Study on the streamflow compensation characteristics of reservoirs in Luanhe River Basin based on Copula function Open Access

Study on the characteristics of wet and dry compensation in different regions is actually a joint probability distribution problem between variables. At present, the commonly used two-variable joint probability distribution models include two-variable normal distribution model, mixed Gumbel model, two-variable Gumbel-logistic model, exponential distribution model, Pearson type III model, Farlier-Gumbel-Morgenstern model, etc. These models are established based on the linear correlation between random variables, and the correlation between variables is measured by the linear correlation coefficient, which will lead to wrong conclusions when used to describe the nonlinear correlation problem. However, various random variables in the field of hydrology science often show complex linear and nonlinear correlations, so the above linear correlation variable distribution model cannot accurately describe the joint distribution between variables. Therefore, Copula function is introduced to describe the correlation between variables. Copula function can connect the joint distribution function and its respective edge distribution function. Based on the nonlinear correlation between variables, Copula function can describe the nonlinear, asymmetric and symmetric correlation between variables, which is widely used in hydrology.

Copula theory was first introduced by Sklar in 1959, and subsequently Nelsen provided a detailed introduction to Copula theory (Nelsen 2000; Kimberling 2016). In recent years, Copula theory has been widely used in the field of hydrology to solve the problem of multivariate joint distribution (Chowdhary et al. 2011; Lee et al. 2013; Fernández Sánchez & Trutschnig 2015; Guo et al. 2016; Razmkhah et al. 2016; Rizwan et al, 2019; Wu et al. 2021). Chen & Guo (2019) established two four-dimensional Copula functions according to the joint distribution of flood magnitude and occurrence date, and calculated the coincidence probability of flood magnitude and occurrence date. Kuai et al. (2014) used Copula theory to construct the two-dimensional joint distribution function of the natural inflow of the two reservoirs in the whole year, flood season and non-flood season, and analyzed the streamflow encounter characteristics of the two reservoirs at three time scales. Yu et al. (2017) analyzed the influence of different Copula function parameter estimation methods on the selection of function types and the encounter of annual streamflow, which provided a theoretical basis for water resources development and water supply dispatching management in Xi’ an. Abdollahi et al. (2019) significantly optimized the modeling of complex hydrological phenomena using the Copula function for multivariate probabilistic analysis of hydrological elements. Based on the long series of streamflow data of Dahuofang Reservoir, the Copula function and mathematical statistics method are used to analyze the streamflow compensation characteristics of Dahuofang Reservoir, and the diversion dispatching line of streamflow compensation conditions in wet, moderate and dry years is formulated (Zhao et al. 2020).

Panjiakou Reservoir is located at the junction of Tangshan City and Chengde City in Hebei Province. It is the source of the Luan River Diversion Project, which brings distributable water to Tianjin and Tangshan through the Luan River Diversion Project; Yuqiao Reservoir is an important hub project in Tianjin, which undertakes the task of water supply from Luan River Diversion Project to Tianjin; Douhe Reservoir is a comprehensive water conservancy hub for domestic water and industrial and agricultural production water in Tangshan City. In recent years, with the development of social economy, the shortage of water resources in Tianjin and Tangshan is becoming more and more serious, and the inflow of each reservoir shows a downward trend. Therefore, it has great significance to analyze the compensation characteristics between Panjiakou, Yuqiao and Douhe reservoirs for the joint scheduling of reservoirs and the rational allocation of water resources.

The joint scheduling between Panjiakou Reservoir, Daheiting Reservoir and Taolinkou Reservoir is mainly reflected in the agricultural irrigation water supply to the downstream of Luan River:

• (1)

  The Taolinkou Reservoir uses the remaining indicator water volume of Qinhuangdao City to increase the water supply to the downstream agriculture of the Luan River through the Qinglong River channel, and reduces the water supply to the downstream agriculture of the Luan River from the Panjiakou Reservoir and Daheiting Reservoir, thus increasing the water supply to Tianjin City or Tangshan City.

• (2)

  Using the remaining index water volume in Tianjin or Tangshan City, increase the agricultural water supply from Panjiakou and Daheiting Reservoirs to the lower reaches of the Luan River, the reduced agricultural water supply from Taolinkou Reservoir to the lower reaches of Luanhe River is supplied to Qinhuangdao City through the water diversion project from Qing to Qin.

Since Daheiting Reservoir is located 30 km downstream of the main dam of Panjiakou Reservoir on the main stream of the Luan River, it is close to the geographical location and climatic conditions, so the flow of the two reservoirs is basically the same abundance and depletion. Therefore, the compensation characteristics of inflow streamflow between Panjiakou Reservoir and Taolinkou Reservoir are analyzed. Figure 1 shows the water relations, geographical location and receiving cities of reservoirs. The hydraulic relationship between reservoirs is shown in Figure 2.

In 1959, Copula function theory was proposed, Sklar suggested that a joint distribution function could be decomposed into a Copula function and K marginal distribution functions, and the decomposed Copula function could be used to describe the correlation between variables; that is, a function that could relate the joint distribution to its respective marginal distribution function. The N-element Copula function has the following properties:

• (1)

  Nondecreasing for any variable ⁠,⁠,

• (2)

  ⁠,

• (3)
  For any variable ⁠, ⁠:

  

  (1)
• (4)

  

There are many kinds of commonly used Copula functions. This paper mainly introduces multivariate normal Copula function, Archimedes Copula function and extreme value Copula function. Commonly used Archimedes Copula functions include Clayton Copula function, Gumbel Copula function and Frank Copula function.

Because the multivariate normal Copula function has the characteristics of symmetry, it requires that the correlation structure between variables is the same, or the correlation coefficient between variables is very close, so it is difficult to fit the asymmetric correlation of variables.

The Clayton Copula function, Gumbel Copula function and Frank Copula function are commonly used bivariate Archimedes Copula functions, which can be expanded as N-ary Archimedes Copula functions respectively:

• (1)
  The expression of N-element Gumbel Copula function is:

  

  (6)
• (2)
  The expression of N-element Clayton Copula function is:

  

  (7)
• (3)
  The expression of N-element Frank Copula function is:

  

  (8)

  where are relevant parameters.

The analysis of the correlation between variables by Gumbel Copula function, Clayton Copula function and Frank Copula function covers various situations of the change of correlation structure, which can basically comprehensively describe the complex correlation between variables. By calculating their relevant parameter values, it is easy to get the correlation measure values that people pay attention to, especially the tail correlation measure values, so they are most widely used in reality.

Extreme Copula function is suitable for analyzing the correlation between variables caused by extreme events, so extreme Copula functions are increasingly used in the measurement of extreme event correlation.

In general, multivariate normal Copula function usually describes the correlation between variables, but due to the symmetry of this function, it is difficult to fit the asymmetric correlation of variables.

The Gumbel Copula function and Clayton Copula function have the characteristics of asymmetry. The Gumbel Copula function is ‘J’ shaped distribution, the lower tail is low and the upper tail is high. It is not very sensitive to the change of the lower tail of the hydrological variables, but sensitive to the distribution of the upper tail. Therefore, it is difficult to describe the relevant changes of the lower tail. When one hydrological variable appears to be a maximum, the probability of the other two hydrological variables also appearing to be a maximum increases.

The Clayton Copula function is an ‘L’ shaped distribution, and the lower tail is high and the upper tail is low. It is sensitive to the change of the lower tail of the hydrological but not sensitive to the change of the upper tail. Therefore, it is difficult to describe the related changes of the upper tail. That is, when a hydrological variable appears to be a minimum, the probability of the other two hydrological variables also appearing to be a minimum increases.

The Frank Copula function is a ‘U’ shape distribution, it has the characteristics of symmetry, so it is difficult to describe the asymmetric relationship between hydrological variables. The function is only suitable for describing the correlation between variables with symmetric correlation structure; that is, the maximum correlation and the minimum correlation between each hydrological variable are symmetric growth, but the distribution of the tail variable is relatively independent, so the Frank Copula function is not sensitive in describing the correlation between the upper tail and the lower tail, so it cannot describe the correlation of the tail change.

Gumbel Copula function, Clayton Copula function and Frank Copula function are three kinds of commonly used Archimedes functions that can capture the tail dependence: the correlation of the upper tail, the correlation of the lower tail, and the symmetric correlation of the upper tail and the lower tail. These Copula functions can describe the relationship between the various hydrological variables, especially the tail correlation. The relationship between the variables in the hydrological system is complex and changeable, not confined to a specific relationship. They can only reflect a certain aspect of the correlation between hydrological variables, so it is difficult to use a simple function to comprehensively describe the correlation pattern between variables in the hydrological system. Therefore, a more flexible function needs to be constructed to describe the relationship between various hydrological variable patterns.

The that makes the maximum value taken by the likelihood function is the maximum likelihood estimate.

According to the inflow streamflow data of Panjiakou, Douhe and Yuqiao reservoirs from 1956 to 2000, the annual average annual distribution of annual streamflow of each reservoir is analyzed, as shown in Figures 2–4.

From Figures 3–5, it can be seen that the streamflow distribution of Panjiakou, Douhe and Yuqiao reservoirs is decreasing year by year. Due to the rapid development of industry and agriculture, the water consumption increases annually, which makes the joint operation of reservoir water supply imminent.

The ⁠, and parameters of streamflow samples from Panjiakou, Douhe and Yuqiao reservoirs are shown in Table 1.

According to the calculation results in Table 1 and formulas (11)-(15), each correlation parameter and test evaluation value of Gumbel-Copula, Clayton-Copula, Frank-Copula and M-Copula functions were calculated for Panjiakou, Steep River and Yuqiao reservoirs, respectively. The calculation, test and evaluation results of the function are shown in Table 2 (K-S test is significant at 0.05 level).

Table 2 shows that Gumbel-Copula, Clayton-Copula, Frank-Copula and M-Copula functions can fit well the edge distribution of reservoir streamflow series by K-S test. According to the minimum criterion of root mean square error RSME, M-Copula function is selected as the connection function to fit the streamflow joint distribution of the three reservoirs.

The wet and dry indexes of the reservoir are divided into three levels : wet, moderate and dry, and the guarantee rate of the corresponding reservoir is divided by the transcendental probability. The wet and dry indexes are respectively and ⁠. When the M-Copula function is used to calculate the encounter of wet and dry, the above indexes are transformed into the cumulative probability; namely, ⁠. The specific division of the three reservoirs' flow wet and dry index values are shown in Table 3.

Based on the above classification criteria, the M-Copula connection function is used to analyze the streamflow series wet and dry compensation of Panjiakou, Douhe and Yuqiao Reservoirs. There are 27 kinds of wet and dry compensation encounter forms. Equations (21)–(23) lists three types of wet-dry encounter calculation formulas for Panjiakou, Douhe and Yuqiao reservoirs respectively, for wet-moderate-dry, dry-wet-wet and moderate-wet-moderate. The same applies to other cases, and the specific forms and calculation results of each are shown in Table 4.

From the analysis of 27 kinds of wet and dry encounters of streamflow series of three reservoirs in Table 4, it is known that :

The probabilities of Panjiakou, Douhe and Yuqiao reservoirs being in the same wet, moderate and dry water years are 18.25%, 4.08% and 16.53%, respectively, which means the probability of synchronous of wetness-dryness is 38.86%, and the probability of asynchronous wetness-dryness is 61.14%. The high probability of asynchronous wetness-dryness relative to synchronous wetness-dryness indicates that the three reservoirs have certain mutual compensation ability, which provides more favorable conditions for joint water supply scheduling of the reservoirs.

When Panjiakou Reservoir is a wet year, and at least one of Douhe Reservoir and Yuqiao Reservoir is a combination of moderate or dry years, Panjiakou Reservoir has a certain compensation capacity for Douhe Reservoir and Yuqiao Reservoir, namely: wet-moderate-moderate, wet-moderate-dry, wet-moderate-wet, wet-wet-moderate, wet-wet-dry, wet-dry-moderate, wet-dry-wet, wet-dry-dry, its probability is 20.77%. When the inflow of Panjiakou Reservoir is a moderate year, and there is at least one combination of moderate or dry years in Douhe Reservoir and Yuqiao Reservoir, Panjiakou Reservoir has a certain compensation ability for Douhe Reservoir and Yuqiao Reservoir, that is: moderate-moderate-moderate, moderate-moderate-dry, moderate-moderate-wet, moderate-wet-moderate, moderate-wet-dry, moderate-dry-moderate, moderate-dry-wet, moderate-dry-dry, its probability is 19.1%.

When the inflow of Panjiakou Reservoir is dry year, and one of Douhe Reservoir and Yuqiao Reservoir appears dry year or two reservoirs are both dry year, Panjiakou Reservoir has no compensation ability for Douhe Reservoir and Yuqiao Reservoir. There are several combinations as follows: dry-dry-dry, dry-moderate-dry, dry-wet-dry, dry-dry-wet, dry-dry-moderate, its probability is 25.77%.

Through the correlation analysis of the long series runoff of the three reservoirs, it is shown that the correlation coefficients of Panjiakou Reservoir to Douhe Reservoir and Yuqiao Reservoir are 0.28 and 0.21 respectively, and the correlation coefficient of Douhe Reservoir to Yuqiao Reservoir is 0.57, which proves that it has a certain correlation (due to limited space, the process of correlation analysis was not listed). However, how the reservoirs are complementary needs to be further calculated by the method proposed in this paper.

The analysis of the wetness-dryness compensation characteristics of the downstream reservoirs in the Luan River basin shows that mixed Copula function can describe the correlation between variables in hydrological system more flexibly, and Copula distribution function can well fit the correlation structure and distribution between reservoirs variables. According to the minimum criterion of RMSE, M-Copula function is selected as the connection function to fit the joint distribution of streamflow between reservoirs. Based on the analysis of the streamflow distribution of Panjiakou Reservoir, Douhe Reservoir and Yuqiao Reservoir, the probability of asynchronous wetness and dryness of each reservoir is 61.14%, indicating that there is a certain mutual compensation ability between reservoirs, which provides favorable conditions for the joint water supply scheduling of reservoirs. The probability that Panjiakou Reservoir has no compensation ability for Douhe Reservoir and Yuqiao Reservoir is 25.77%. Therefore, in order to maximize the protection of water supply security in Tianjin, Tangshan and other cities, it is necessary to give full play to the annual regulation function of reservoirs, strengthen the joint optimal operation of reservoirs in Luanhe River Basin, so as to improve the utilization rate of water resources and reduce the contradiction between supply and demand.

All relevant data are included in the paper or its Supplementary Information.

The research was supported by Major Science and Technology Special Projects in Henan Province (201300311400).