Model fusion approach for monthly reservoir inflow forecasting

Reservoirs are significant components of water resource systems, providing effective irrigation, water supply, hydropower, flood and drought control, and other functions. Hence the effective operation and scheduling of reservoirs are important tasks in water resource management. An accurate and reliable inflow forecast is a vital reference for making decisions in reservoir management. During the past few decades, a great deal of research has been devoted to the formulations and model developments in reservoir inflow forecasting.

Autoregressive (AR) model, autoregressive moving average and autoregressive integrated moving average (ARIMA) (Box & Jenkins 1976; Pankratz 1983; Brockwell & Davis 1991; Commandeur & Koopman 2007) are all general time series models in hydrological forecasting (Valipour 2012, 2015; Valipour et al. 2012b, 2013; Wang et al. 2015). All these approaches are very useful for forecasting changes in the assumption process that the data have an internal structure over time (such as autocorrelation, trend or seasonal variation). However, the real time series of the reservoir inflow often features dynamic, nonlinear and non-stationary characteristics, which limit the applicable scope of these general models.

With the developments in computer technologies and new theories, researchers have proposed various advanced methods, e.g., artificial neural network (Asaad 2010; Othman & Naseri 2011; Kale et al. 2012; Sattari et al. 2012; Valipour et al. 2012a; Prakash et al. 2014), support vector machine (SVM) (Lin et al. 2009a; Wang et al. 2010), Bayesian regression (Ticlavilca & Mckee 2011), decision support system (Regonda et al. 2014), fuzzy inference systems (FIS) (Luna et al. 2011; Lohani et al. 2011, 2014; Jaiswal et al. 2015), model tree (Jothiprakash & Kote 2011; Maestre et al. 2013) and grey model (GM) (Lin et al. 2012). In addition to the direct time series forecast using the inflow data themselves, external factors such as weather conditions were also taken into account to improve the inflow forecasting accuracy (Zhang et al. 2009; Lin et al. 2009b, 2010; Wang et al. 2012; Tsai et al. 2014).

The application of a sole forecast model is often ineffective in forecasting different reservoir inflow series. Hence, some hybrid methods consisting of two or more models or criteria (Li et al. 2015) have been reported recently. Chen (2011) proposed a nonlinear model which incorporated improved real-coded grammatical evolution with a genetic algorithm to predict the 10-day inflow of the De-Chi Reservoir in central Taiwan. Budu (2013) studied the capability of two pre-processing techniques, i.e., wavelet transform and moving average, in combination with feed forward neural networks, radial basis and multiple linear regression models for the prediction of the daily inflow values of Malaprabha reservoir, Belgaum, India. Okkan & Ali Serbes (2013) applied wavelet transform technique to remove nonlinear dynamic noise of raw data, and established three hybrid wavelet-based box models, i.e., multiple linear regression, feed forward neural networks and least square SVM, for monthly inflow forecasts. Toro et al. (2013) presented a hybrid hydrologic estimation model incorporating both statistical and artificial intelligence techniques for calculating the mean daily inflow of the Salvajina reservoir, Colombia. Li et al. (2014) used a couple method, SVM and ensemble Kalman filter, to forecast flooding in the Luo River Basin, China. Awan & Bae (2014) investigated the ability of adaptive neuro-fuzzy inference system (ANFIS) method to improve the accuracy of reservoir inflow forecasting. Bai et al. (2014) proposed an additive model for forecasting monthly reservoir inflow, combining AR for trend term, SVM for periodic term and ANFIS for stochastic term, respectively. Wang et al. (2015) studied the ARIMA model coupled with the ensemble empirical mode decomposition for forecasting annual runoff time series, improving the accuracy based on a deep insight into the data characteristics. These hybrid models have shown better forecasting ability than that of the mono-models. These hybrid models are capable of sufficiently reflecting the characteristics of the real time series. Some techniques, such as nonlinear and linear model integration, multiple model combination, data pre-processing, were used for taking advantage of various methods.

Combining data from multiple sources to improve the potential values and interpretation performances of the source data, data fusion is as a general and popular multi-discipline approach to produce a high-quality representation of the data (Zhang 2010). Considering the complexity of hydrologic systems, data fusion approaches have been used in hydrological engineering (Azmi et al. 2010; Ababaei et al. 2013). Inspired by the idea of data fusion (Liggins et al. 2008), we propose in this paper a model fusion approach for monthly reservoir inflow forecasting. Due to the different features in the different forecasting time scales, the time scales of the real time series should be determined first. In each scale, an appropriate model selected by the scale's time characteristics is used to train and forecast the scale's information. The results forecasted at all scales are employed to obtain the final forecasting results through scale-normalized fusion. In this paper, according to different inflow information represented, the historical data of the reservoir are divided into two time series, namely, yearly-scale series and monthly-scale series. To accommodate different characteristics of the two scales, the even grey model (EGM) and the ANFIS are adopted for the forecasts at the two scales, respectively. The grey relational analysis (GRA) is subsequently used as a scale-normalized model fusion tool to integrate the yearly-scale model and the monthly-scale model to obtain final forecasting results. In this way, a model fusion approach is constructed to improve the forecasting accuracy of the monthly reservoir inflow. The proposed approach is applied to the Three Gorges reservoir, China. The forecasting results are also compared to those of some published peer approaches.

As mentioned above, the monthly-scale pattern describes the nonlinear changes information of the adjacent months. This calls for a nonlinear model to express the nonlinear relationship F₁(.) between the input and the output variables. The ANFIS, integrating both the neural networks and the fuzzy logic principles, has capabilities in handling nonlinear, complexity and discontinuity problems (Lohani et al. 2012). It also has some other advantages, such as fast convergence speed, good repeatability and high forecasting precision. Moreover, the results in our previous research (Bai et al. 2014) have shown that the ANFIS has good performance in single application on the monthly inflow forecasting. Hence, the ANFIS is chosen as the monthly-scale regression model.

As shown in Equation (2), for the monthly-scale forecasting pattern (ANFIS), there are two parameters (τ₁ and d₁) to be specified first. As pointed out in the literature (Sivakumar & Berndtsson 2010), hydrological sequence has generally chaotic characteristics. Thus, chaos phase space reconstruction technique is chosen to determine the ANFIS inputs. In this paper, mutual information (MI) method and false nearest neighbor (FNN) method are applied to determine τ₁ and d₁, respectively.

The yearly-scale pattern reflects the similarity of the same month's inflows in the adjacent years. Generally, the weather and the hydrologic conditions are roughly the same in a certain month for different years. Hence, the observations of the history series in the yearly-scale have a quasi-linear and quasi-stationary nature. Moreover, the modeling samples of the history series in the year's scale are usually less (only one sample for one year). Therefore, we adopt the EGM to model the yearly-scale pattern (F₂(.)) shown in Equation (2).

The EGM, first proposed by Deng (2005), has been widely used in forecasting fields. The EGM can be utilized to forecast the time series with only four or more observations. It overcomes the disadvantage that the forecasting accuracy of the statistical model depends on the data amount. In this paper, we employ the EGM (1, 1) algorithm that is described as follows (Liu et al. 2011).

Assume that the yearly-scale series X₂ = (x(t-τ₂), x(t-2τ₂), …, x(t-d₂τ₂)) as the modeling series X⁽⁰⁾ = (x⁽⁰⁾(1), x⁽⁰⁾(2), …, x⁽⁰⁾(n)) (n = 1,2,…,d₂), and X⁽¹⁾ is the 1-AGO sequence of X⁽⁰⁾, X⁽¹⁾ = (x⁽¹⁾(1), x⁽¹⁾(2), …, x⁽¹⁾(n)), where ⁠.

Data fusion integrates multiple data and knowledge representing the same real-world object into a consistent, accurate and useful representation. Its operating principle is using various corresponding methods to process data which reflect different characteristics of the time series. In this paper, we employ the GRA (Liu et al. 2011) as a fusion tool to normalize monthly-scale and yearly-scale models for the reservoir inflow forecast. The GRA method makes up the defects of the system analysis caused by the mathematical statistics, and inconsistencies will not exist between the qualitative analysis and the quantitative results. In addition, it is applicable to handling problems with less samples with the characteristics of generality and less calculation burden. The basic principle of the GRA is based on the geometrical relation comparison between multiple data sequences in the system. The much more similar geometry shapes mean the greater correlation degree. In other words, the smaller the forecasting error is, the higher the grey correlation degree will be. Therefore, model fusion based on the GRA can avoid amplification of different forecasting sequences' errors, thus improving the final forecasting precision.

According to the theory of the data fusion, the reservoir inflow X = (x(1), x(2), …, x(N)) (N represents the length of time) can be divided into two relevant series, named monthly-scale series ⁠, and yearly-scale series ⁠.

Step 1: Analysis of original data series X, and construct new series containing different characteristics (in this paper, we construct two relevant series, monthly-scale series X₁ and yearly-scale series X₂).

Step 2: Forecast the different series' reservoir inflow using corresponding methods (in this paper, Y₁ was obtained by ANFIS, and Y₂ was achieved using EGM (1,1)).

Step 3: Scale-normalized model fusion through GRA between Y₁, Y₂ and X₀, respectively.

Step 4: Output the forecasting results according to Equation (18). End.

From Figure 4, it can be seen that the reservoir inflow has obvious periodicity in monthly-scale with the wet season from June to September, the dry season from December to March, and normal season in the rest of the months. In addition, the reservoir inflow also features the seasonality in yearly-scale, that the same monthly inflow in each year is maintained at a certain amount because of the same hydrologic condition and weather changes. One can also observe singular inflow patterns in year 2006 and year 2011 (when drought events occurred in the Yangtze River watershed).

In this section, the proposed model fusion approach is evaluated using the reservoir inflow data of the Three Gorges, as shown in Figure 4. The forecasting performances are also compared with the sole application of the ANFIS, the EGM (1,1) and other peer methods to demonstrate the superiority.

In this subsection, the proposed model fusion will be employed to forecast the monthly inflow of the Three Gorges reservoir. First, the two scales' constituting models (the ANFIS and the EGM (1,1)) are established for the monthly- and yearly-scales, respectively. Then, the scale-normalized model fusion based on the GRA is subsequently used to integrate the yearly-scale and the monthly-scale results.

The ANFIS can be trained with the optimal input variables (three inputs and one output) of the model. After training, an ANFIS model with forecasting function will be obtained for output forecasting. In this paper, the monthly-scale series are divided into two sets, the training set and the testing set. The first 132 records, as shown in Figure 4, are employed for training, and the rest for testing. Subtractive clustering is used for training the number of MFs and rules with different value of influence range, squash factor, accept ratio, and reject ratio. A total of 25 models, designed by the orthogonal design, are trained as shown in Table 1. Note that the initial values of the four parameters are 0.5, 1.25, 0.5, 0.15, respectively. In addition, ANFIS training needs at least two rules. Therefore, we chose the range of four parameters [0.2, 0.3, 0.5, 0.8, 1], [0.5, 0.75, 1, 1.25, 1.5], [0.2, 0.4, 0.6, 0.8, 1], [0.1, 0.15, 0.3, 0.4, 0.5], respectively.

The different FIS are trained by hybrid method and all performance indices are computed. To determine the effective networks, the testing data are used for validation. Therefore, only nine models shown in Table 2 are suitable according to the criteria. According to the performances of both training and testing data as shown in Table 2, one can determine the optimal ANFIS networks (M22) with three inputs and one output in this paper. Each input has seven Gaussian membership functions with seven rules.

The second constituting model is the EGM (1,1). For a given sequence, the feasibility test for the EGM modeling is a key process that supplies information about whether an accurate EGM model can be built. Usually, if ratio ⁠, then the data of the reservoir inflow can be used to establish the EGM model, where ⁠.

In this study, the yearly-scale series are divided into two sets: the records from year 2000 to year 2010, as shown in Figure 4, are employed for training and the rest for testing (that is to say d₂ = 11). In practice, we transform the data of the training set into the logarithm form before modeling. The ratio σ (k) ∈[0.964, 1.076] ⁠, so the training set conforms to the requirements of the EGM (1,1) modeling. Through modeling 12 groups' series, respectively, we can obtain the yearly-scale's results in the periods from January 2011 to December 2012 (model parameters shown in Table 3).

From Figure 7(a), it can be seen that the modeling results can follow the trend of the training data successfully except for a time in the period 2006, and the forecasting results of the testing data are roughly good except for that of the inflow peak. As shown in Figure 7(b) and 7(c), the REs have a little fluctuation both in the training and testing processes, except for several values. There only exist six and three forecasted outliers in the training and the testing processes, respectively. Since the interval around the RE does not contain zero, this indicates that the nine REs are caused by the unfortunate forecasting, beyond the 95% confidence interval. In other words, the number of effective forecasted values accounts for 95% of the training data, and 87.5% of the testing data, respectively. Hence the thorough processes of the training and the testing are successful, and the results are acceptable. Furthermore, the scatter plot as shown in Figure 7(d) and 7(e), demonstrate that the correlations between the recorded and the forecasted data of the peak inflow are relatively poor, and other data are concentrated in the ideal fit. Besides the intuitive observation, Table 4 shows the quantitative evaluations using the MAPE, NRMSE, R, and PPTS of the training and testing data, respectively.

For quantitative evaluation, Table 5 shows the forecasting results using the ANFIS and the EGM (1,1), respectively. Compared with Table 4, it more obviously illustrates the fact that the proposed model fusion method has the best forecasting performance in the MAPE, NRMSE, and R. For the PPTS, the lower the value is, the better the capability for forecasting peak inflow will be (Lohani et al. 2014). Compared with Table 4, the PPTS values (PPTS (5) of the ANFIS, and PPTS (5) and PPTS (10) of the EGM (1,1)) confirm that the proposed model is capable of forecasting the peak inflow more accurately.

In addition to the comparisons with the ANFIS and the EGM (1,1), the proposed approach is also compared with two methods researched and applied more widely, i.e., back propagation neural network (BPNN) and ARIMA, and the authors' peer method (Bai et al. 2014), using the same data set as shown in Figure 4.

Table 6 shows the performance criteria of the BPNN and the ARIMA methods. Compared with Table 4, the BPNN and the ARIMA can provide forecasting accuracy up to 20% in MAPE (11.2858%, 11.3975% reductions for the testing data, respectively), more than 0.4 in NRMSE (0.2424, 0.1411 reductions, respectively), and lower index in R (0.1603, 0.0851 reductions, respectively). Moreover, the PPTS values also demonstrate that the present model is capable of forecasting the peak inflow more accurately.

As above, these criteria shown in Tables 4–6 clearly illustrate that the proposed method in this paper has the best forecasting performance among all the peer methods.

In addition, compared with the additive model proposed by the authors (Bai et al. 2014), the differences are evident: (a) the combination structures (sum-up strategy in the peer paper; GRA in this paper); (b) the features extraction methods (ensemble empirical mode decomposition in the peer paper; scale feature analysis and classification in this paper); (c) the temporal scales (three sub-terms all in monthly-scale in the peer paper; yearly-scale and monthly-scale in this paper); (d) the computational complexities (the peer paper involves four forecasting models and two time-frequency analysis methods, numerous model parameters; this paper contains only two models and one fusion method, especially, the structures of the GRA and EGM (1,1) are simple).

In the case of little gaps in evaluation criteria (compared with the fusion model in this paper, the MAPE, NRMSE, and R are better than 0.0985%, 0.1023%, and 0.0526% in the peer paper, respectively), the computational burden has obviously dropped.

In this paper, a model fusion method has been constructed for forecasting monthly reservoir inflow. Considering the characteristics of the reservoir inflow in different time scales (nonlinearity in monthly-scale and quasi-stability in yearly-scale), the ANFIS and the EGM (1,1) are applied to follow the monthly-scale and the yearly-scale inflow patterns, respectively. Using the GRA, the patterns at the two scales are normalized and integrated to generate the final forecasting results. The present method is applied to forecast the Three Georges reservoir inflow. The proposed method is also compared with four peer methods, the ANFIS, the EGM (1,1), the ARIMA, and the BPNN. The results show that, having integrated both monthly-scale and yearly-scale patterns into a grey fashion, the proposed approach exhibits the best forecasting performance in terms of the criteria. Due to the hydrological similarities of the reservoirs, the proposed model fusion method can also be applied for other reservoirs' inflow forecasts.

This work is supported in part by the Natural Science Foundation of China (51375517, 71271226), and the Project of Key Discipline Construction of Anhui Science and Technology University (AKZDXK2015B01). The authors would also like to thank the editors/reviewers for their valuable suggestions and comments that have helped improve this paper.