one of the most expensive energy equipment in a power system, has become a research hotspot toward solving the bearing limit and operation capacity under abnormal working conditions. Because of many factors affecting the distribution of negative sequence loss and temperature rise that have an extremely complex nonlinear relationship, the traditional calculation and prediction methods of negative sequence conditions cannot suitably reflect the time accumulation effect. Therefore, a prediction method of rotor steady-state negative sequence heating based on radial basis function process neural network is proposed in this paper, and a negative sequence working condition prediction model is established. Accordingly, this study focuses on the study of the relationship among negative sequence heating characteristics, negative sequence component proportion, and transverse slot; additionally, their influence degree, variation relationship, and main principles are further explored that provide a theoretical basis for the design and operation of large generators. As observed from the test results, the steady-state negative sequence condition prediction method based on the improved genetic algorithm radial basis function process neural network features high accuracy; it is a feasible prediction method, specifically for negative sequence conditions of large generators.


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## Body

## 1. Introduction

Identifying the negative sequence working condition and accurately obtaining the negative sequence loss distribution and its maximum temperature rise point is an indispensable part of the design and operation of large generators. Hence, the analysis and calculation of rotor negative sequence heating are particularly important as the core part of determining and evaluating the negative sequence operation capacity of generators. By effectively predicting and analyzing the performance parameters of large generators, the operation state can be estimated in the future under negative sequence conditions [1]. Furthermore, it is of immense significance to prevent and eliminate generator faults and potential safety hazards, plan overall power generation tasks, reasonably allocate various support resources, optimize operation strategies, and effectively implement the operation concept of long-term steady-state negative sequence conditions [2, 3].Initially, the identification of negative sequence heating is performed in power plants or professional laboratories. Negative sequence current is applied directly and physical or direct detection methods are adopted for qualitative and quantitative description of rotor heating. Although accurate negative sequence temperature rise evaluation results are obtained, the disadvantages are extremely prominent. This method is not only extremely likely to cause serious damage to the generator but also consumes extensive time and is extremely costly [4]. With the development of science and technology, power monitoring and detection technology improve this situation. Using generator operation data to predict the negative sequence conditions has become a speedy, low-consumption, extensive method. Monitoring and detection technology measures the main operation data of generators that are closely related to the negative sequence electromagnetic loss distribution, maximum temperature rise point, and negative sequence operation capacity. From the current research and development results, it is practical to identify the negative sequence temperature rise distribution using operation and experimental data [5].A large generator is a complex system, and different performance parameters dynamically reflect the operation state of the generator from different levels and angles. Owing to its simplicity in engineering, most performance predictions are mainly based on a single parameter; nonetheless, the useful information contained in other parameters is ignored. Predicting a single parameter may lead to a large deviation in the analysis results. Multiparameter prediction objectively reflects the complex mechanism of the generator composed of multiple systems and coordinated work output of multiple functional modules and is becoming a research hotspot of generator operation state prediction [6–8]. The prediction results demonstrate the characteristics of coupling, high dimensionality, and nonlinearity. It is extremely important to further systematically and effectively analyze the useful information contained in them and draw reasonable and reliable prediction conclusions. Zheng et al. have addressed the problem that the traditional VSG control strategy does not consider the negative sequence components and cannot eliminate them, and a comprehensive control strategy of VSG under unbalanced voltage conditions is proposed [9]. By simulating asymmetrical external failures in the developed simulation model, noisy and unbalanced fluctuations that carry the effects of positive, negative, and zero sequences in currents were realized by Bayar et al. [10]. The component of the negative sequence represents the unbalance influence, and the fifth and seventh harmonic components represent the nonlinear influence [11]. At present, multiparameter prediction mainly adopts information fusion. Although the final result of fusion analysis contains more information than any single parameter, because its effectiveness is limited by the accuracy of the extracted feature information, it cannot avoid the defect of detailed information in the fusion process, and the introduction of multiple information sources may lead to the decline of prediction accuracy that renders uncertainty to the prediction outcome. In addition, this method is limited by the complex and cumbersome process, high computational overhead, and difficulty in interpretability of the prediction results.Condition monitoring data is the external expression of the internal state of the generator that indicates the trend of its future operation state. However, because the negative sequence performance parameters are nonlinear, nonstationary, high-dimensional, uncertain, chaotic, and time-varying, using the traditional linear time series prediction model causes large errors [12–14]. Radial basis function process neural network (RBFPNN) can approach any continuous function within any error accuracy and features strong nonlinear mapping ability. Therefore, it has obvious advantages for nonlinear time series prediction [15–20]. In particular, it considers the cumulative effect of the parameters in the time dimension and relaxes the synchronous instantaneous limit of the neurons on the input variables. For the mathematical model of the permanent magnet synchronous generator (PMSG), Wu et al. have analyzed the maximum wind power tracking control strategy without wind speed detection and designed a controller based on the cloud RBF neural network and approximate dynamic programming to track the maximum wind power point [21]. Cecati et al. have investigated the effectiveness of some of the newest designed algorithms in machine learning to train typical radial basis function (RBF) networks for 24 h electric load forecasting [22]. Liu and Fei presented an adaptive fractional sliding mode control scheme based on dual RBF neural networks (NNs) to enhance the performance of a three-phase shunt active power filter [23]. With continuous improvement and development, the RBF network has been widely used in various applications of the power system [24–30]. However, the RBF network has certain problems—the network structure, particularly the number of hidden layer nodes, is difficult to determine and the training process easily falls into the local minima that affects its practical application [31–37].Considering the above analysis, an improved RBFPNN is used to establish a dynamic model for the steady-state negative sequence process of a 300-MW large turbo-generator unit using the practical operation data and experimental calculation results, in order to predict the main operation parameters, compare the modeling effects of different network structures, and discuss the variation law of negative sequence loss and temperature rise with the transverse slot. Owing to the advantages of intuitive and clear information and strong integration of RBFPNN, the heating state of the large generator is speedily predicted at the system level, enabling further improvement of the negative sequence operation capacity.


## 2. Characteristic Analysis of Negative Sequence Temperature Rise for Large Generators

Under steady-state symmetrical conditions, many performance indexes such as efficiency and torque of large generators are effective. Therefore, most generators are designed according to such normal conditions and used under such conditions as far as possible. However, in actual operation, the asymmetric operation always occurs—while certain asymmetric operations are caused by load asymmetry or temporary measures, others are caused by accidents. The asymmetric operation of large generators is often accompanied by a series of negative sequence problems. Therefore, it is of immense practical significance to analyze and understand the negative sequence heating problem of large generators.The damage to the symmetrical operation state of the system or generator causes a three-phase voltage or current asymmetry. This asymmetric operation state may be long-term or short-term. If the generator operates asymmetrically, the stator voltage and current will be asymmetric. In particular, the current asymmetry has the most significant impact on the generator. Further, once a negative sequence current traverses stator winding, a negative sequence magnetic field rotating at a synchronous speed and opposite to the rotor direction is established in the air gap. It induces twice the power frequency current in all parts of the rotor surface (big tooth, small tooth, slot wedge, and damping winding). The rotor of a turbo-generator is a single forged body with a strong damping effect; therefore, the induced component of double frequency current in excitation winding is extremely small. In addition, due to the strong skin effect of the double frequency current, the penetration depth to the rotor surface is only a few millimeters, and the equivalent active resistance in the flow path is large. Therefore, the additional electromagnetic loss on the rotor surface may be enormous when the generator operates asymmetrically. The negative sequence current path is concentrated near the transverse slot on the surface of the big tooth at its two ends; it passes through the slot wedge and flows to the small teeth at the but joint of each slot wedge. The research results show that the negative sequence current and loss at the aforementioned parts are the largest, resulting in local overheating or even burning [4–6].It can be concluded from the above analysis that the ability of the generator to bear negative sequence current (hereinafter referred to as negative sequence capacity) is determined by the maximum allowable temperature of the rotor surface structural parts. Within the allowable temperature range, the mechanical properties of each metal structure will not decrease significantly, and the insulating materials will not exceed its maximum allowable temperature. Therefore, it is of immense significance to calculate the negative sequence temperature rise of the rotor body; but the influencing factors under negative sequence conditions are nonlinear. However, many modeling parameters and complex rotor structures should be considered in the electromagnetic dynamic process, and the training process necessitates extensive calculations that have low convergence speed and easily fall into local optimum. In this study, the improved genetic algorithm RBFPNN is used to effectively predict the electromagnetic loss and maximum temperature rise of the main structural parts on the rotor surface under steady-state negative sequence conditions.


## 3. Improved RBFPNN for Negative Sequence Condition Prediction

On the basis of the above-detailed analysis, it is known that the negative sequence heating of a large turbo-generator rotor is caused by the electromagnetic loss on the rotor surface, and the negative sequence capacity is determined by the highest negative sequence temperature rise. Hence, in this study, the electromagnetic loss and the highest negative sequence temperature rise are considered as the main research parameters to predict the negative sequence current amplitude and the working state that the generator can withstand for a long time, in order to provide a theoretical basis for power operation practice. In order to overcome the effects of uncertain factors in nonlinear chaotic time series samples of electromagnetic loss and maximum negative sequence temperature rise, the phase space reconstruction theory is introduced to mine the corresponding relationship between the input and output. The phase space reconstruction theory maps the point sequence in the time series to the corresponding points in the phase space according to the embedding dimension [15, 17]. The trajectory composed of these points can reproduce or inherit the operation law and operation state characteristics of the original system variables and improve the generalization ability of the nonlinear time series model.Based on the embedding theorem, for the time series {xi} (i = 1, 2, …, n), the information of the current state can be expressed as an m-dimensional vector:(1)xi+τ=fxi,xi−τ,…,xi−m−1τ,where m is the embedding dimension, τ is the delay time, and the sampling period interval is often taken. Formula (1) constructs the mapping relationship f between the input and output; that is, as long as the specific previous m parameter values x(i), x(i-τ), …, x(i-(m-1)τ) are known, x(i + τ) can be determined. Thus, it provides a research foundation for the construction of a training sample set and the design of a network structure. Because the mapping function f is difficult to obtain as an accurate mathematical analytical form, RBFPNN is selected to approximate f in this study.


### 3.1. Structure and Principle of RBFPNN

RBFPNN was first proposed in 1985. It realizes nonlinear mapping by changing the parameters of neuron nonlinear change function that leads to the linearization of connection weight adjustment, thereby improving the learning speed of the network, and it can approach any continuous function with any accuracy. After repeated analysis, verification, and improvement, RBFPNN has been considerably developed [15–17]. The network model is a three-layer feedforward neural network model that is composed of the input layer, network hidden layer, and output layer from left to right. The input layer is usually composed of m node elements, enabling the input of the time-varying function into the model. The middle layer is the hidden layer of the network that has q RBF process neurons, and the RBF kernel function is used as the network excitation function. The third layer is called the output layer which is the linear weighted sum of the output signals of the network hidden layer. The radial basis process neuron contains a two-dimensional aggregation of time and space and radial basis kernel function transformation. A typical m-q-1 three-layer RBFPNN is shown in Figure 1.Figure 1
m-q-1 RBFPNN.The input–output relationship of the network can be expressed as follows:(2)y=∑j=1qwjK∫0TXt−Xjtdt,where wjj=1,2,…,q is the weight coefficient of the output layer and the adjustable parameter of the network. x(t) = (x1(t), x2(t), ..., xm(t)) is the network input function, Xj(t) is the nucleus center function of the j-th RBF process neuron, and [0, T] is the input process interval. The training of RBFNN mainly includes the adjustment of neutral mass parameters of the RBF kernel function K, the determination of RBF kernel center function Xj(t), and the iterative correction of the output layer weight coefficient, such that the network may meet the mapping relationship between the input and output of the training samples in the supervised mode. In this paper, the trigonometric function system is selected as the orthogonal basis function, the number of basis functions is 50, and the radial basis kernel function is the Gauss function. Given k groups of learning examples, the i-th input is Xi = (xi-1(t), xi-2(t), ..., xi-m(t)). Meanwhile, di and yi are used to represent the expected output and actual output of the network for the i-th time, respectively, and the network error function is expressed as follows:(3)E=∑i=m+1k+myi−di2.


### 3.2. Structural Optimization Design of RBFPNN Based on Genetic Algorithm

RBFPNN structure parameters, such as the node number of each layer, affect the generalization ability and prediction accuracy of the network to a certain extent. Because there is no unified design rule for the structural parameters of the network, the trial and error method is usually used for parameter selection that is difficult to accept in the case of multiparameter prediction, and particularly, the result of trial and error selection may not be the best. Therefore, it is necessary to design the network structure reasonably.For the parameters to be predicted,m continuous data can be fitted into a time-varying function as the input of RBFPNN. The parameters themselves are the output target; therefore, the node numbers of the input and output layers are simultaneously set to 1. The empirical formula is only used to limit the range of the number of hidden layer nodes in this study. Because genetic algorithm (GA) features powerful global search and parallel processing ability, it does not easily fall into the local minima and can speedily search for the global best point. Hence, GA can be used to optimize the node number of hidden layers of RBFPNN. Based on the combination of convergence accuracy and convergence speed, the precision convergence ratio is used as the individual fitness function.(4)F=lgE0−E1lgt,where E0 and E1 are the set convergence accuracy and actual convergence accuracy, respectively, and t is the network training time.The process is as follows: the initial value ofq is binary coded, and the precision convergence ratio is used to represent the individual fitness function. The population number, evolutionary algebra, crossover rate, and mutation rate are set, subsequently, the network is trained with some samples, tested with other samples, and the average fitness value of all individuals is calculated. The individuals in the population are selected, crossed, and mutated to obtain the optimized fitness value. When the average fitness value of all individuals reaches the requirements or the maximum evolutionary algebra, the iteration is terminated; otherwise, the selection, crossover, and mutation operations continue. Once q is determined, the topology optimization design of RBFPNN is completed.


### 3.3. Structural Optimization Design of RBFPNN Based on Genetic Algorithm

It is found that, for the same training sample set and initial training speed, the trained network is tested with test samples, and the results do not reach the expected accuracy, indicating that the network falls into a local minimum during the training process. In order to eliminate the influence of the initial weight and threshold on the convergence accuracy of RBFPNN, an improved genetic algorithm is introduced again to optimize the initial weight and threshold of the network.For the conventional GA, the crossover ratePc and mutation rate Pm remain unchanged during the algorithm implementation that is easy to converge prematurely and falls into the local extremum. Hence, it is necessary to adaptively and dynamically adjust the crossover probability and mutation probability according to the current individual fitness and iteration times. The crossover rate Pc and mutation rate Pm are adopted according to the following calculation formulas:(5)Pc=Pc1−Pc1−Pc2B−BavgBmin−Bavg,whenB′<Bavg,(6)Pc=Pc1,whenB′≥Bavg,(7)Pm=Pm1−Pm1−Pm2Bmin−BBmin−Bavg,whenB<Bavg,(8)Pm=Pm1,whenB≥Bavg,where Pc1 = 0.8, Pc2 = 0.6, Pm1 = 0.1, and Pm2 = 0.01; B′ is the smaller fitness value of the two individuals to be crossed; B is the individual fitness value to be mutated; Bavg is the average fitness value of each generation, and Bmin is the minimum fitness value in the population. The optimal individual obtained by the improved GA optimization is considered the optimal parameter value of RBFPNN, and the simulation test of the highest negative sequence temperature rise prediction is conducted to verify the effectiveness of the algorithm. The fitness function in this part is the network error function E.


### 3.4. Output Steps and Flow of Negative Sequence Condition Prediction Results Based on Improved RBFPNN Algorithm

RBFPNN structure parameters, such as the node number of each layer, affect the generalization ability and prediction accuracy of the network to a certain extent. Because there is no unified design rule for the structural parameters of the network, the trial and error method is usually used for parameter selection that is difficult to accept in the case of multiparameter prediction, and particularly, the result of trial and error selection may not be the best. Therefore, it is necessary to design the network structure reasonably.For the parameters to be predicted,m continuous data can be fitted into a time-varying function as the input of RBFPNN. The parameters themselves are the output target; therefore, the node numbers of the input and output layers are simultaneously set to 1. The empirical formula is only used to limit the range of the number of hidden layer nodes in this study. Because GA has powerful global search and parallel processing ability, it does not easily fall into the local minima and can speedily search for the global best point. Hence, GA can be used to optimize the node number of hidden layers of RBFPNN. Based on the combination of convergence accuracy and convergence speed, the precision convergence ratio is used as the individual fitness function.(9)F=lgE0−E1lgt,where E0 and E1 are the set convergence accuracy and actual convergence accuracy, respectively, and t is the network training time.The specific process is as follows: the initial value ofq is binary coded, and the precision convergence ratio is used to represent the individual fitness function. The population number, evolutionary algebra, crossover rate, and mutation rate are set; subsequently, the network is trained with some samples, tested with other samples, and the average fitness value of all individuals is calculated. The individuals in the population are selected, crossed, and mutated to obtain the optimized fitness value. When the average fitness value of all individuals reaches the requirements or reaches the maximum evolutionary algebra, the iteration is terminated; otherwise, the selection, crossover, and mutation operations continue. Once q is determined, the topology optimization design of RBFPNN is completed.The previous analysis shows that the hidden layer node number, weight, and threshold have a significant impact on the prediction effect of the RBFPNN model. Therefore, GA is selected to optimize the number of hidden layer nodes, initial weight, and threshold of RBFPNN, respectively. The main steps of optimization are as follows:Step 1: construct the training sample set of parameterj. Set the parameters such as population number, evolutionary algebra, crossover probability, and mutation probability. Initialize the population of hidden layer node number q and code it.Step 2:Q is assigned by individuals, and the fitness of each individual is calculated according to Formula (4) through training. Select individuals according to the Roulette method, and implement single-point crossover, mutation, and other operations to obtain the optimized fitness value.Step 3: if the set number of iterations is reached or the fitness value meets the requirements, the optimized number of hidden layer nodes is output. Otherwise, go to Step 2 and continue the iteration.Step 4: after establishing the optimal structure of RBFPNN, set the parameters of GA. Initialize and code the initial weight and threshold of the network.Step 5: individuals are used to assign values to the weights and thresholds, and the fitness of each individual after training is calculated according to Formula (3). Perform selection, crossover, mutation, and other operations to obtain the optimized fitness value according to Formulas (5)–(8).Step 6: if the set stop training condition is reached, output the optimized initial weight and threshold, and exit the optimization program. Otherwise, go to Step 5 and continue the iteration.The main steps of the algorithm are as follows:Step 1. The RBFPNN structure, weight, and threshold are optimized by GA. Set the training convergence accuracy of the network ε, maximum number of iterations M, cumulative number of learning iterations s, and the other parameters.Step 2. Expand the input function and weight function. Input all training samples into the network, and use Formula (2) to calculate the output yi of the network.Step 3. The error function E of the network is calculated using Formula (3). If E< ε, or s >M, go to Step 5.Step 4. Update the network weight and threshold, enable s+1⟶s, and move to Step 4.Step 5. Save the current network learning results, and use the trained network to calculate the predicted value of parameters.After all parameters are predicted, output the predicted performance results and exit the program.


## 3.1. Structure and Principle of RBFPNN

RBFPNN was first proposed in 1985. It realizes nonlinear mapping by changing the parameters of neuron nonlinear change function that leads to the linearization of connection weight adjustment, thereby improving the learning speed of the network, and it can approach any continuous function with any accuracy. After repeated analysis, verification, and improvement, RBFPNN has been considerably developed [15–17]. The network model is a three-layer feedforward neural network model that is composed of the input layer, network hidden layer, and output layer from left to right. The input layer is usually composed of m node elements, enabling the input of the time-varying function into the model. The middle layer is the hidden layer of the network that has q RBF process neurons, and the RBF kernel function is used as the network excitation function. The third layer is called the output layer which is the linear weighted sum of the output signals of the network hidden layer. The radial basis process neuron contains a two-dimensional aggregation of time and space and radial basis kernel function transformation. A typical m-q-1 three-layer RBFPNN is shown in Figure 1.Figure 1
m-q-1 RBFPNN.The input–output relationship of the network can be expressed as follows:(2)y=∑j=1qwjK∫0TXt−Xjtdt,where wjj=1,2,…,q is the weight coefficient of the output layer and the adjustable parameter of the network. x(t) = (x1(t), x2(t), ..., xm(t)) is the network input function, Xj(t) is the nucleus center function of the j-th RBF process neuron, and [0, T] is the input process interval. The training of RBFNN mainly includes the adjustment of neutral mass parameters of the RBF kernel function K, the determination of RBF kernel center function Xj(t), and the iterative correction of the output layer weight coefficient, such that the network may meet the mapping relationship between the input and output of the training samples in the supervised mode. In this paper, the trigonometric function system is selected as the orthogonal basis function, the number of basis functions is 50, and the radial basis kernel function is the Gauss function. Given k groups of learning examples, the i-th input is Xi = (xi-1(t), xi-2(t), ..., xi-m(t)). Meanwhile, di and yi are used to represent the expected output and actual output of the network for the i-th time, respectively, and the network error function is expressed as follows:(3)E=∑i=m+1k+myi−di2.


## 3.2. Structural Optimization Design of RBFPNN Based on Genetic Algorithm

RBFPNN structure parameters, such as the node number of each layer, affect the generalization ability and prediction accuracy of the network to a certain extent. Because there is no unified design rule for the structural parameters of the network, the trial and error method is usually used for parameter selection that is difficult to accept in the case of multiparameter prediction, and particularly, the result of trial and error selection may not be the best. Therefore, it is necessary to design the network structure reasonably.For the parameters to be predicted,m continuous data can be fitted into a time-varying function as the input of RBFPNN. The parameters themselves are the output target; therefore, the node numbers of the input and output layers are simultaneously set to 1. The empirical formula is only used to limit the range of the number of hidden layer nodes in this study. Because genetic algorithm (GA) features powerful global search and parallel processing ability, it does not easily fall into the local minima and can speedily search for the global best point. Hence, GA can be used to optimize the node number of hidden layers of RBFPNN. Based on the combination of convergence accuracy and convergence speed, the precision convergence ratio is used as the individual fitness function.(4)F=lgE0−E1lgt,where E0 and E1 are the set convergence accuracy and actual convergence accuracy, respectively, and t is the network training time.The process is as follows: the initial value ofq is binary coded, and the precision convergence ratio is used to represent the individual fitness function. The population number, evolutionary algebra, crossover rate, and mutation rate are set, subsequently, the network is trained with some samples, tested with other samples, and the average fitness value of all individuals is calculated. The individuals in the population are selected, crossed, and mutated to obtain the optimized fitness value. When the average fitness value of all individuals reaches the requirements or the maximum evolutionary algebra, the iteration is terminated; otherwise, the selection, crossover, and mutation operations continue. Once q is determined, the topology optimization design of RBFPNN is completed.


## 3.3. Structural Optimization Design of RBFPNN Based on Genetic Algorithm

It is found that, for the same training sample set and initial training speed, the trained network is tested with test samples, and the results do not reach the expected accuracy, indicating that the network falls into a local minimum during the training process. In order to eliminate the influence of the initial weight and threshold on the convergence accuracy of RBFPNN, an improved genetic algorithm is introduced again to optimize the initial weight and threshold of the network.For the conventional GA, the crossover ratePc and mutation rate Pm remain unchanged during the algorithm implementation that is easy to converge prematurely and falls into the local extremum. Hence, it is necessary to adaptively and dynamically adjust the crossover probability and mutation probability according to the current individual fitness and iteration times. The crossover rate Pc and mutation rate Pm are adopted according to the following calculation formulas:(5)Pc=Pc1−Pc1−Pc2B−BavgBmin−Bavg,whenB′<Bavg,(6)Pc=Pc1,whenB′≥Bavg,(7)Pm=Pm1−Pm1−Pm2Bmin−BBmin−Bavg,whenB<Bavg,(8)Pm=Pm1,whenB≥Bavg,where Pc1 = 0.8, Pc2 = 0.6, Pm1 = 0.1, and Pm2 = 0.01; B′ is the smaller fitness value of the two individuals to be crossed; B is the individual fitness value to be mutated; Bavg is the average fitness value of each generation, and Bmin is the minimum fitness value in the population. The optimal individual obtained by the improved GA optimization is considered the optimal parameter value of RBFPNN, and the simulation test of the highest negative sequence temperature rise prediction is conducted to verify the effectiveness of the algorithm. The fitness function in this part is the network error function E.


## 3.4. Output Steps and Flow of Negative Sequence Condition Prediction Results Based on Improved RBFPNN Algorithm

RBFPNN structure parameters, such as the node number of each layer, affect the generalization ability and prediction accuracy of the network to a certain extent. Because there is no unified design rule for the structural parameters of the network, the trial and error method is usually used for parameter selection that is difficult to accept in the case of multiparameter prediction, and particularly, the result of trial and error selection may not be the best. Therefore, it is necessary to design the network structure reasonably.For the parameters to be predicted,m continuous data can be fitted into a time-varying function as the input of RBFPNN. The parameters themselves are the output target; therefore, the node numbers of the input and output layers are simultaneously set to 1. The empirical formula is only used to limit the range of the number of hidden layer nodes in this study. Because GA has powerful global search and parallel processing ability, it does not easily fall into the local minima and can speedily search for the global best point. Hence, GA can be used to optimize the node number of hidden layers of RBFPNN. Based on the combination of convergence accuracy and convergence speed, the precision convergence ratio is used as the individual fitness function.(9)F=lgE0−E1lgt,where E0 and E1 are the set convergence accuracy and actual convergence accuracy, respectively, and t is the network training time.The specific process is as follows: the initial value ofq is binary coded, and the precision convergence ratio is used to represent the individual fitness function. The population number, evolutionary algebra, crossover rate, and mutation rate are set; subsequently, the network is trained with some samples, tested with other samples, and the average fitness value of all individuals is calculated. The individuals in the population are selected, crossed, and mutated to obtain the optimized fitness value. When the average fitness value of all individuals reaches the requirements or reaches the maximum evolutionary algebra, the iteration is terminated; otherwise, the selection, crossover, and mutation operations continue. Once q is determined, the topology optimization design of RBFPNN is completed.The previous analysis shows that the hidden layer node number, weight, and threshold have a significant impact on the prediction effect of the RBFPNN model. Therefore, GA is selected to optimize the number of hidden layer nodes, initial weight, and threshold of RBFPNN, respectively. The main steps of optimization are as follows:Step 1: construct the training sample set of parameterj. Set the parameters such as population number, evolutionary algebra, crossover probability, and mutation probability. Initialize the population of hidden layer node number q and code it.Step 2:Q is assigned by individuals, and the fitness of each individual is calculated according to Formula (4) through training. Select individuals according to the Roulette method, and implement single-point crossover, mutation, and other operations to obtain the optimized fitness value.Step 3: if the set number of iterations is reached or the fitness value meets the requirements, the optimized number of hidden layer nodes is output. Otherwise, go to Step 2 and continue the iteration.Step 4: after establishing the optimal structure of RBFPNN, set the parameters of GA. Initialize and code the initial weight and threshold of the network.Step 5: individuals are used to assign values to the weights and thresholds, and the fitness of each individual after training is calculated according to Formula (3). Perform selection, crossover, mutation, and other operations to obtain the optimized fitness value according to Formulas (5)–(8).Step 6: if the set stop training condition is reached, output the optimized initial weight and threshold, and exit the optimization program. Otherwise, go to Step 5 and continue the iteration.The main steps of the algorithm are as follows:Step 1. The RBFPNN structure, weight, and threshold are optimized by GA. Set the training convergence accuracy of the network ε, maximum number of iterations M, cumulative number of learning iterations s, and the other parameters.Step 2. Expand the input function and weight function. Input all training samples into the network, and use Formula (2) to calculate the output yi of the network.Step 3. The error function E of the network is calculated using Formula (3). If E< ε, or s >M, go to Step 5.Step 4. Update the network weight and threshold, enable s+1⟶s, and move to Step 4.Step 5. Save the current network learning results, and use the trained network to calculate the predicted value of parameters.After all parameters are predicted, output the predicted performance results and exit the program.


## 4. Application Research on Negative Sequence Temperature Rise Prediction Using RBFPNN

### 4.1. Research Object of 300-MW Large Turbo-Generator

In order to verify the effectiveness of the improved RBFPNN algorithm, this study deploys two 300 MW large turbo-generators as the research object; the real operation data are obtained as the training samples for algorithm comparison and experimental testing. The generator is a large power generation equipment jointly developed and designed by many companies. It is characterized by reasonable structure, advanced performance, stable operation, and a large capacity margin. At present, dozens of such generators have been operated safely in the power plants in China. This type of generator has won high praise from users for its reliable operation. The schematic and sectional view of the large generator is separately shown in Figures2 and 3, and the electrical parameters are shown in Table 1.Figure 2
Schematic of 300-MW large turbo-generator.Figure 3
Sectional view of the plant unit layout.Table 1
Main parameters of the 300-MW large turbo-generator.
ParameterUnitValueRated powerMW300Rated voltagekV20Rated currentA10190Rated speedr/min3000Rated frequencyHz50Steady-state negative sequence capability0.1Transient negative sequence capabilitys10


### 4.2. Performance Analysis of the Prediction Model

Comprehensively considering the actual operation data of a large generator, the principle of negative sequence heating, the technical manual, and the experience of engineers, six parameters are selected to characterize the negative sequence heating state of the rotor. They are the highest negative sequence temperature rise of the big tooth (BTT), the highest negative sequence temperature rise of the small tooth (STT), the highest negative sequence temperature rise of small tooth wedge (STWT), the electromagnetic loss density of big tooth (BTLD), the electromagnetic loss density of small tooth (STLD), and the electromagnetic loss density of small tooth wedge (STWLD).In this study, 130 groups of steady-state negative sequence operation data, including the above six parameters of No.1 300-WM steam turbine generator in a power plant for six months, are selected to predict the rotor heating data under the condition of long-term negative sequence current. For any parameterj (j = 1, 2, ..., 6), its continuous data mj is fit into a time-varying function as the input of the prediction model (mj represents the embedding dimension of parameter j), the mj + 1 is considered as the corresponding expected output of the prediction model, and 130-mj groups of samples are constructed. The first 110-mj groups of samples are used to train the model, and the last 20 groups of samples are used to verify the prediction. The initial population size is set to 35, the training error is 0.001, and the maximum number of iterative steps is 1000. According to the steps in Sections 3.2 and 3.3, the number of hidden layer nodes, initial weight, and threshold are optimized, and subsequently, the optimized model is trained by using the training samples of the parameter based on the algorithm in Section 3.4. Finally, the test data is substituted into the trained network to predict the last 20 predicted values of the parameter. When all the parameters are predicted, the prediction results are combined, and the output is a 20 × 6 negative sequence heating prediction performance matrix. The results are shown in Table 2.Table 2
Embedding dimension and optimal topology of the improved RBFPNN model for each parameter.
ParametermOptimal topologyBTT71-13-1STT51-11-1STWT71-12-1BTLD61-14-1STLD71-13-1STWLD71-14-1For comparison, the simple RBFPNN model, the RBFPNN model with only the structure optimized based on conventional GA (CGA-RBFPNN), the RBFPNN model with the structure, network initial weight, and threshold optimized based on conventional GA (AGA-RBFPNN), the RBFPNN model with the structure, network initial weight, and threshold optimized based on the improved GA (IGA-RBFPNN), and the RBFPNN model optimized based on both the conventional GA optimization structure and the improved GA optimization network initial weight and threshold (GA-IGA-RBFPNN), are used to predict the above parameters for three rounds. The number of input and output layer nodes of all the models is 1. For the same prediction parameter, the number of hidden layer nodes of RBFPNN is determined by the empirical formula. The last four networks adopt their optimized optimal topology. The first two types of networks use the same initial weights and thresholds in each round of training, but the initial weights and thresholds of the different rounds are different. The last three types use their optimized initial weights and thresholds. In this study, mainly the average relative errorAE, maximum relative error AEmax, similarity R, and single round cumulative time T are predicted by each model. The results are shown in Table 3.Table 3
Comparative analysis of the model prediction results.
Test no.ModelAE/%AEmax/%RT/s1RBFPNN5.8923.210.91051.23CGA-RBFPNN4.0315.460.95740.14AGA-RBFPNN2.8611.470.96239.31IGA-RBFPNN1.966.030.97938.89GA- IGA-RBFPNN1.785.740.98536.962RBFPNN5.7427.650.91649.56CGA-RBFPNN4.1514.690.96243.61AGA-RBFPNN3.0810.350.97041.58IGA-RBFPNN1.926.810.97740.72GA- IGA-RBFPNN1.746.620.98338.173RBFPNN5.4820.170.92554.38CGA-RBFPNN3.8511.480.96441.57AGA-RBFPNN2.9810.160.96841.35IGA-RBFPNN1.637.360.98141.32GA- IGA-RBFPNN1.576.820.98638.14It can be seen from Table3 that in the three rounds of tests, the average error of the GA-IGA-RBFPNN model is 1.78% at the highest and 1.57% at the lowest which is better than the other models. Meanwhile, the average error of the RBFPNN model is 5.89% at the highest and 5.48% at the lowest which is in the last of the five. In terms of the maximum relative error, the maximum value of the GA-IGA-RBFPNN model is 6.82% which is significantly lower than the other models, reflecting the suitable stability of this model. In particular, the minimum similarity of GA-IGA-RBFPNN is 0.983 which is higher than the other models, indicating that the predicted value is extremely close to the real value overall. In addition, among the three models, GA-IGA-RBFPNN also has certain advantages in terms of being time-effective. It can be seen that although the initial training conditions are different, the prediction effect of the GA-IGA-RBFPNN model with two different optimizations is better than that of the CGA-RBFPNN model with single optimization, AGA-RBFPNN and IGA-RBFPNN with two same optimizations, and simple RBFPNN model. The reason why the GA-IGA-RBFPNN model has the most ideal accuracy, generalization, and stability is mainly due to the more effective optimization method for different structural parameters. GA is used to search in parallel in a large range, to obtain the optimal structure of the network, which improves the generalization ability of the network, while IGA is used to optimize the initial weight and threshold that further modifies the convergence direction of the network, and subsequently, the defect of falling into the minimum is eliminated. Because other models adopt the optimized network structure to varying degrees, the effect is relatively better than that of the RBFPNN model. It should be noted that although the other four models above have certain shortcomings, the overall consideration of the above indicators does not imply that they cannot be used to predict the performance parameters of rotor negative sequence heating.


### 4.3. Experimental Verification and Application

In addition to the experiments and comparisons above under normal operation, in this study, a number of temperature rise tests and prediction calculations are conducted under no-load conditions. The practicality of the optimization algorithm was further studied through the analysis of typical characteristics under different negative sequence conditions. The negative sequence capacity of a large generator depends on the rotor temperature rise; that is, the temperature of the hot spot does not exceed the rotor temperature rise in the allowable temperature range of the material at this point. According to theoretical analysis and practical experience, the hot point is on the polar surface of the rotor, particularly the end of the transverse slot. During the experiment, the Electric Machinery Company installs several temperature measuring points on the rotor, and the specific distribution of certain measuring points on the large tooth surface is shown in Figure4. The interface structure of the rotor big tooth is provided in Figure 5.Figure 4
Sectional view of the plant unit layout.Figure 5
Diagram of rotor big tooth interface structure.On the polar surface, the lead shall is arranged at the shortest distance from each measuring point to the lead slot. After leaving the lead slot, the body is routed along; the rotor center hole is entered through the straight hole, along the conductive rod to the rotor end, and finally connected with the external small slip ring. Because it is a hydrogen cooled generator, in order to prevent hydrogen leakage, it is also necessary to effectively seal the lead. The cold end of the thermocouple is placed between the rotor end and the small slip ring, where a temperature measuring element is placed to monitor the temperature change of the cold end of the thermocouple. The temperature measuring element adopts a thermocouple, and the two wires are twisted together to reduce the error. The polar thermocouple and lead are fixed by bonding. Figure6 shows a schematic of the lead arrangement. Some results of the temperature rise test and prediction calculation are provided in Table 4.Figure 6
Schematic of the lead arrangement.Table 4
Negative sequence experiment results and calculation results.
Measuring pointNo-load operationU0 = 1.05UNU0 = 1.2UNIf = 1030 AIf = 1564 ANo. 1Measured value34.839.3Predicted value35.241.5Error1.1%5.6%No. 2Measured value34.745.4Predicted value35.446.6Error2.0%2.6%Through the analysis and comparison of the test and simulation results in Table4, it can be found that the calculated results of the temperature distribution under no-load conditions are extremely close to the measured data, all errors are within the allowable range of engineering errors, and the change trend of temperature rise is basically the same. Therefore, the overall comparison of the results has completely proved that the prediction model established in this study is accurate and effective; it can accurately reflect the real operation of the generator. Furthermore, the heating change of the rotor is consistent with the actual operation trend as well. Therefore, the prediction results in this paper are accurate and reliable.In order to further test the generalization and reliability, the GA-IGA-RBFPNN model is used to synchronously predict the operation data of another No. 2 generator in the same power plant under negative sequence conditions. The total number of data is 160 × 6; that is, 160 data are recorded for each of the six parameters in the same period. The continuousm data of each parameter are fitted into a time-varying function as the input of the prediction model (m represents the embedding dimension of the parameter), and the m + 1 data is the corresponding expected output of the prediction model that would enable constructing 160-m groups of samples. The first 140-m groups of samples are used for GA-IGA-RBFPNN modeling, and the last 20 groups of samples are adopted for prediction verification. The setting and selection of GA parameters are the same as in the above example. After all parameters are predicted, the prediction results of six parameters are combined and a 20 × 6 prediction performance matrix is output. The prediction results also show that the GA-IGA-RBFPNN model can meet the requirements of generator performance monitoring under steady-state negative sequence conditions. Tables 5 and 6 show the embedding dimension, optimal topology, and prediction results of the GA-IGA-RBFPNN model for each parameter of the No. 2 generator.Table 5
Optimal topology of GA-IGA-RBFPNN.
ParametermOptimal topologyBTT61-13-1STT51-12-1STWT71-12-1BTLD51-14-1STLD61-13-1STWLD71-14-1Table 6
Prediction result analysis based on the GA-IGA-RBFPNN model.
AE/%AEmax/%RT/s1.857.160.97943.57


## 4.1. Research Object of 300-MW Large Turbo-Generator

In order to verify the effectiveness of the improved RBFPNN algorithm, this study deploys two 300 MW large turbo-generators as the research object; the real operation data are obtained as the training samples for algorithm comparison and experimental testing. The generator is a large power generation equipment jointly developed and designed by many companies. It is characterized by reasonable structure, advanced performance, stable operation, and a large capacity margin. At present, dozens of such generators have been operated safely in the power plants in China. This type of generator has won high praise from users for its reliable operation. The schematic and sectional view of the large generator is separately shown in Figures2 and 3, and the electrical parameters are shown in Table 1.Figure 2
Schematic of 300-MW large turbo-generator.Figure 3
Sectional view of the plant unit layout.Table 1
Main parameters of the 300-MW large turbo-generator.
ParameterUnitValueRated powerMW300Rated voltagekV20Rated currentA10190Rated speedr/min3000Rated frequencyHz50Steady-state negative sequence capability0.1Transient negative sequence capabilitys10


## 4.2. Performance Analysis of the Prediction Model

Comprehensively considering the actual operation data of a large generator, the principle of negative sequence heating, the technical manual, and the experience of engineers, six parameters are selected to characterize the negative sequence heating state of the rotor. They are the highest negative sequence temperature rise of the big tooth (BTT), the highest negative sequence temperature rise of the small tooth (STT), the highest negative sequence temperature rise of small tooth wedge (STWT), the electromagnetic loss density of big tooth (BTLD), the electromagnetic loss density of small tooth (STLD), and the electromagnetic loss density of small tooth wedge (STWLD).In this study, 130 groups of steady-state negative sequence operation data, including the above six parameters of No.1 300-WM steam turbine generator in a power plant for six months, are selected to predict the rotor heating data under the condition of long-term negative sequence current. For any parameterj (j = 1, 2, ..., 6), its continuous data mj is fit into a time-varying function as the input of the prediction model (mj represents the embedding dimension of parameter j), the mj + 1 is considered as the corresponding expected output of the prediction model, and 130-mj groups of samples are constructed. The first 110-mj groups of samples are used to train the model, and the last 20 groups of samples are used to verify the prediction. The initial population size is set to 35, the training error is 0.001, and the maximum number of iterative steps is 1000. According to the steps in Sections 3.2 and 3.3, the number of hidden layer nodes, initial weight, and threshold are optimized, and subsequently, the optimized model is trained by using the training samples of the parameter based on the algorithm in Section 3.4. Finally, the test data is substituted into the trained network to predict the last 20 predicted values of the parameter. When all the parameters are predicted, the prediction results are combined, and the output is a 20 × 6 negative sequence heating prediction performance matrix. The results are shown in Table 2.Table 2
Embedding dimension and optimal topology of the improved RBFPNN model for each parameter.
ParametermOptimal topologyBTT71-13-1STT51-11-1STWT71-12-1BTLD61-14-1STLD71-13-1STWLD71-14-1For comparison, the simple RBFPNN model, the RBFPNN model with only the structure optimized based on conventional GA (CGA-RBFPNN), the RBFPNN model with the structure, network initial weight, and threshold optimized based on conventional GA (AGA-RBFPNN), the RBFPNN model with the structure, network initial weight, and threshold optimized based on the improved GA (IGA-RBFPNN), and the RBFPNN model optimized based on both the conventional GA optimization structure and the improved GA optimization network initial weight and threshold (GA-IGA-RBFPNN), are used to predict the above parameters for three rounds. The number of input and output layer nodes of all the models is 1. For the same prediction parameter, the number of hidden layer nodes of RBFPNN is determined by the empirical formula. The last four networks adopt their optimized optimal topology. The first two types of networks use the same initial weights and thresholds in each round of training, but the initial weights and thresholds of the different rounds are different. The last three types use their optimized initial weights and thresholds. In this study, mainly the average relative errorAE, maximum relative error AEmax, similarity R, and single round cumulative time T are predicted by each model. The results are shown in Table 3.Table 3
Comparative analysis of the model prediction results.
Test no.ModelAE/%AEmax/%RT/s1RBFPNN5.8923.210.91051.23CGA-RBFPNN4.0315.460.95740.14AGA-RBFPNN2.8611.470.96239.31IGA-RBFPNN1.966.030.97938.89GA- IGA-RBFPNN1.785.740.98536.962RBFPNN5.7427.650.91649.56CGA-RBFPNN4.1514.690.96243.61AGA-RBFPNN3.0810.350.97041.58IGA-RBFPNN1.926.810.97740.72GA- IGA-RBFPNN1.746.620.98338.173RBFPNN5.4820.170.92554.38CGA-RBFPNN3.8511.480.96441.57AGA-RBFPNN2.9810.160.96841.35IGA-RBFPNN1.637.360.98141.32GA- IGA-RBFPNN1.576.820.98638.14It can be seen from Table3 that in the three rounds of tests, the average error of the GA-IGA-RBFPNN model is 1.78% at the highest and 1.57% at the lowest which is better than the other models. Meanwhile, the average error of the RBFPNN model is 5.89% at the highest and 5.48% at the lowest which is in the last of the five. In terms of the maximum relative error, the maximum value of the GA-IGA-RBFPNN model is 6.82% which is significantly lower than the other models, reflecting the suitable stability of this model. In particular, the minimum similarity of GA-IGA-RBFPNN is 0.983 which is higher than the other models, indicating that the predicted value is extremely close to the real value overall. In addition, among the three models, GA-IGA-RBFPNN also has certain advantages in terms of being time-effective. It can be seen that although the initial training conditions are different, the prediction effect of the GA-IGA-RBFPNN model with two different optimizations is better than that of the CGA-RBFPNN model with single optimization, AGA-RBFPNN and IGA-RBFPNN with two same optimizations, and simple RBFPNN model. The reason why the GA-IGA-RBFPNN model has the most ideal accuracy, generalization, and stability is mainly due to the more effective optimization method for different structural parameters. GA is used to search in parallel in a large range, to obtain the optimal structure of the network, which improves the generalization ability of the network, while IGA is used to optimize the initial weight and threshold that further modifies the convergence direction of the network, and subsequently, the defect of falling into the minimum is eliminated. Because other models adopt the optimized network structure to varying degrees, the effect is relatively better than that of the RBFPNN model. It should be noted that although the other four models above have certain shortcomings, the overall consideration of the above indicators does not imply that they cannot be used to predict the performance parameters of rotor negative sequence heating.


## 4.3. Experimental Verification and Application

In addition to the experiments and comparisons above under normal operation, in this study, a number of temperature rise tests and prediction calculations are conducted under no-load conditions. The practicality of the optimization algorithm was further studied through the analysis of typical characteristics under different negative sequence conditions. The negative sequence capacity of a large generator depends on the rotor temperature rise; that is, the temperature of the hot spot does not exceed the rotor temperature rise in the allowable temperature range of the material at this point. According to theoretical analysis and practical experience, the hot point is on the polar surface of the rotor, particularly the end of the transverse slot. During the experiment, the Electric Machinery Company installs several temperature measuring points on the rotor, and the specific distribution of certain measuring points on the large tooth surface is shown in Figure4. The interface structure of the rotor big tooth is provided in Figure 5.Figure 4
Sectional view of the plant unit layout.Figure 5
Diagram of rotor big tooth interface structure.On the polar surface, the lead shall is arranged at the shortest distance from each measuring point to the lead slot. After leaving the lead slot, the body is routed along; the rotor center hole is entered through the straight hole, along the conductive rod to the rotor end, and finally connected with the external small slip ring. Because it is a hydrogen cooled generator, in order to prevent hydrogen leakage, it is also necessary to effectively seal the lead. The cold end of the thermocouple is placed between the rotor end and the small slip ring, where a temperature measuring element is placed to monitor the temperature change of the cold end of the thermocouple. The temperature measuring element adopts a thermocouple, and the two wires are twisted together to reduce the error. The polar thermocouple and lead are fixed by bonding. Figure6 shows a schematic of the lead arrangement. Some results of the temperature rise test and prediction calculation are provided in Table 4.Figure 6
Schematic of the lead arrangement.Table 4
Negative sequence experiment results and calculation results.
Measuring pointNo-load operationU0 = 1.05UNU0 = 1.2UNIf = 1030 AIf = 1564 ANo. 1Measured value34.839.3Predicted value35.241.5Error1.1%5.6%No. 2Measured value34.745.4Predicted value35.446.6Error2.0%2.6%Through the analysis and comparison of the test and simulation results in Table4, it can be found that the calculated results of the temperature distribution under no-load conditions are extremely close to the measured data, all errors are within the allowable range of engineering errors, and the change trend of temperature rise is basically the same. Therefore, the overall comparison of the results has completely proved that the prediction model established in this study is accurate and effective; it can accurately reflect the real operation of the generator. Furthermore, the heating change of the rotor is consistent with the actual operation trend as well. Therefore, the prediction results in this paper are accurate and reliable.In order to further test the generalization and reliability, the GA-IGA-RBFPNN model is used to synchronously predict the operation data of another No. 2 generator in the same power plant under negative sequence conditions. The total number of data is 160 × 6; that is, 160 data are recorded for each of the six parameters in the same period. The continuousm data of each parameter are fitted into a time-varying function as the input of the prediction model (m represents the embedding dimension of the parameter), and the m + 1 data is the corresponding expected output of the prediction model that would enable constructing 160-m groups of samples. The first 140-m groups of samples are used for GA-IGA-RBFPNN modeling, and the last 20 groups of samples are adopted for prediction verification. The setting and selection of GA parameters are the same as in the above example. After all parameters are predicted, the prediction results of six parameters are combined and a 20 × 6 prediction performance matrix is output. The prediction results also show that the GA-IGA-RBFPNN model can meet the requirements of generator performance monitoring under steady-state negative sequence conditions. Tables 5 and 6 show the embedding dimension, optimal topology, and prediction results of the GA-IGA-RBFPNN model for each parameter of the No. 2 generator.Table 5
Optimal topology of GA-IGA-RBFPNN.
ParametermOptimal topologyBTT61-13-1STT51-12-1STWT71-12-1BTLD51-14-1STLD61-13-1STWLD71-14-1Table 6
Prediction result analysis based on the GA-IGA-RBFPNN model.
AE/%AEmax/%RT/s1.857.160.97943.57


## 5. Influence Prediction of Transverse Slot on Negative Sequence Loss and Heating under Different Negative Sequence Conditions

Under negative sequence conditions, there are many influencing factors that affect the distribution of negative sequence loss and temperature rise. In this study, the optimal prediction model of the improved GA-IGA-RBFPNN algorithm is used to analyze the negative sequence heating degree under different negative sequence conditions. For example, in general, multiple transverse crescent slots are evenly distributed along the axial direction on the center line of the big tooth of the rotor body, to balance the stiffness of the orthogonal two axes, and the frequency doubling vibration can be reduced. However, balancing the stiffness causes new problems. Under negative sequence conditions, a large number of eddy currents collect on the big teeth of the rotor body, particularly at the edge of the transverse crescent slot. Thereafter, the sharp rise of the temperature in this part is caused by the negative sequence eddy current heating at the edge of the transverse slot. Therefore, the problem of negative sequence temperature rise increasingly becomes serious and the temperature rise in this part directly determines the negative sequence capacity of the generator. In this section, for different negative sequence conditions, the influence of transverse slots on rotor negative sequence loss and heating is studied, and its important role in rotor loss and heating is emphasized.Based on the negative sequence heating prediction model of the GA-IGA-RBFPNN algorithm, through the systematic research and analysis of the different negative sequence component proportions of the turbo-generator, this study attempts to summarize the extent to which the specific distribution of eddy current loss and temperature of rotor components are affected by different transverse slot number. For the generator structure with a big tooth with no slot wedge, the influence of rotor negative sequence loss and heating under different transverse slot spacing and different negative sequence component proportions is analyzed. Simultaneously, the variation law of negative sequence loss and heating of large turbo-generator rotor under different negative sequence component proportion operation conditions is calculated. Accordingly, an effective method to improve the negative sequence capacity is found.For more accurate analysis and prediction, different transverse slot spacing is adopted, and considering eddy current loss penetration depth, the rotor surface is divided into different areas. Thus, the negative sequence current eddy current loss and the highest negative sequence temperature rise can be predicted and compared effectively under the conditions of different negative sequence component proportions, different transverse slot spacings, and different positions. Specifically, in the calculation and prediction process, the big tooth area, slot wedge area, and small tooth area are eventually divided into different calculation areas according to the rotor structural position. The calculation area of the slot wedge is divided into three layers: upper, middle, and lower. The starting position is the big tooth part adjacent to the slot wedge, and the positions of all other components are numbered clockwise.Internationally, the requirement for the generator to bear the steady-state negative sequence current is that, after the rated operation, the negative sequence current generated by asymmetry does not exceed the specified value, and the generator can still operate stably and continuously under this condition. For the No. 1 generator with 10% actual steady-state negative sequence capacity, the negative sequence current with the proportion of 3%, 7%, and 11% is applied to the stator winding, respectively. In order to determine the influence law of transverse slots more clearly, there are no transverse slots (the polar surface is a smooth surface); 10 transverse slots and 20 transverse slots on each polar surface are discussed separately in this paper.


### 5.1. Influence of Big Tooth without Transverse Slot on Negative Sequence Loss

For the case of no transverse slot on the big tooth, the optimal topology of GA-IGA-RBFPNN will be used to predict the rotor loss and variation law when the negative sequence components are 3% In, 7% In, and 11% In, respectively. The prediction results are shown in Figures7–11.Figure 7
Loss density distribution of the top part of the small tooth wedge.Figure 8
Loss density distribution of the middle part of the small tooth wedge.Figure 9
Loss density distribution of the bottom part of the small tooth wedge.Figure 10
Loss density distribution of the big tooth.Figure 11
Loss density distribution of the small tooth.As observed in the prediction results in Figures7–11, the loss density of the slot wedge gradually decreases with the increase in the penetration depth, in which the density value of the upper slot wedge is the largest and the lower layer is the smallest. The reason is that the electromagnetic field intensity decreases with the depth increase, and when it reaches a certain depth, the magnetic field becomes considerably weak. Therefore, with the depth increase, the induced eddy current decreases, and the loss is reduced accordingly. In Figures 7–9, the loss density of slot wedges 1, 16, 17, and 32 is significantly higher than that of the other slot wedges. This is because these slot wedges are close to the big teeth; therefore, the induced electric density value is significantly higher.In order to study the influence of negative sequence component on rotor loss under different transverse slot spacings, similar to having a transverse slot, the rotor big tooth surface is equally divided into three parts in the case of no transverse slot on the pole surface. It can be seen from Figures10 and 11 that the eddy current loss density of the large tooth is greater than that of the small tooth. This is because a large number of eddy currents will be induced due to the large area of the big tooth, resulting in large losses. Positions 3 and 6 in Figure 8 refer to the polar surface where the center line of the big tooth is located; therefore, there are heavy eddy currents induced at these two positions, and accordingly, the loss increases.


### 5.2. Influence of 10 Transverse Slots on Negative Sequence Loss

For the case of 10 transverse slots on each big tooth of the rotor, the optimal topology of GA-IGA-RBFPNN will be used to predict the rotor loss and variation law when the negative sequence components are 3% In, 7% In, and 11% In respectively. The prediction results are shown in Figures12–16.Figure 12
Loss density distribution of the top part of the small tooth wedge.Figure 13
Loss density distribution of the middle part of the small tooth wedge.Figure 14
Loss density distribution of the bottom part of the small tooth wedge.Figure 15
Loss density distribution of the big tooth.Figure 16
Loss density distribution of the small tooth.It can be seen from the prediction results that when the rotor has 10 transverse slots, the loss density of the slot wedge decreases gradually with the increase in depth, the loss density of the slot wedge close to the big tooth is significantly greater than that of all other slot wedges, and the eddy current loss density of the big tooth is greater than that of the small tooth. All prediction results and variation trends are highly consistent with the experimental results that completely prove that the GA-IGA-RBFPNN optimal topology used in this paper is highly feasible and effective for the accurate prediction of the large generator operating parameters under negative sequence conditions.


### 5.3. Influence of 20 Transverse Slots on Negative Sequence Loss

There are 20 transverse slots on each big tooth of the generator, and the GA-IGA-RBFPNN topology can be adopted to predict the rotor loss and variation law, when the negative sequence components are 3%, 7%, and 11% respectively. Calculation results are shown in Figures17–21.Figure 17
Loss density distribution of the top part of the small tooth wedge.Figure 18
Loss density distribution of the middle part of the small tooth wedge.Figure 19
Loss density distribution of the bottom part of the small tooth wedge.Figure 20
Loss density distribution of the big tooth.Figure 21
Loss density distribution of the small tooth.Comparing the calculation results of 20 transverse slots with those of zero and 10 transverse slots, it can be found that the loss of each component is directly related to the number of transverse slots. The more the number of transverse slots, the larger the eddy current loss generated on the rotor surface. Simultaneously, the loss of the other parts, particularly the slot wedge and small tooth close to the big tooth, is also sharply increased. In general, the more the number of transverse slots, the greater the negative sequence eddy current loss.


### 5.4. Effect of Transverse Slot on Negative Sequence Maximum Temperature Rise

From the loss distribution prediction results above, the three transverse slot spacing, the proportion of negative sequence components, and the number of transverse slots have a significant impact on the rotor heating under negative sequence conditions. Table7 shows the maximum negative sequence temperature rise results of the different parts in the rotor domain under three negative sequence conditions and for three different transverse slot numbers.Table 7
Maximum temperature rise comparison caused by negative sequence current under three transverse slot conditions.
Structural positionTransverse slot numberNegative sequence component proportion3%7%11%Big tooth023.1925.2331.571024.3126.3633.482026.3928.8237.03Slot wedge013.4515.3621.591014.3116.4723.162016.1818.8326.96Small tooth012.3214.2420.451013.1915.5122.172015.0817.6224.74From Table7, it can be found that when the negative sequence component increases proportionally, the negative sequence temperature rise does not increase proportionally, even if the same transverse slot spacing is adopted. When the proportion of negative sequence component increases from 3% to 7%, the temperature rise is approximately 2°C. When the proportion of negative sequence components increases from 7% to 11%, the temperature rise is more than 6°C. The negative sequence component also increases by 4%, but the temperature rise rate of the latter is approximately three times that of the former. It can be seen that in the actual operation process, it is extremely necessary to suppress the generation of negative sequence components and reduce the amplitude of negative sequence components for rotor loss and heating.In addition, it can be clearly seen from Table7 that the different numbers of transverse slots have a significant impact on the rotor negative sequence loss and heating. If the negative sequence component is the same, the temperature rise in the case of zero transverse slots on the polar surface is the smallest, and the temperature rise for 20 transverse slots is the largest. Hence, under the same negative sequence condition, the rotor negative sequence loss and heating caused by 20 transverse slots on the pole surface are the most significant. Therefore, for the selection of transverse slot spacing, from the perspective of negative sequence loss and heating, the actual structure with a slightly smaller number of transverse slots on the polar surface should be preferred, and the negative sequence temperature rise is significantly reduced, in order to effectively improve the negative sequence capacity.Meanwhile, through the analysis of the data in Table7, it is concluded that the maximum temperature rise in the entire domain of the rotor occurs when the negative sequence component is 11% and each polar surface has 20 transverse slots; the maximum temperature rise on the surface of the big tooth is 37.03°C. Because the limit temperature of each rotor component under steady-state negative sequence working conditions is 130°C, there is still a large temperature rise margin to meet the requirements of negative sequence operation. Irrespective of the type of transverse slot spacing, the maximum temperature rise in all cases is less than the limit steady-state temperature of the material itself. Therefore, the generator has the steady-state ability to bear negative sequence current I2 = 0.1 for a long time. For the transverse slot spacing of large generators, various factors, such as stiffness balance and the impact on negative sequence loss and heating, should be comprehensively considered. On the premise of reducing the frequency doubling vibration, the rotor body structure with larger transverse slot spacing is a suitable choice for 300-MW large generators.


## 5.1. Influence of Big Tooth without Transverse Slot on Negative Sequence Loss

For the case of no transverse slot on the big tooth, the optimal topology of GA-IGA-RBFPNN will be used to predict the rotor loss and variation law when the negative sequence components are 3% In, 7% In, and 11% In, respectively. The prediction results are shown in Figures7–11.Figure 7
Loss density distribution of the top part of the small tooth wedge.Figure 8
Loss density distribution of the middle part of the small tooth wedge.Figure 9
Loss density distribution of the bottom part of the small tooth wedge.Figure 10
Loss density distribution of the big tooth.Figure 11
Loss density distribution of the small tooth.As observed in the prediction results in Figures7–11, the loss density of the slot wedge gradually decreases with the increase in the penetration depth, in which the density value of the upper slot wedge is the largest and the lower layer is the smallest. The reason is that the electromagnetic field intensity decreases with the depth increase, and when it reaches a certain depth, the magnetic field becomes considerably weak. Therefore, with the depth increase, the induced eddy current decreases, and the loss is reduced accordingly. In Figures 7–9, the loss density of slot wedges 1, 16, 17, and 32 is significantly higher than that of the other slot wedges. This is because these slot wedges are close to the big teeth; therefore, the induced electric density value is significantly higher.In order to study the influence of negative sequence component on rotor loss under different transverse slot spacings, similar to having a transverse slot, the rotor big tooth surface is equally divided into three parts in the case of no transverse slot on the pole surface. It can be seen from Figures10 and 11 that the eddy current loss density of the large tooth is greater than that of the small tooth. This is because a large number of eddy currents will be induced due to the large area of the big tooth, resulting in large losses. Positions 3 and 6 in Figure 8 refer to the polar surface where the center line of the big tooth is located; therefore, there are heavy eddy currents induced at these two positions, and accordingly, the loss increases.


## 5.2. Influence of 10 Transverse Slots on Negative Sequence Loss

For the case of 10 transverse slots on each big tooth of the rotor, the optimal topology of GA-IGA-RBFPNN will be used to predict the rotor loss and variation law when the negative sequence components are 3% In, 7% In, and 11% In respectively. The prediction results are shown in Figures12–16.Figure 12
Loss density distribution of the top part of the small tooth wedge.Figure 13
Loss density distribution of the middle part of the small tooth wedge.Figure 14
Loss density distribution of the bottom part of the small tooth wedge.Figure 15
Loss density distribution of the big tooth.Figure 16
Loss density distribution of the small tooth.It can be seen from the prediction results that when the rotor has 10 transverse slots, the loss density of the slot wedge decreases gradually with the increase in depth, the loss density of the slot wedge close to the big tooth is significantly greater than that of all other slot wedges, and the eddy current loss density of the big tooth is greater than that of the small tooth. All prediction results and variation trends are highly consistent with the experimental results that completely prove that the GA-IGA-RBFPNN optimal topology used in this paper is highly feasible and effective for the accurate prediction of the large generator operating parameters under negative sequence conditions.


## 5.3. Influence of 20 Transverse Slots on Negative Sequence Loss

There are 20 transverse slots on each big tooth of the generator, and the GA-IGA-RBFPNN topology can be adopted to predict the rotor loss and variation law, when the negative sequence components are 3%, 7%, and 11% respectively. Calculation results are shown in Figures17–21.Figure 17
Loss density distribution of the top part of the small tooth wedge.Figure 18
Loss density distribution of the middle part of the small tooth wedge.Figure 19
Loss density distribution of the bottom part of the small tooth wedge.Figure 20
Loss density distribution of the big tooth.Figure 21
Loss density distribution of the small tooth.Comparing the calculation results of 20 transverse slots with those of zero and 10 transverse slots, it can be found that the loss of each component is directly related to the number of transverse slots. The more the number of transverse slots, the larger the eddy current loss generated on the rotor surface. Simultaneously, the loss of the other parts, particularly the slot wedge and small tooth close to the big tooth, is also sharply increased. In general, the more the number of transverse slots, the greater the negative sequence eddy current loss.


## 5.4. Effect of Transverse Slot on Negative Sequence Maximum Temperature Rise

From the loss distribution prediction results above, the three transverse slot spacing, the proportion of negative sequence components, and the number of transverse slots have a significant impact on the rotor heating under negative sequence conditions. Table7 shows the maximum negative sequence temperature rise results of the different parts in the rotor domain under three negative sequence conditions and for three different transverse slot numbers.Table 7
Maximum temperature rise comparison caused by negative sequence current under three transverse slot conditions.
Structural positionTransverse slot numberNegative sequence component proportion3%7%11%Big tooth023.1925.2331.571024.3126.3633.482026.3928.8237.03Slot wedge013.4515.3621.591014.3116.4723.162016.1818.8326.96Small tooth012.3214.2420.451013.1915.5122.172015.0817.6224.74From Table7, it can be found that when the negative sequence component increases proportionally, the negative sequence temperature rise does not increase proportionally, even if the same transverse slot spacing is adopted. When the proportion of negative sequence component increases from 3% to 7%, the temperature rise is approximately 2°C. When the proportion of negative sequence components increases from 7% to 11%, the temperature rise is more than 6°C. The negative sequence component also increases by 4%, but the temperature rise rate of the latter is approximately three times that of the former. It can be seen that in the actual operation process, it is extremely necessary to suppress the generation of negative sequence components and reduce the amplitude of negative sequence components for rotor loss and heating.In addition, it can be clearly seen from Table7 that the different numbers of transverse slots have a significant impact on the rotor negative sequence loss and heating. If the negative sequence component is the same, the temperature rise in the case of zero transverse slots on the polar surface is the smallest, and the temperature rise for 20 transverse slots is the largest. Hence, under the same negative sequence condition, the rotor negative sequence loss and heating caused by 20 transverse slots on the pole surface are the most significant. Therefore, for the selection of transverse slot spacing, from the perspective of negative sequence loss and heating, the actual structure with a slightly smaller number of transverse slots on the polar surface should be preferred, and the negative sequence temperature rise is significantly reduced, in order to effectively improve the negative sequence capacity.Meanwhile, through the analysis of the data in Table7, it is concluded that the maximum temperature rise in the entire domain of the rotor occurs when the negative sequence component is 11% and each polar surface has 20 transverse slots; the maximum temperature rise on the surface of the big tooth is 37.03°C. Because the limit temperature of each rotor component under steady-state negative sequence working conditions is 130°C, there is still a large temperature rise margin to meet the requirements of negative sequence operation. Irrespective of the type of transverse slot spacing, the maximum temperature rise in all cases is less than the limit steady-state temperature of the material itself. Therefore, the generator has the steady-state ability to bear negative sequence current I2 = 0.1 for a long time. For the transverse slot spacing of large generators, various factors, such as stiffness balance and the impact on negative sequence loss and heating, should be comprehensively considered. On the premise of reducing the frequency doubling vibration, the rotor body structure with larger transverse slot spacing is a suitable choice for 300-MW large generators.


## 6. Conclusions

The negative sequence loss and heating prediction method based on the GA-IGA-RBFPNN algorithm is proposed in this paper, to address the RBFPNN shortcomings of the difficulty of designing and training the structure and the tendency to easily fall into the minima when it is used to predict the performance parameters of large turbo-generator under negative sequence conditions. Moreover, the health information obscured in the prediction results is difficult to be clearly and rapidly identified. Using the phase space reconstruction theory to construct the training sample set, combined with a genetic algorithm, the RBFPNN structure, initial weight, and threshold are optimized, and subsequently, the prediction performance matrix is generated. The verification results show that the network structure of the GA-IGA-RBFPNN model is reasonable and reliable that effectively improves the accuracy, stability, and generalizability of the model. With the assistance of the prediction results, it can clearly, intuitively, and systematically reflect the nonlinear, coupling relationship and spatiotemporal distribution of multiple prediction parameters of a large generator, rapidly mine the negative sequence condition information obscured in the prediction performance matrix, and improve the identification efficiency of the abnormal parameters and operation model. The prediction method realizes the purpose of accurately, efficiently, and intuitively predicting the rotor negative sequence loss and heating state at the system level and provides a novel approach for the prediction, monitoring, and decision-making of the complex system operation state.Based on the GA-IGA-RBFPNN algorithm, the influence degree of the different negative sequence component proportions on the rotor negative sequence loss and heating is intensively analyzed and calculated, and the general principle and characteristics of negative sequence component affecting rotor heating are determined. From the perspective of negative sequence loss and heating, the influence of the transverse slot on the steady-state negative sequence capacity of the turbo-generator is analyzed. The calculation and analysis of eddy current loss on rotor surface with different proportions of negative sequence components and different transverse slot spacing are conducted. The specific distribution of eddy current loss density is calculated in detail when there are no transverse slots, 10 transverse slots, and 20 transverse slots on each polar surface. The prediction results show that irrespective of the type of transverse slot spacing adopted, the loss density of the slot wedge decreases with the increase in the penetration depth. Moreover, the transverse slot on the polar surface can significantly increase the maximum loss of the rotor and affect the loss distribution of each component of the rotor.According to the analysis of the negative sequence loss, the influence of the transverse slot on rotor heating is studied. The analysis of the prediction results indicates that the maximum negative sequence temperature rise of the rotor is different for different transverse slot spacings. When the big tooth has no transverse slot, the temperature value of the entire region is the lowest. Therefore, it is concluded that different transverse slot spacings have a significant influence on rotor heating. The area of the large tooth is larger than that of the small tooth which is more seriously affected by the synthetic magnetic field of the stator. Under different negative sequence operating conditions, the surface of the large teeth is still the region with the highest temperature. Although the maximum temperature rise in the three cases occurs at the big tooth position, the temperature values are different and the specific temperature distribution is also different. It can be seen that the negative sequence component and the specific structure of the rotor have a significant impact on the unbalanced energy distribution on the rotor surface. Therefore, adopting a more intelligent and efficient prediction and calculation method for the negative sequence problem of turbo-generator has important research value and development prospects.



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*Source: 1865380-2022-04-13.xml*