Displacement is an important physical quantity of hydraulic structures deformation monitoring, and its prediction accuracy is the premise of ensuring the safe operation. Most existing metaheuristic methods have three problems: (1) falling into local minimum easily, (2) slowing convergence, and (3) the initial value’s sensitivity. Resolving these three problems and improving the prediction accuracy necessitate the application of genetic algorithm-based backpropagation (GA-BP) neural network and multiple population genetic algorithm (MPGA). A hybrid multiple population genetic algorithm backpropagation (MPGA-BP) neural network algorithm is put forward to optimize deformation prediction from periodic monitoring surveys of hydraulic structures. This hybrid model is employed for analyzing the displacement of a gravity dam in China. The results show the proposed model is superior to an ordinary BP neural network and statistical regression model in the aspect of global search, convergence speed, and prediction accuracy.
Dam cracks and displacement monitoring that reflect the structural aging and disease are widely used in various forms of dams (e.g., Xianghongdian Dam [
Statistical regression models [
Researchers further studied and proposed deterministic models [
In the last 10 years, artificial intelligence algorithms, such as the grey system, the fuzzy mathematic theory, the time series, the wavelet theory, and bionics algorithm, were also gaining popularity. The grey system has been proposed and applied to the dam stress grey forecasting model [
Artificial neural network, which possesses strong ability of nonlinear function approximation and self-organizing and self-adaptive function, has been applied to the data of dams safety monitoring analysis and forecasting to remove data irregularity. Backpropagation neural networks have been proposed to monitor and predict the dam deformation while based on the actual values of a concrete gravity dam’s horizontal displacement [
Many studies [
As a new type of search methods, genetic algorithm (GA) has many advantages, such as simple general, high global searching capability, strong robustness, and wide application range. Many studies have shown that these advantages can be used to optimize BP neural network’s structure, the weight, threshold value, and parameters and improve the prediction accuracy. Fu et al. [
The objective of this study is to analyze the feasibility of GA-BP in dam deformation, to explore the usefulness of proposed multiple population genetic algorithm backpropagation (hybrid MPGA-BP model), and to compare it with statistical regression model and conventional BP network model with the same parameter for dam deformation analysis. The rest of the paper is arranged as follows. Section
Dam structure is influenced by hydraulic, environmental, and geomechanical factors. Therefore, the situation requires us to study the variables that affect the dam behavior before applying the improved artificial neural network approaches. Based on the results of the study of [
As an important factor of deformation, hydrostatic pressure can be expressed as a polynomial function for the reservoir water level
The displacement contributed by temperature variation can be modeled in two ways. If temperature measurements within the dam body and foundation are adequate and available, then
If temperature measurements are inadequate and unavailable, the form of measured temperature cannot be used to describe temperature field’s variation. When dam temperature field closes to the quasi-stable temperature field, we can describe approximatively the changes of temperature field within dam body through the changes of temperature outside. But there is a lag effect on the dam body internal temperature variation which was influenced by air temperature changes. So, the influence of air temperature variation on monitoring effect-quantity also causes a lag effect. As a consequence, average temperature of several days before the day of monitoring effect-quantity was served as temperature factors. Therefore the form of temperature temporal loadings can be described as follows:
W. McCulloch and W. Pitts opened an era of neuroscience research in 1943 since they created the mathematical model of neuron formation and imitation of biological neuron activity function. As a mathematical analogue of the biological system, it can be used to highlight processes, to deal with fuzzy information, or to display chaotic properties. BP neural network is by far the most widely used neural network, whose training method is based on the error backpropagation (BP) to the multilayer neural network. The topology of the BP neural network structure is as in Figure
BP neural network topology structure.
Assuming the input value of neural network is
As the output layer excitation function,
The BP neural network training process is mainly divided into the following two steps.
Obtain the input values of the
Because the threshold value of the
If the difference value between output value of forward propagation and that we expect is bigger than set value, the corresponding weights need to be adjusted backwards constantly. The error function of the neural network can be expressed as follows:
Through the update incremental formula above, neurons connection weights and thresholds can be iterated and updated for the next network learning and training. Subsequently, circuit training the above steps many times, the neural network connection weights and thresholds are continually updated. Meanwhile, output error will also tend to be minimum. When the minimum reaches set value, the loop is completed; if not reached, the cycle training continues for many times. The BP neural network calculation flow diagram was shown in Figure
Calculation flow chart of BP neural network.
Although the genetic algorithm shows excellent characteristics of its global efficiency, it also has its disadvantages in practical application. One of the important is premature convergence which is considered the common phenomenon in the GA. It has much effect on the solution of the optimal value, whose main characteristic is that all individuals in the population present a trend and terminate evolution. Thus, the satisfied solution cannot be obtained. The multiple population structure has been introduced in order to solve the problem of premature convergence of the GA. This paper uses multiple population genetic algorithm replacing conventional standard genetic algorithm. The structure diagram was shown in Figure
MPGA structure diagram.
MPGA is mainly done through the following optimization based on SGA: MPGA introduces multiple populations to optimize searching simultaneously, while SGA has only a single population. Different population can achieve different searching purposes with control different parameters. Through the migration operator contacting various populations, the multiple population coevolution can be realized. It will eventually obtain the optimal solution. Artificial selection operator will save the best individual of each population in evolution, which is regarded as criterion of the algorithm convergence.
In MPGA, each population selects different control parameters. Meanwhile, the value of crossover probability and mutation probability decided the algorithm’s global search and local search ability. Although many articles and scholars advised choosing a larger crossover probability
As aforesaid, the BP neural network and the standard genetic algorithm (SGA) have their own shortcomings. This study suggests using multiple population genetic algorithm (MPGA) to solve the problem of premature convergence in SGA and to optimize the BP neural network weights and threshold for speeding up the learning speed of network. MPGA-BP model includes the advantages of the BP neural work, such as nonlinear mapping ability, self-learning and adaptive ability, and strong fault-tolerant ability. It uses genetic algorithm to overcome the BP neural network easy falling into local minimum value, slow convergence speed, initial threshold and weight values difficult to determine, and other shortcomings. Meanwhile, it uses concept of multiple population to solve the problem of premature convergence of genetic algorithm itself retaining the global search ability of genetic algorithm. The modeling steps of MPGA-BP generally include the following.
Determine the topological structure of BP network (determine the number of weights and thresholds to be optimized).
Genetic operation is based on each population’s initialized code (put the norm of prediction error matrix as objective function; design the fitness function).
Set crossover and mutation parameters for each population (introduce multiple population concept to genetic algorithm).
Collect the best individual of various populations to form an elite population through immigration artificial selection operation (realize best individual with the optimal weights and thresholds of the network).
Assign the best individual weights and threshold to the BP neural network.
Get the final result through the network training, testing, and verifying.
Set the size of the population as
Each individual population needs encoding because solutions that need to be optimized in initial populations are BP network’s weights and thresholds. This study adopts binary encoding method, which encodes every individual into a binary string. Here, the size of the individual consists of four parts; there are connection weights of input layer and hidden layer, thresholds of implicit layer, connection weights of hidden layer and connection layer, and thresholds of output. Each individual’s weights and thresholds use
In the genetic algorithm, the solution of issue is shown as the values of population individual and survival of the fittest through fitness function. In this paper, we are going to get the optimal weight and threshold of BP network, so the individual values are expressed as weights and thresholds of BP network and the error norm of the matrix can be chosen as target function. There will be a positive and negative value of the target function, so we need to design a fitness function. We need to ensure that fitness value is nonnegative based on the relationship between fitness function and target function. At the same time, the optimization direction of target function is the increased direction of fitness.
For the optimization problem of target function’s minimum, we need to add a minus sign to transform it into the optimization problem of target function’s maximum in theory as follows:
For the problem of target function’s maximum, we can directly set up fitness function as equal to the target function, when the target function is always positive as follows:
This paper mainly involves the target function’s minimum, so we choose the fitness function as formula (
MPGA includes SGA genetic operation: selection, crossover, and mutation. Selection operation is mainly to choose excellent individual with a certain probability from old population to form new population in order to reach the goal of breeding the next generation individuals. Crossover operation is to select two individuals at random from the population and to pass parents’ excellent genetics to offspring through exchanging and combination of two chromosomes for generating new excellent individuals. The main purpose of the mutation operation is to maintain the diversity of population.
In MPGA model, different population has different crossover probability
Multiple population genetic neural network calculation flow diagram.
A series of observations including the water level, temperature, and aging of a gravity dam in China were used building the MPGA-BP model. The crest length of the dam is 1080 m, the crest elevation is approximately 91.7 m, the upstream slope is 95%, the downstream slope is 78%, the profile is approximately the triangle, and the total reservoir capacity is 11.5 billion cubic meters. Observation wire system has been set within crest cable corridor. In this study, we examined the recorded data of the monitoring point No. 16, which is located at the centre of impervious reinforced concrete face of the overflow dam section as shown in Figure
Variables used to forecast dam deformation.
Description | Variable | Calculation |
---|---|---|
Water level | Var. 1 | |
Var. 2 | | |
Var. 3 | | |
Temperature | Var. 4 | |
Var. 5 | | |
Var. 6 | | |
Var. 7 | | |
Var. 8 | | |
Aging | Var. 9 | |
Var. 10 | |
Arrangement of monitoring points.
There exists a Kolmogorov theorem, namely, the continuous function representation theorem, ensuring that any continuous function or mapping can be used in a three-layer neural network to realize. The network includes input layer, output layer, and hidden layer. Then, the number of hidden layer nodes has a great impact on the generalization and the training speed. If the number of nodes were too small, network nonlinear mapping ability would be low, prediction accuracy would be not high, and the network would be thin. If the number of nodes were too large, the learning time would be too long and the training speed would be slow. So the appropriate number of nodes is crucial to a strong network. At present, there is no clear theoretical guidance to selection method of the number of hidden layer nodes. The empirical formula commonly used in the application is as follows:
In addition, the number of nodes can be found out through the test algorithm. We eventually get the appropriate value through either training from smaller number of nodes and increasing gradually the number of nodes based on the change of network output error or training from larger number of nodes and reducing gradually the number of nodes. This method consumes more time and energy for a large number of calculations. And not only that, it is able to quickly find suitable number of hidden layer nodes on account of strong randomness.
This study uses the method of combining the empirical formula and test algorithm to determine the number of nodes in the hidden layer. In the case,
Relationships between the number of hidden layer nodes and the MSE value.
| 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| ||||||||||
MSE | 0.00286 | 0.00247 | 0.0023 | 0.00254 | 0.00242 | 0.00251 | 0.00241 | 0.00243 | 0.00228 | 0.00231 |
Through observing the relationship between the number of hidden layer nodes and the MSE value we concluded that if the number of hidden layer nodes were 13, the MSE value of network would be smaller. So we determine that the number of hidden layer nodes is 13 and the network structure is 10-13-1.
On account of the difference of units and dimension within sample data (displacement, water pressure, temperature, and aging), we need to carry out normalization processing with sample values of input factors and output factors. Retaining original nature of sample data makes the values in
Common encoding includes binary code, grey code coding, and real coding. The binary coding is one of the most common kinds of coding way. It not only facilitates implementation of the genetic operations such as crossover and mutation, but also causes encoding and decoding operation to be easy. This study uses a binary code, which is set up of binary notation
Fitness function is a method to judge whether individual population is good or bad. We decode the code representing weights and threshold on the code string and substitute into the BP neural network. By using training samples to train the network and using testing samples to test the network the test error
In MATLAB, Sheffield genetic algorithm toolbox has a fitness distribution function named ranking based on the sort to achieve the goal as follows:
The optimization process of MPGA certainly contains the inherent optimization operation of standard genetic algorithm (SGA), which has selection, crossover, and mutation operation. Selection operation adopts roulette selection operator (RWS) in the Sheffield genetic algorithm toolbox. Crossover operation adopts single-point crossover operator (xovsp). Crossover probability
Parameters used in implementing MPGA-BP variable selection.
Hidden layer transfer function | tansig |
Output layer transfer function | purelin |
Training function | trainlm |
Times of training | 2000 |
Training goal | 0.001 |
Learning rate | 0.1 |
Number of populations | 10 |
Population size | 100 |
Generation gap | 0.9 |
Binary digits | 10 |
In this paper, we established MPGA-BP model compiled by MATLAB software to train the sample data and forecast dam deformation. From the error changing curve of the inducing generation (see Figure
Error evolution process curve of MPGA-BP network.
Training process curve of MPGA-BP network.
On the other hand, stepwise regression and ordinary BP have the same topological structure 10-13-1 and were also carried out in this study for comparison with MPGA-BP model. The same data has been applied to train and forecast. Details of the investigation are listed in Table
Comparison of measured value under the conditions and predicted values for three models.
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Through the contrast in Table
The related parameters for three models.
Model | Prediction | Prediction error | Prediction | Training | Validation |
---|---|---|---|---|---|
MPGA-BP network | 0.4581 | 0.4778 | 2.35% | 0.4379 | 0.4769 |
BP network | 0.6049 | 0.7545 | 8.04% | 0.4855 | 0.5071 |
Statistical | 1.6821 | 1.8539 | 18.38% | 1.5478 | 1.7456 |
By contrast, it is not difficult to find that error matrix norms of MPGA-BP model are smaller than other two models in the process of the training, validation, and prediction. This reflects that MPGA-BP model is superior to other two models from another aspect. From Figure
Predicted values of three models compared with the measured values.
Prediction error curve of three models.
Figures
Comparison of predicted and measured values of horizontal displacement (unit: mm).
Error distribution curve (unit: mm).
To sum up, we suggest that MPGA-BP model is far superior to statistical regression model and BP network model. It can be proved from three angles of prediction accuracy, convergence speed, and error matrix norm. It means that MPGA-BP model that has been built is scientific and effective. It is feasible that predicting gravity dam deformation uses this model.
This study extends the methods of genetic algorithm with backpropagation neural network (GA-BP) and multiple population genetic algorithm (MPGA) to gravity dam deformation analysis. A hybrid multiple population genetic algorithm backpropagation (MPGA-BP) neural network algorithm has been proposed to cope with premature convergence problem and local extremum problem. A case study using the observations of a gravity dam in China has been presented and discussed to examine the performance of the proposed model. Compared with the statistical regression model and the traditional neural network algorithm, this model prediction accuracy is improved greatly. The applicability of MPGA-BP for the prediction of gravity dam deformation has been highlighted. The results show that the proposed model can be used as an excellent tool instead of that commonly used in gravity dam deformation analysis and prediction.
The authors declare that they have no competing interests.
This study was partly supported by the National Basic Research Program of China (973 Program) (2012CB417006) and National Science Support Plan Project of China (2009BAC56B03). The authors would like to thank Professor Shen for his generous sharing of the deformation-observation data of a gravity dam. Special thanks are due to Professor Wan (University of Houston Victoria, USA) for his constructive comments, which improved the manuscript substantially.