Deformation Prediction of Dam Based on Optimized Grey Verhulst Model

: Dam deformation monitoring data are generally characterized by non-smooth and no-saturated S-type ﬂuctuation. The grey Verhulst model can get better results only when the data series is non-monotonic swing development and the saturated S-shaped sequence. Due to the limitations of the grey Verhulst model, the prediction accuracy will be limited to a certain extent. Aiming at the shortages in the prediction based on the traditional Verhulst model, the optimized grey Verhulst model is proposed to improve the prediction accuracy of the dam deformation monitoring. Compared with those of the traditional GM (1,1) model, the DGM (2,1) model, and the traditional Verhulst (1,1) model, the experimental results show that the new proposed optimized Verhulst model has higher prediction accuracy than the traditional gray model. This study offers an effective model for dealing with the non-saturated ﬂuctuation sequence to predict dam deformation under uncertain conditions.


Introduction
The hydropower station is composed of the water conservancy system, mechanical system and electric energy generation device.It is a hydro-junction project to realize the conversion from water energy to electric energy.The sustainability of electric energy production requires the uninterrupted utilization of water energy in the hydropower station.Through the construction of dams, the distribution of water resources in time and space can be adjusted and changed artificially to realize the sustainable use of water resources [1].As the main structure of water conservancy project, some dams have some potential safety problems, due to the reasons such as construction quality, design defects, hydrology, geology and aging during the operation of the dam, which not only weakens the engineering benefits of the dam, but also directly threatens the safety of people's lives and property in the downstream area [2].Dam monitoring data can reflect the operation behavior and structure state of the dam in real-time, as well as the dynamic information, which is considered to be the most important factor for diagnosing dam health and providing early warning [3,4].Therefore, the analysis and prediction of the original observation data of dam deformation become an important research content in the field of dam safety monitoring.
At present, some theoretical methods are used to analyze and predict deformation monitoring data, such as the Gray model [5], neural network [6], Machine Learning Model [7], Time Series [8], support vector machines [9] and so on.However, in the actual dam construction, the evolution trend of dam deformation often follows the development law of the non-monotonic system.The development trend of deformation is usually from high-speed settlement to a slow growth period, and finally gradually returns to a stable state.This special evolution process can show a saturated S-shaped curve with data [4,7].Mathematical models fitting saturation S-curve, such as the Malthus model [10], logistic model [11], Richards growth model [12], Weibull model [13], order differential Equation (ODE) [14], and traffic burst-sensitive model (TBSM) [15] have certain requirements for sample size.When the modeling sample size is small, the higher the gray level of the information, the more difficult it is for the general statistical model to simulate and predict.There are some shortcomings in nonlinear models, which usually requires training models with large sample size, and some of them also have problems of sample overfitting or improper parameter setting [16].The prediction of dam deformation considering saturated S-shaped characteristics is a typical multivariable system problem, and the grey multivariable model can work well even with small samples, poor information, and no statistical assumptions, which has attracted widespread attention.
As one of the important prediction models in grey system theory, unlike the grey univariable GM (1,1) model, the grey Verhulst model considers the relationship between variables describing system behavior and other relevant variables, which has good characteristics for describing nonmonotonic oscillatory development sequences and saturated S-shaped sequences [17,18].It is widely used to simulate and predict oscillatory development sequences and saturated S-shaped sequences.The dam deformation monitoring data usually has the annual periodicity characteristic, showing as S-shaped, but the S-shaped data is not saturated, so the traditional Verhulst model may be limited to the S-shaped data with general fluctuations, and the prediction accuracy of the grey Verhulst model is relatively low.The GM (1,1) model applies to the sequence with strong exponential law, and can only describe the monotonic change process.For non-monotonic swing development sequence or saturated S-shaped sequence, the DGM (2,1) and Verhulst models can be considered.The gray Verhulst model is similar to the gray model GM (1,1).It also calculates the number sequence generated by accumulation and then calculates the median number sequence.However, the Verhulst model is to establish nonlinear ordinary differential equations.The DGM (2,1) gray model does not need to accumulate the generated sequence and can be operated directly using the original sequence [19].Therefore, an optimized Grey-Verhulst model is established by the discretization method based on the inverse transformation of the original data accumulation sequence in this paper.
In this study, a reciprocal sequence (RS) of accumulated generating operation (AGO) is adopted to structure the identification parameter of the traditional grey Verhulst model, the newly proposed Verhulst model can realize the unbiased simulation for no-saturated S-type data sequences.Meanwhile, the new proposed model solves the problem of the initial value optimization and parameter dislocation substitution.By comparing with the traditional GM (1,1) model, the DGM (2,1) model and the traditional Verhulst (1,1) model, the result indicated that the new proposed Verhulst model could offer superior performance than several different kinds of grey forecasting models.This study provides an effective model for dealing with unsaturated fluctuation series.

Methodology 2.1. Traditional GM (1,1) Model
The traditional GM (1,1) model expresses the time series in the form of a differential equation, where 1-AGO data from the original series is used as intermediate information.
Based on the least squares (LS) estimation method, the parameter estimators of GM (1,1) model can be obtained as follows: we can get the time responded function of GM (1,1) model is The restored values of raw data are given below [22], x(0 where, x(1) (k) is the simulative value of x (1) , and x(0) is the simulative value of x (0) .

Traditional Grey Verhulst Model
The grey Verhulst model can effectively describe some growth processes, such as Scurve which is a non-linear differential equation [16,23].Grey Verhulst model is as follows: dx (1)  dt The parameters a and b are estimated by the least square method [19,24]: where The solution of the whitening equation is as follows: x The time response equation of the grey Verhulst model is as follows: x (1) The restored values of raw data are given below, where, x( 1)

Evaluation Criteria
RMSE (root mean square error), MAE (mean absolute error), APE (absolute percentage error), and MAPE (mean complete percentage error) are commonly used to evaluate the simulation and prediction accuracy of models to investigate the effectiveness of the model [19,28,29].
The RMSE is as the formula where, e(k) is the simulative or predicted values sequence of the original data sequence.
The MAE is as the formula The APE is as the formula The MAPE is as the formula

Data Sources
The monitoring of dam external deformation plays an important role in the monitoring of hydraulic structures.For this study, taking the settlement monitoring data of ZLD-01 point in Changzhou Hydropower Project as an example, the original settlement data sequences are selected to establish the traditional GM (1,1) model, the DGM (2,1) model [30], the traditional Verhulst (1,1) model, and the optimized grey Verhulst (1,1) model to predict the settlement of dam.Each month is one monitoring period, and the cumulative settlement data of 50 periods (from Aug-2007 to Sep-2011) are selected (see Table 1).To highlight the forecast performance of the optimized grey Verhulst (1,1) model, the first 40 data as the training sample are first used to establish the forecast model, and the last testing sample 10 data are used to test and compare with the effectiveness of the forecast models.Figure 1 shows the distribution of training and testing sample data.

Fitting Accuracy Analysis of Training Samples
The original data series (first 40 training sample data) and the fitted values of the optimized grey Verhulst (1,1) model, the DGM (2,1) model, the traditional GM (1,1) model, and the traditional Verhulst (1,1) model from Aug-2007 to Sep-2011 are illustrated in Figure 2. From Figure 2, it is obviously seen that the fitted curve of the optimized grey Verhulst (1,1) model generally agrees better with the originally observed curve than those of the three models, and the extreme effect of the optimized grey Verhulst (1,1) model has been somewhat removed and the fitted curves are on the right track.Besides, the traditional Verhulst (1,1) model is better than the traditional GM (1,1) model or DGM (2,1) model that often requires a large number of training samples to get a higher fitted precision.The accuracy comparison results between fitted and actual values of the four models are listed in Table 2.According to Table 2, in the fitting stage, the MAPE, MAE, and RMSE of the traditional GM (1,1) model in the training set are the largest (64.663%, 5.375 mm, 7.029 mm) among the four models.The RMSE of the Optimized Verhulst (1,1) model is the smallest (1.593 mm) and its MAE and MAPE are also the lowest (1.215 mm, 19.328%) in the four models, whose evaluation criteria are lower than all other models, and the fitted effect is the best.After the Optimized Verhulst (1,1) model is improved, the MAPE, MAE and RMSE decreased 8.938%, 2.671 mm and 3.136 mm compared with the traditional Verhulst (1,1) model, which fully shows that the fitted accuracy of the Optimized Verhulst (1,1) model is much higher than that of the three models.

Comparison of Prediction Accuracies
The predictive value (PV) and absolute percentage errors (APE) of the optimized grey Verhulst (1,1) model, the DGM (2,1) model, the traditional GM (1,1) model, and the traditional Verhulst (1,1) model Dec-2010 to Sep-2011 (next 10 testing sample data) in the testing set are shown in Table 3.As shown in Table 3, in the test/prediction stage, the APEs of the optimized grey Verhulst (1,1) model are the smallest, the minimum value of APE is only 0.089%, and the APEs of the traditional Verhulst (1,1) model is also small.The APEs of the DGM (2,1) model is higher than that of the two models, which is better than that of the traditional GM (1,1) model.There is a large deviation between the predicted values of the DGM (2,1) model and the traditional GM (1,1) model and the original observed values, with the maximum APE reaching 47.080 mm and 167.553 mm, respectively, and only the optimized grey Verhulst (1,1) model is the closest to the trend and distance of the actual curve.The results of all evaluation indexes of various models in the prediction stage are shown in Table 4 and Figure 5. From Table 4, it is easy to see that the predictive performance of the optimized Verhulst (1,1) model outperforms other models in the process of forecasting.When through the test values, we find that the optimized Verhulst (1,1) model gives more satisfactory performances in RMSE, MAE, and MAPE than other models.The MAPE, MAE and RMSE of the traditional GM (1,1) model are the highest, reaching 95.638%, 59.522 mm and 64.018 mm, respectively, which cannot meet the needs of dam settlement prediction.However, the MAPE of the optimized Verhulst (1,1) model is the lowest, only 1.479%, which can meet the demand for dam settlement monitoring.The results validate the effectiveness of the optimized Verhulst (1,1) model.In Figure 5, The MAE and RMSE of the optimized Verhulst (1,1) model are also the smallest, only 0.886 mm and 1.180 mm, far lower than other models.From the above analysis, it is concluded that the optimized Verhulst (1,1) model has the best accuracy from different aspects of data fitting and prediction, and these results prove that the optimized Verhulst (1,1) model improves the forecast accuracy of the Verhulst model after optimizing the background value, and the forecast effect is the best.

Discussion
The point ZLD-02 data from Aug-2007 to Nov-2010 is used for analysis to verify the effectiveness of the optimized Verhulst (1,1) model.The fitting curves of the four models are shown in Figure 6.According to the results, it can be seen from the given Figure 6 that the simulation values of the optimized Verhulst model are more stable and better.The optimized Verhulst forecasting model can achieve complementary advantages and can better improve the efficiency of the traditional Verhulst modeling and the prediction accuracy of the model.In these models, the trend of the simulated curves of the GM (2,1) model and the traditional Verhulst model are basically the same.the prediction accuracy of the traditional Verhulst model is higher than that of the GM (2,1) model and the traditional GM (1,1) model, but the prediction accuracy is lower than the optimized Verhulst model.It can be seen from Figure 7 that the average relative error of the settlement simulation value of the traditional GM (1,1) model, the GM (2,1) model, the traditional Verhulst model and the optimized Verhulst model are 62.964%, 15.022%, 11.741%, and 2.049%, respectively.The simulation values errors of the optimized Verhulst model are significantly reduced and the fluctuation is very small compared with other models.Through comparative analysis, the proposed optimized Verhulst model can well capture the overall development and individual trends of the original data.This is a reliable and stable prediction model for the future evolution of dam settlement.With the rise of the water level, the dam body is generally affected by hydrostatic pressure, water gravity, soil consolidation, temperature effect, continuous crest subsidence and lateral extrusion (Figure 8) [31].The hydrostatic pressure will create a vertical force on the upstream slope, causing the dam surface to sink [31,32].Secondly, the effect of gravity on the water mass changes the shape of the bedrock and foundation under the dam body (Figure 8).With further impoundment from August 2007(Figure 9), uncontrolled seepage flowed through the dam, and the highest part of the dam could not withstand the increased mass.The upstream section and the top section of the slope continue to accelerate the subsidence, causing the landslide to move in the downstream direction, which finally resulted in the uplift movement of the downstream slope surface.After the impoundment, the overall displacement of the dam body, especially the displacement of the upstream part of the dam body is in the downstream direction.The distribution of the whole deformation of the dam body and its change with time accord with the general rule.The rearrangement of soil particles will lead to slight deformation of the dam in the consolidation process, and the soil accumulation in the dam will make the creep effect of the dam more significant [33].The deformation of the dam body is also related to the seasonal temperature variation, the thermal expansion effect of soil particles and the uneven thermal response will also lead to the inconsistent deformation of the dam body [34,35].In addition, the combined effects of other major deformation sources (i.e., water pressure and consolidation) may also superimpose thermal effects, resulting in uneven deformation of the dam body.Thus, it is necessary to conduct joint monitoring of dam water level, temperature and water pressure to improve the monitoring accuracy of deformation.During the regular monitoring of the dam, the deformation of the dam is analyzed and predicted based on previous monitoring data to prevent dangerous situations in advance.

Conclusions
The optimized gray Verhulst (1,1) model introduces the background value of the extrapolation method and the RS algorithm to optimize the variable parameters in the background value.The method makes the proposed model able to estimate the curve trend accurately and improve the precision.The optimized Verhulst (1,1) model includes the advantages of the traditional grey Verhulst (1,1) model, which could fully excavate the grey information of data and could better improve the stability and prediction accuracy of the model.Compared with the traditional GM (1,1) model, the DGM (2,1) model and the traditional Verhulst (1,1) model, the MAPE, MAE, and RMSE of the optimized Verhulst (1,1) model are very smaller, and its prediction accuracy is higher.The optimization model proposed in this paper can enhance better forecasting results and minimize predicted errors.In addition, the proposed optimization Verhulst (1,1) model is particularly suitable for predicting the periodic trend due to the introduction of the traditional Verhulst model.
Given the inherent variation trend and evolution law of dam excavation deformation in the time series data, some uncertain factors such as water pressure and temperature in the dam deformation process are not taken into comprehensive consideration in this model, which may affect the prediction accuracy to a certain extent.In the future, the hybrid prediction model which considers the influence of uncertain factors in the construction process will be an important research direction of dam settlement and displacement combination prediction.

Figure 1 .
Figure 1.Distribution of the training samples and testing samples from Aug-2007 to Sep-2011.
t l e m e n t v a l u e s / m m D a t e / m m -y y y y O r i g i n a l v a l u e s O p i m i z e d V e r h u l s t ( 1 , 1 ) t l e m e n t v a l u e s / m m D a t e / m m -y y y y O r i g i n a l v a l u e s D G M ( 2 , 1 ) ( b ) S e t t l e m e n t v a l u e s / m m D a t e / m m -y y y y O r i g i n a l v a l u e s G M ( 1 , 1 ) ( c ) S e t t l e m e n t v a l u e s / m m D a t e / m m -y y y y O r i g i n a l v a l u e s V e r h u l s t ( 1 , 1 ) ( d )

Figure 3
Figure 3 shows the APE comparative analysis of different prediction models.From the values of the APE comparison, it can be found that the prediction accuracy of the optimized Grey Verhulst (1,1) model is significantly lower than other forecast models.The comparison results between predicted and original values in the testing set by four models are shown in Figure 4.It can be seen from the given Figure 4 that the prediction consequences of the traditional grey Verhulst (1,1) model are more stable and better, whose predicted values are close to the trend of the actual values.The prediction accuracy of the DGM (2,1) model and the traditional GM (1,1) model is lower than that of the optimized grey Verhulst (1,1) model and the traditional grey Verhulst (1,1) model.Moreover, there is a large deviation between the trend of the predicted value of the two models and the actual curve.Compared with other models, the prediction error of the optimized grey Verhulst (1,1) model is significantly reduced with little fluctuation.The optimized grey Verhulst (1,1) model can complement each other's advantages and better improve the efficiency and prediction accuracy of the traditional grey Verhulst model.Through comparative analysis, the proposed optimized grey Verhulst (1,1) model can well capture the overall development and individual trend of the original data.

Figure 3 .
Figure 3. APE histogram between the original measured value and prediction values of four prediction models from Dec-2010 to Sep-2011 (next 10 testing sample data).

Figure 4 .
Figure 4.The curve of the original measured value and prediction values of four models.

Figure 5 .
Figure 5. Error histogram between the original measured value and prediction values of four prediction models from Dec-2010 to Sep-2011 (next 10 testing sample data).(a) MAPE; (b) RMSE and MAE.

Figure 6 .
Figure 6.Comparison between measured original values and fitted values of the four models for the monitoring point ZLD-02 from Aug-2007 to Nov-2010.

Figure 9 .
Figure 9.The upstream water level of the dam from August 2007 to November 2010.

Table 1 .
The accumulated settlement value of monitoring point ZLD-01 from Aug-2007 to Sep-2011.

Table 2 .
The accuracy comparison between the fitted and actual value of four models for the monitoring point ZLD-01 from Aug-2007 to Nov-2010 (first 40 training sample data).

Table 3 .
The comparison results between predicted and actual value by four models from Dec-2010 to Sep-2011 (next 10 testing sample data).

Table 4 .
The accuracy comparison between predicted and actual value of four models for the monitoring point ZLD-01 from Dec-2010 to Sep-2011 (next 10 testing sample data).