An improved two-step parameter adjustment method for the optimization of a reservoir operation function model based on repeated principal component analysis and a genetic algorithm

Owing to climate change and intensive human activities, river runoff faces reduction, and hydrological extremes occur more frequently. This causes water scarcity worldwide (Gain & Giupponi 2015), affecting the sustainable development of the social economy. Reservoirs have an important role in water conservation by regulating supply through storage, alleviating regional water shortages. They can prevent flooding, and can provide high volumes of water for irrigation and power generation. Reservoirs are particularly useful for runoff regulation; by storing floodwater for dry seasons, natural water can meet the requirements of the water sectors, allowing efficient regional water resource utilization and social economic growth. Thus, efficient reservoir operation is of interest to governments and to researchers (Li et al. 2013).

Scheduling problems relate to reservoirs' ecological environment. Reservoir operation changes water level, which causes environmental damage to the water-level-fluctuating zone and its surroundings. Meanwhile, changes in the release of water from the dam, and instream flow downstream of the dam determine whether the ecological water requirement is adequate. The problems of reservoir operation are also socio-economic issues; the volume of incoming water, the water quantity required for irrigation and power generation directly affect the social and economic benefits and there is a trade-off of interests among water sectors. Reservoir operation rules guide the use of reservoirs, and promote coordinated development of industrial and agricultural production, national economy, and ecological environment (Bai et al. 2015). Therefore, the study of optimized reservoir dispatching methods not only has general theoretical value, but also has practical significance.

A discrete stochastic dynamic programming model was first applied to reservoir operation in the 1940s (Liu 2006). Existing reservoir models can help to plan reservoir operation, providing efficient distribution of the water use and the water storage (Bai et al. 2015). Current methods for formulating reservoir operation rules mostly use a long-term runoff dataset. A deterministic optimization method is used to solve the optimal reservoir scheduling problem, then a variety of mathematical methods (e.g., regression analysis) are used to fit functions to the results, which then guide the operation of the reservoir (Liu et al. 2006). The reservoir operation function model is a scheduling tool which can incorporate a variety of scheduling information, and it usually reflects the relationship among some reservoir operation elements, such as surface runoff, reservoir storage capacity (or water level), and water discharge (or the output of hydropower stations), and so on. In general, the conventional regression analysis requires parameters to remain independent. However, parameters of the scheduling function often exhibit high linear correlation with each other, such as water volume for power generation and water discharge, ecological water demand and water discharge downstream of the dam. If all of these parameters are used in the regression analysis, then an unstable regression equation will be produced due to the existence of the co-linear parameters (Abdul-Wahab et al. 2005). Therefore, it has become the consensus that it is necessary to exclude unwanted interference from co-linear parameters when establishing a reservoir operation function model (Labadie 2004; Farrokhnia & Karimib 2016).

In recent years, the commonly used methods to solve co-linear problems in multiple regression have included ridge regression, partial least squares (PLS) regression, and principal component analysis (PCA) regression. Ridge regression, in essence, is a modified least squares estimation method. It can obtain relatively realistic and reliable regression coefficients at the expense of partial information loss and accuracy reduction. In addition, it has certain subjectivity, since ridge parameters are determined according to experience (Xie & Hawkes 2015). The PLS regression is to find the best function relationship for a set of data by minimizing the sum of errors squared. It can be seen as a combination of the least squares regression and principal components (De Pauloa et al. 2016). The outstanding characteristic of PLS is that it can implement regression modeling, predigest the data structure and analyze the correlation between two groups of parameters simultaneously with the same arithmetic. It brings great convenience to multiple linear regression analysis. However, ordinary PLS cannot get satisfactory results for dealing with the problem that the number of parameters are more than that of sample points (Song 2009). As a widely used dimensionality reduction technique, the PCA is a multiple statistical method, which detects the correlation among multiple parameters and has a data compression effect. It can reveal the simple data structure involved in the complex data (Çamdevýrena et al. 2005). In spite of the advantages of easy operation, repeated use of PCA may be needed in order to achieve a good effect for solving co-linear problems when there are linear correlations among regression parameters.

In addition, the results obtained from the regression methods for solving the reservoir operation function model may be inaccurate, because they omit some important constraints. Researchers have found that intelligent algorithms can be used to extract reservoir operation rules and solve optimizations, saving time and avoiding redundant work (Neelakantan & Pundarikanthan 2000; Chandramouli & Deka 2005; Liu et al. 2006). To be specific, intelligent algorithms can overcome the limitations of traditional optimization methods in dealing with complex problems, such as problems regarding multi-objective, uncertainty, non-linearity, discontinuousness, and discreteness in water resources and reservoir scheduling (Simonovic 2009). As a kind of widely used intelligent algorithm, the genetic algorithm (GA) directly uses the objective function value to globally search the problem space, and can improve solution accuracy and convergence; thus, it is suitable for solving reservoir operation function models. For example, through comparative analysis, Vasan & Srinivasa (2009) analyzed three methods including simulated annealing, simulated quenching, and genetic algorithms for optimal reservoir operation, and found that all these non-traditional optimization techniques can be utilized for efficient planning of any irrigation system with suitable modifications. In addition, Chang et al. (2010) researched constrained genetic algorithms for optimizing multi-use reservoir operation and concluded that they would be powerful tools in searching for the optimal strategy. Considering the ease of using repeated PCA and the GA's ability to quickly approach optimal solutions for complex system problems, it is advantageous to jointly use these methods to optimize parameters in conventional multiple regression models and finally get a more accurate model.

Therefore, the objective of this research is to construct a new two-step parameter adjustment method to exclude the co-linear parameters in multiple regression expression, and get an optimized reservoir operation function model. This objective entails the following tasks: (1) combine repeated PCA with a multiple regression model, to produce an initial reservoir operation function model; (2) use GA to optimize parameters in that initial scheduling function to get a more accurate model for supporting reservoir operation management. After carrying out two-step adjustments to the reservoir operation function model, the co-linear factors in the reservoir scheduling data can be effectively eliminated, and reasonable relationships among the decision variable and the impact factors are reflected more accurately. Therefore, this paper provides a scheduling model resulting in better economic efficiency and ecological benefits, producing optimal operation of the reservoir.

Danjiangkou Reservoir is in the upper reaches of Hanjiang River in central China, and is located across the waters of Hubei and Henan provinces (see Figure 1). It is Asia's largest artificial freshwater lake. Danjiangkou Dam was constructed in early 1973. Covering an area of 745 km², the reservoir adjoins Danjiangkou City, Yun County, Yunxi County, and Shiyan City of Hubei Province and Xichuan County of Henan Province, covering a total area of 15,900 km². Danjiangkou's designed water level is 157 m, and the corresponding storage capacity is 17.45 billion m³. The place at which the reservoir is located has a north subtropical monsoon climate, with a mild climate and abundant rainfall. The average temperature is 15.9 °C, while the average annual rainfall of the reservoir area is 850–950 mm, and the annual water-surface evaporation is 860 mm. The annual water-inflow of the reservoir is an average of 39.48 billion m³; the water comes from the Han River and its tributary, Dan River. Danjiangkou Reservoir's runoff frequency distribution and the dam's discharged water from 1980 to 2005 are shown in Figures 2 and 3. After the heightening project of the Danjiangkou Dam was built in 2005, both the top elevations of the concrete dam and the embankment dam were increased from 162 m to 176.6 m; the normal water level of the reservoir was increased from 157 m to 170 m. This resulted in an increase of the reservoir area from 745 km² to 1,050 km² and the reservoir capacity from 1.75 × 10⁸ m³ to 2.91 × 10⁸ m³.

Danjiangkou Reservoir is responsible for flood control, power generation, water supply, and shipping affairs. As a resource for the South–North Water Transfer Project in China, the reservoir will provide domestic water, industrial water, and agricultural water for more than 20 cities around Beijing City, Tianjin City, Henan and Hebei Provinces in the future. The total installed capacity of the hydroelectric power plants in Danjiangkou is currently 900,000 kilowatts. The cumulative energy generation was up to 141.82 billion kwh (at the end of 2011), and each kilowatt-hour of electricity could create 3.26 yuan (i.e., 0.48 USD) for Hubei Province's gross domestic product (GDP) in 1990; the cumulative power generation can hit 429.70 billion yuan (i.e., 62.59 billion USD) of GDP.

In this study, a total of 26 years of daily average inflow data of Danjiangkou Reservoir from 1980 to 2005 was collected as time series data for the optimization analysis of reservoir operation function model. Related hydrological, meteorological and reservoir data were derived from the South–North Water Transfer Project Construction Committee under China's State Council, and local water authorities.

The main task of reservoir operation is to increase economic and environmental benefits, and thus the utilization benefits of the reservoir as much as possible by coordinating the relations of upstream runoff, socio-economic water use, and the discharged ecological flow. Therefore, the water volume for power generation is chosen as the decision variable for reservoir operation, and other parameters (e.g., reservoir inflow, water level, and ecological flow) are used as its impact factors to build the reservoir operation function model. Then, the model parameters are optimized with the proposed two-step parameter adjustment method. The process is designed as follows (see Figure 4): (1) collect data, including historical hydrological conditions, and reservoir performance data over the years; (2) implement Step 1 to adjust parameters: apply repeated PCA to a multiple regression model to establish an initial reservoir operation function model; and (3) implement Step 2 to adjust parameters: use GA to optimize the initial reservoir operation function model, and achieve better economic benefits in the premise of ensuring the ecological water demand. Meanwhile, how the optimized scheduling rules can impact the objective function is compared.

Then, a repeated PCA is used to eliminate collinearity of Equation (1). The PCA can extract several major components of the original parameters, including as much information as possible to preserve the original parameter information while keeping them independent (Petersen et al. 2001). There is a relationship between the parameters for the reservoir operation schedule. For example, storage capacity is a function of the water level; ecological water demand downstream of the dam is calculated by using the historical data for annually discharged water; and the water volumes for power generation and discharge are usually linear in the case of very small amounts of discarding water from the reservoir to the river. If all of these parameters are used in the regression analysis, it will produce an unstable regression equation, which is highly influenced by the co-linear parameters. Thus, all parameters in the regression equation are first of all analyzed by using PCA repeatedly to decide if there is collinearity, and to eliminate some co-linear parameters. The first step is to carry out first PCA on all of the system's parameters (including both dependent and independent parameters) to get the principal component 1 (PC 1); then, to keep the most contributive parameter from the group of parameters that have the most significant contribution to PC 1 (i.e., excluding the remaining parameters that are highly relevant to PC 1 in this group, which are also called the excluded parameters). The second step is to carry out a second PCA on all parameters except for the excluded parameters to get the current PC 1, then to keep the most contributive parameter from the group of parameters that have the most significant contribution on this PC 1, excluding remaining parameters that are highly relevant to this PC 1 in this group, and so on. These steps are repeated until all the parameters are uncorrelated. Then, the remaining parameters are used to do a scatterplot analysis for further testing correlation between every two parameters.

The graphical method can be used to analyze whether the residuals are independent or normal distributed. The more close to zero, or the more close to the normal distribution for the residuals, the less obvious co-linear characteristics for the data used for regression analysis, indicating it being better than the correlation between the calculated results with the proposed model and the actual data.

A GA is further used to optimize the initial reservoir operation function model developed in Equation (2). The GA is an efficient, parallel, and global search method. It can automatically acquire and accumulate information related to the search space during the process, and adaptively control the search process to achieve the optimal solution (Zandieh & Karimi 2011). The users first determine the control parameters, set the number of chromosomes in the population, and then determine the probabilities of selection, crossover probability and mutation probability by a roulette method. Genetic Algorithms Toolbox in MATLAB software developed by the University of Sheffield in the UK is used to produce solutions of the model. The algorithm design process is shown in Figure 4. The GA parameters are set as follows: population size is 20; mutation probability is 0.2; crossover probability is 0.8; and the number of generations is 100, indicating that if generations are greater than 100 then the iteration terminates.

As for the proposed optimal reservoir operation function model, the objective function is the maximization of water volume for power generation. The following constraints are considered: (1) the water storage for the reservoir from a certain initial period to final period should meet the water balance relationship; (2) the value for the storage capacity should be between the dead storage capacity and the normal storage capacity (or the flood storage capacity in the flood season); (3) the ecological water demand should be maintained; (4) the power generation guarantee rate should be met. Since intelligent algorithms randomly generate the individuals in a certain space, part of individuals can inevitably not meet the constraint of power generation guarantee rate. Thus, the solutions that fail to meet the guarantee rate are handled by adding penalty coefficients.

According to results of the frequency analysis for the runoff, 1993 is found to be a typical year where the annual runoff guarantee rate is most close to 50%. Thus, the historical data of this year are selected as the data of 50% runoff guarantee rate. The water volume for power generation is used as the dependent parameter, and reservoir inflow ⁠, water level (H), stored water ⁠, surplus water ⁠, reservoir outflow ⁠, and the ecological water demand under 95% runoff guarantee rate are chosen as the independent parameters. The above indicators of daily mean values are selected for PCA, and the results are shown in Table 1. The correlation between operation parameters of the reservoir is shown in Figure 5. It can be seen that the upper triangular area is a scatter plot, which gives a visual representation about whether there is a correlation between two variables, and the strength of the correlation. For example, the scatter plot of E_95% and VO can be found at the intersection of these two variables. The lower region is a smooth curve fitting, and confidence ellipses. The confidence ellipse can be used to describe the confidence region with its long axis and short axis as the parameters of the confidence region. The semi-major axis and the semi-minor axis of the confidence ellipse represent the standard deviation of the two-dimensional coordinate component, respectively. The main diagonal panel in Figure 5 contains parameters with maximum and minimum values; the rank of the matrix is reordered using PCA. As can be seen in Table 1, the contributions of the dependent parameter and the reservoir outflow to the PC 1, were both greater than 0.85, when using all the parameters to do PCA for the first time. Thus, collinearity is exhibited, and in the next step of regression analysis the independent parameter should be excluded. Then, it can be seen that the contributions of the stored water and the water level (H) to the PC 1 are both greater than 0.90 when doing PCA for the second time. It means that collinearity is also exhibited. Then, the independent parameter V is excluded while keeping H. Then, the remaining parameters are used for the third PCA, and the reservoir inflow exhibits collinearity with the surplus water ⁠. It also can be seen that made a significant contribution to the PC 2 alone for each PCA analysis, and is left alone in the process of repeated analysis for the PC 1, so it is kept. Finally, the water level (H), reservoir inflow ⁠, and ecological demand water are chosen to conduct multiple regression analysis. It should be noted that because of fewer parameters for the reservoir operation, a relationship between the parameters is exhibited. Under this circumstance, if all of the parameters are selected directly to do regression analysis, it is inevitable that incorrect results would be produced, regardless of the choice of method used to propose parameters (such as perfect fit, forward, backward, enter). Repeated PCA is used to avoid the occurrence of this problem, and more accurate results are produced by correctly excluding parameters with collinearity.

Using a graphical method to analyze the residuals' statistical distribution, the results are shown in Figures 6–8. As can be seen in Figure 6, the mean residual 1.87 × 10⁻¹⁴ is very close to 0; the standard deviation is 0.996; most of the regression residuals are normally distributed, and they are in the range (−3, +3), with no significant deviation. Figure 7 presents the regression analysis between observed residuals and expected normal residuals. It can be seen that the observed residuals' distribution is well under the assumption of a normal distribution regression, and the correlation between the calculated values of the model and the actual data is good. Figure 8 shows the distribution of residuals, taking the standardized predictive value as the x-axis, the standardized residuals as the y-axis, and the number of days as the label. Most of the residuals scatter within the vicinity of the origin and they are within two standard deviations. Large deviation values are very small, and only the data on the 238th day show a deviation within the vicinity of three standard deviations.

To investigate the accuracy of the above methods, the regression results and the actual values are used to calculate the Nash coefficient, and the result is 0.599. Also, the principal component regression and the ridge regression are used to do a united regression, and the Nash coefficients are calculated as 0.464 and 0.475, respectively. Thus, it can be concluded that the repeated PCA combined with multiple regression can better reflect the correlations among the decision variable for reservoir operation (power generation) and impact factors such as the reservoir inflow, water level, ecological demand water, etc., and it can guide the actual operation of the reservoir.

It should be noted here that the year of 1993 with annual runoff guarantee rate most close to 50% is selected as a typical year for the analysis in this research, with the aim to verify the application effects of the proposed optimal operation function model. Actually, the scheduling model can be combined with other models for water resources planning of local development. For example, it can be combined with runoff prediction models such as the improved Thomas–Fiering and wavelet neural network models (Cui et al. 2016) to work out the reservoir operation planning under a certain amount of runoff in the next year, or it could be combined with an ecological planning model to determine the ecological water demand guarantee programs. With the joint operation of the above model, water resources can be used rationally. Therefore, it is a useful technical tool for reservoir management.

A reservoir operation function model was developed through applying repeated PCA to exclude co-linear parameters in multiple regression. Then, GA is used to optimize the reservoir scheduling parameters. The results show that the proposed parameter optimization method for reservoir operation can better reflect the correlation between the reservoir operation elements, and show that the accuracy is higher than other regression methods. Compared with former operation planning methods, the optimized approach not only increases the amount of water used to generate electricity, but also economic benefits, by assuring downstream ecological flow. Therefore, the constructed model is practicable and feasible, and can provide a scientific basis for reservoir operation.

In fact, the relationship between the outflow, inflow, and the water level is usually connected by a water balance equation; the regression model does not reflect this point. In future reservoir scheduling models, planning for water diversion and the effects that the water transfer project have on the amount of water need to be further considered in Danjiangkou Reservoir, one of the most important water resources for the South–North Water Transfer Project in China. In addition, some runoff forecast models and ecological planning models will be combined with the proposed two-step parameter adjustment method to increase economic or ecological benefits during the reservoir operation in the future. Nevertheless, this paper presents a simple method to optimize reservoir operation, which can also be applied to other reservoirs.

This research was financially supported by the National Natural Science Foundation of China (Grant No. 51439001, 51721093, 51679008), Chinese National key research and development program (Grant No. 2016YFC0401302), and National Science and Technology Support Program (Grant No. 2011BAC12B02). We would like to extend special thanks to the editor and the anonymous reviewers for their valuable comments in greatly improving the quality of this paper.