Simulating thermal stratification and modeling outlet water temperature in reservoirs with a data-mining method

The thermal stratification of lakes and reservoirs implies a change in the temperature at different depths in the lake or reservoir, and occurs because of the change in density with temperature. Thermal stratification develops two layers called epilimnion and hypolimnion separated by a layer of rapid temperature changes called metalimnion. The hypolimnion layer consists of water that is generally denser and colder than water in the epilimnion layer. Thermal stratification and heat budget significantly affect the water quality and ecological characteristic of lakes and reservoirs (Wetzel 1983; Wang et al. 2012), contaminant transport and hydrodynamic mixing in reservoir and lakes (Fischer et al. 1979; Kennedy et al. 1982), downstream irrigation area of reservoirs (Yang et al. 2012), and aquatic environment of lakes and reservoirs (Hanna et al. 1999). For instance, hypolimnion layers in stratified reservoirs contain anoxic water with poor quality which might contain dissolved iron, manganese, sulfide, ammonium, and phosphate (Dortch 1997). Rising water temperature in water reservoirs increases the speed of chemical and biological reactions, which have a strong influence on reservoir nutrient cycling and initial productivity (Sahoo & Schladow 2008).

Elci (2008) considered the mixing and thermal stratification effects in the Tahtali reservoir in Turkey using multivariate analyses on qualitative data of the reservoir. Wei et al. (2011) developed a three-dimensional hydrodynamic simulation model and considered the effects of reservoir inflow temperature on thermal structure in a reservoir located in south China. Yang et al. (2012) simulated the thermal stratification effects on downstream irrigation area of the Xiahushan reservoir in China with a three-dimensional hydrodynamic model, environmental fluid dynamics code (EFDC). Wang et al. (2012) used a three-dimensional hydrodynamic simulation model (ELCOM) to consider the effects of local climate and hydrologic conditions on the thermal stratification in the Liuxihe Reservoir, China. Kerimoglu & Rinke (2013) estimated the response of the Bautzen reservoir, Germany, to different combinations of external factors such as the hydrological regime, water level fluctuation, dewatering depth and meteorological variables with the one-dimensional DYRESM model. Lugg & Copeland (2014) assessed the impacts of the location, quantity, and extent of cold-water pollution (CWP) in the downstream Murray-Darling basin in Australia. Bermudez et al. (2018) considered the impacts of a pumped-storage hydroelectric power plant on the reservoirs' thermal stratification. Gelda & Effler (2007) linked the CE-QUAL-W2 simulation model to an evolutionary optimization algorithm and showed that using selective withdrawal can decrease the epilimnion and metalimnion in the reservoir. All the models applied in the cited studies were solved based on differential equations and numerical methods called physically based models.

Data-mining methods can be applied as surrogates of physically based models to reduce the computational burden. Data mining is a process for selecting, identifying, and modeling based on large databases searched to discover relations among data providing useful results for database analysts (Giudici 2003). There are many data-mining methods such as linear and nonlinear regression, Kriging method, artificial neural network (ANN), genetic programming (GP), multilayer perceptron (MLP), and support vector machine (SVM). Xiang & Jiang (2009) applied least-squares support vector regression (LSSVR) method to predict water quality in the Liuxi River located in China. They predicted chemical oxygen demand (COD) and DO with an integrated algorithm consisting of the LSSVR method and the particle swarm optimization (PSO) algorithm. Raghavendra & Deka (2014) reviewed the application of support vector regression (SVR) in hydrology.

There have not been, to our knowledge, reported applications of the SVM method to modeling outlet water temperature in reservoirs even in recent investigations. This study simulates outlet temperature in a reservoir with the combined application of water-quality simulation with the CE-QUAL-W2 model and data mining with the LIBSVM, a variant of the SVM method. LIBSVM serves as the data-mining method for its capacity to model complex and non-linear water-quality dynamics present in the physically based model CE-QUAL-W2. The LIBSVM yields substantial reductions in the computational burden of water-temperature modeling while preserving predictive accuracy.

The CE-QUAL-W2 model and the SVM method are briefly summarized in this section. The statistical criteria used to evaluate the LIBSVM method's performance are also defined in this section.

A physically based hydrologic model consists of a mathematical description of surface and subsurface processes, external and internal boundary conditions, and initial conditions (Furman 2008). Physically based models have been applied in numerous studies to address a wide range of water-quality questions (McCuen 1973; Rango & Martinec 1995; Montanari & Grossi 2008; Mendoza et al. 2015). The CE-QUAL-W2 model is a two-dimensional water quality and hydrodynamic simulation model developed by the U.S. Army Corps of Engineers' Waterways Experiment Station (WES). The temporal and spatial changes of water surface level and water temperature are modeled using CE-QUAL-W2 version 3.71 (Cole & Wells 2008). The required model data consist of reservoir geometric data, initial temperature conditions, boundary conditions (time series of meteorological, hydrological, water quality, released reservoir discharge, and hydraulic and kinetic parameters). The reasons for choosing the CE-QUAL-W2 version 3.71 are: (1) the model simulates flow, water level, horizontal and vertical velocities, water temperature, ice cover, and the concentrations of constituents such as ammonia, nitrate, phosphate; (2) in case of thermally stratified reservoirs the model allows reservoir operators to define water release from multiple floating or fixed elevational outlets that access waters at several depths with distinct temperature; and (3) it allows the operators to set priority allocations for each outlet to optimize water releases.

SVM is a machine-learning system based on constrained optimization theory (Vapnik et al. 1996; Vapnik 1998). The SVM method has been applied in many fields of inquiry successfully. SVM regression determines a relational function between dependent or output variables (y) and independent or input variables (x). The relational function, f(x), is obtained by the SVM model trained on a data set.

This risk function ignores risks that are smaller than ɛ and considers risk values according to or data in which the difference between observed values and estimated values exceeds the threshold ɛ. Bold font expresses vector values in the adopted notation.

The LIBSVM model parameters are ⁠, which expresses the ɛ-insensitive risk function, C (expresses functional capability), and the kernel function coefficient that appears in the RBF kernel function, Equation (12). These parameters can be obtained by sensitivity analysis resorting to the trial and error method which is commonly applied based on prior knowledge of the analyst (Vapnik 1998).

The Karkhe Reservoir is the sixth largest earthen reservoir in the world and the largest earthen reservoir in Iran. The catchment area of the Karkhe River is approximately about 44,000 km². It is located between 46°57′–49°10′ eastern longitudes and 31°48′–34°58′ northern latitudes. The reservoir volume equals 5 × 10⁹ m³, its length equals 64 km, and its surface area equals 162 km² at normal water pool (220 m above sea level). The average and maximum depth of the reservoir are 61.8 and 117 m, respectively, and the maximum and minimum elevation of the reservoir are 230 and 113 m above sea level, respectively. The reservoir has three outlets. One is a longitudinal outlet whose outlet water temperature is not considered in this study, one agricultural outlet and one hydropower outlet located at 163 and 181 m above sea level, respectively. The reservoir supplies water for 180,000 hectares of agricultural land and domestic water for cities located in Khozestan province. Therefore, the reservoir is strategic for supplying reliable and high-quality water for irrigation and municipal consumption. According to field observations the reservoir is affected by eutrophication and is susceptible to thermal stratification. In recent years, the reservoir has exhibited algal blooms and water quality degradation. The reservoir average retention time is estimated about 0.74 years. The relative long retention time causes increasing nutrient load and water quality degradation, which call for detailed modeling, monitoring, and assessment of thermal stratification formation in the reservoir (Afshar & Saadatpour 2009; Saadatpour et al. 2017).

The set up of the CE-QUAL-W2 model involves: (1) specifying reservoir geometry; (2) adding component structures in the reservoir; (3) adding meteorological and hydrological data; (4) defining the simulation time period and time step; and (5) calibrating and verifying the model.

In previous research, the Karkhe Reservoir was described with 66 longitudinal segments each 1,000 m long (Figure 3), and up to 55 vertical layers depending on water depth, layers' thicknesses ranging from 2 to 5 m in each segment (Figure 4) (Afshar & Saadatpour 2009; Saadatpour et al. 2017). The same description is applied in this study. Reservoir components include a reservoir spillway, one longitudinal outlet, one agricultural oulet, and one hydropower outlet located at 163 and 181 m above sea level; two proposed outlets located at 120 and 140 m above sea level used to simulate water releases blending waters from different levels. Figure 4 specifies the reservoir's minimum and maximum elevations and the outlet elevations. Meteorological, hydrological, and water quality data were obtained from previous studies of the Karkhe system project by the Mahab Qods Consulting Engineers Company and the Meteorological Organization (Iran). Thermal stratification and outlet water temperature were simulated for the 15-year period 1981–1995. The first simulation day is September 19th, 1981, and the last day is September 15th 1995. The minimum and maximum simulation time steps are equal to one second and one hour, respectively. The CE-QUAL-W2 model was calibrated and verified by Afshar et al. (2011).

The implementation steps of the LIBSVM model are depicted in Figure 2. Using the LIBSVM model as a surrogate model for CE-QUAL-W2 model for water-quality simulations requires careful choice of the input variables. The input variables having the greatest impact on system response must be chosen so that the LIBSVM model's predictive skill is accurate.

The key variables governing surface heat exchange and reservoir thermal stratification in a reservoir are air temperature, dew point temperature, cloud cover, wind velocity, thermal inflow and outflow fluxes, water depth, and water volume (Edinger 2002; Afshar & Saadatpour 2009). An information-theoretic approach based on the mutual information (MI) concept proposed by Tourassi et al. (2001) is applied herein to select the most significant variables among the cited governing variables to simulate Karkhe reservoir outlet water temperature and reduce the computational burden of the LIBSVM method (Saadatpour et al. 2017). The MI quantifies the ‘amount of information’ gained about outlet water temperature through other variables such as air temperature, cloud cover, and so on. Complex cause-and-effect relations govern feedbacks and time delays on large reservoir thermal responses. Therefore, appropriate time delays must be added to the input data. Moreover, the application of fractional withdrawals at various outlets must be taken into account in the LIBSVM input data to model outlet water temperature due to the withdrawals from different elevations in a lake.

The training (calibration) and testing input data of the LIBSVM model are obtained from inputs and outputs of 18 defined operational scenarios based on the authors' knowledge from simulations with the CE-QUAL-W2 model. This led to choosing the results of 12 and 6 operational scenarios among the 18 scenarios as training and testing data, respectively. Training and testing scenarios were chosen based on random selection (Liu & Motoda 2012). The withdrawal fractions at each outlet for different operational scenarios are listed in Table 1. Moreover, the LIBSVM model's performance is evaluated for two different structural states defined as follows: (1) the LIBSVM method is trained by operating scenarios 1 through 12, and tested with operating scenarios 13 through 18 for all the simulated years (1981–1995); (2) the LIBSVM method is trained with operating scenarios 1 through 12, and tested with operating scenarios 13 through 18 for similar months (12 months) during the simulation period (1981–1995).

The present study has the two main goals of: (1) simulating reservoir water temperature (both outlet water temperature and internal reservoir water temperature); and (2) implementing the LIBSVM model to estimate outlet water temperature at several outlet levels of the reservoir.

The thermal stratification of Karkhe Reservoir was simulated with the CE-QUAL-W2 model for the period 1981–1995. Figure 5 shows vertical water temperature profiles in seasons of 1982, 1985, 1988, 1991, and 1994, which displays that thermal stratification begins in spring in every year of simulation (1981–1995). The surface of the reservoir warms up with the onset of spring and the density of the upper layer decreases. This prevents vertical mixing in the reservoir. The calculated water temperature on May 17th of all the simulation years in the lowest and highest layers averaged 14.18 °C and 26.66 °C, respectively. This implies a temperature difference of 12.48 °C over 88 m of depth. With the onset of summer, the water surface continues to warm up and the vertical temperature gradient increases. The surface water temperature on August 15th of all the simulation years in the lowest and highest layers averaged 14.18 °C and 33.58 °C, respectively, producing a temperature difference equal to 19.39 °C over 88 m of depth. With the onset of autumn, the epilimnion, metalimnion and hypolimnion layers become clearly delineated by November 13th of all years. The depths of the epilimnion and hypolimnion layers in this season are approximately 17 and 35 m with temperatures equal to 25.50 °C and 15 °C, respectively. Autumn exhibits the steepest thermal slope of the metalimnion layer among all seasons. The surface layer cools with advancing autumn. This lowers the temperature difference between the epilimnion and metalimnion layers causing vertical mixing. Complete vertical mixing develops through February 16th of all years. In the winter season of all simulated years, the difference between the lowest and highest layer is almost 1.2 °C over 88 m of depth. Thermal stratification does not occur during winter.

Figure 5 establishes that Karkhe Reservoir is subjected to strong thermal stratification which prevents vertical mixing in the reservoir. Therefore, applying selective withdrawal to counter thermal stratification is necessary.

The input data to the LIBSVM model are a ten-day average of air temperature, wind speed, heat flux input (obtained by multiplying reservoir inflow discharge by inflow water temperature), water surface level of the reservoir with an average three-month lag from the current condition (average lag of 1, 2 and 3 months), reservoir outflow, and reservoir withdrawal fractions (at elevations equal to 120, 140, 163, and 181 m). These variables were selected as the most important input variables to the LIBSVM model based on MI criteria results reported by Saadatpour et al. (2017). Selection of the variables, averaging time intervals, and time lags are based on prior knowledge and sensitivity analyses (Saadatpour 2012; Saadatpour & Afshar 2013; Saadatpour et al. 2017).

The ɛ, C and LIBSVM model parameters were obtained based on sensitivity analysis shown in Table 2. Accordingly, the RMSE and number of iterations were captured for different sets of LIBSVM parameters. According to Table 2 the RMSE decreases and the number of iterations increases with decreasing ⁠, which leads to an increase of the model run time. In addition, the RMSE value decreases and the number of iterations increases with increasing C. The best value of ɛ equals 0.075. The optimal ɛ, C, and which minimize the RMSE and the number of iterations were equal to 0.075, 500, and 0.0006, respectively. Figure 6 portrays the calculated outlet water temperature with LIBSVM and the CE-QUAL-W2 models for corresponding to scenarios 9 and 14 as training and testing scenarios, respectively.

The LIBSVM model parameters were obtained from sensitivity analysis and are listed in Table 2. For all months except September these are ɛ, C, and equal to 0.075, 5000, and 0.006, respectively, and for September they are equal to 0.75, 5000, and 0.006, respectively. Figure 7 depicts the outlet water temperature obtained from CE-QUAL-W2 and LIBSVM models for April and September obtained from Scenario 9 and 14 as the training and testing scenarios, respectively.

Figures 6 and 7 demonstrate the LIBSVM performance in modeling outlet water temperature was excellent for the two structural states of the model. Moreover, the accuracy of the model in approximating maximum and minimum outlet water temperature is also good for each of two modeling states.

Figure 7(b) shows that in 1978 through 1990 the difference between outlet temperature predicted with the CE-QUAL-W2 and LIBSVM models is larger than in other years. The input variables to the LIBSVM model consist of time series data that vary substantially within a specified month through different years. Therefore, the input data for September of the years 1987 through 1990 were different from the input data of this same month in other years because of natural or man-made changes in data series. The LIBSVM identified fitting functions for each particular scenario. The performance of the LIBSVM to model outlet temperature in some years was not as good as in other years. The LIBSVM model predictions exhibit some error, but, generally, the predictive performance of the model for outlet water temperature is excellent overall.

The LIBSVM model performance based on statistical criteria is listed in Table 3, in which Max Error was the maximum difference between obtained the CE-QUAL-W2 and the LIBSVM model outputs. Table 3 lists the maximum values of the RMSE, MAE, and Max Error being equal to 0.76 °C, 0.65 °C, and 4.80 °C, respectively, and the minimum value of the NS is 0.92 for both structural states. It shows that both structural states had excellent performance modeling outlet water temperature. The averaged value of RMSE, MAE, Max Error, and NS for structural state 2 are 0.41 °C, 0.30 °C, 1.88 °C, and 0.96, respectively. Comparing the averaged statistical results of structural state 2 to the statistical results of structural state 1 established the LIBSVM model's excellent performance in predicting outlet water temperature for each of two structural states.

The run times for calculating outlet water temperature with CE-QUAL-W2 and LIBSVM model were about 20 minutes and less than two minutes, respectively. The LIBSVM model calculations were about 10 times faster than those for the CE-QUAl-W2 model. Evidently, the LIBSVM model exhibited accurate and relatively rapid performance for extracting optimal selective withdrawal rules for thermal control of the environmental demand.

This research simulated the thermal stratification in the Karkhe Reservoir in the period 1981–1995 employing the CE-QUAL-W2 model. The LIBSVM model was applied as a surrogate of the CE-QUAL-W2 model to approximate outlet water temperature of the reservoir. The application of LIBSVM reduces the computational time, and the number of input data needed for simulation purposes. Results demonstrated the run time of the LIBSVM model was approximately 10 times shorter than the run time of CE-QUAL-W2 model in this study. The simulation of reservoir thermal stratification showed a stratification cycle during each simulated year, in which the thermal stratification began in spring and complete vertical mixing occurred in the mid-winter of all the simulated years. The statistical results of the data-mining model performance measured the capacity of the LIBSVM model to approximate outlet water temperature, with the minimum and maximum values of the NS criteria equaling 0.92 and 0.99, respectively. These encouraging results indicate the LIBSVM model can be used instead of the more complex CE-QUAL-W2 model and be coupled with an optimization model to calculate operating policies at reservoir outlets to control outflow temperature from the reservoir. The trained algorithm performed well in predictions for Karkhe Reservoir. Such performance would have to be evaluated in other reservoirs.

The authors thank Iran's National Science Foundation (INSF) for its financial support of this research.