194. Evaluation of a NYC School-Based Health Center Providing Comprehensive Reproductive Health Care

Water Resour Manage DOI 10.1007/s11269-016-1515-3

The Dynamic Control Bound of Flood Limited Water Level Considering Capacity Compensation Regulation and Flood Spatial Pattern Uncertainty Qiao-feng Tan 1 & Xu Wang 2 & Pan Liu 3 & Xiao-hui Lei 2 & Si-yu Cai 2 & Hao Wang 2 & Yi Ji 4

Received: 5 June 2016 / Accepted: 10 October 2016 # Springer Science+Business Media Dordrecht 2016

Abstract The dynamic control bound of flood limited water level (FLWL) is a fundamental and key element for implementing reservoir FLWL dynamic control. Due to the uncertainty of the inflow and the dimensional increase of the reservoirs, the calculation of the dynamic control bound of FLWL becomes more and more complicated. A new model that considers capacity compensation regulation and the uncertainty of flood spatial pattern (FSP) for a serial multipurpose reservoir system is developed to calculate the dynamic control bound of FLWL. This model consists of three modules: a compensation regulation module to analyze the feasibility to raise the FLWL and calculate the probable maximum upper bound of the FLWL, a risk control module containing a risk constraint to control flood risk and a Copula function to describe the uncertainty of the FSP, and a simulation operation module to simulate the flood control operation for cascade reservoirs. The proposed model was applied to Pankou-Huanglongtan cascade reservoirs in Du River basin. The application results showed that: 1) the proposed model could give a sufficient consideration about the uncertainty of the FSP, thus a safe and reasonable dynamic control bound of FLWL was derived. 2) the upper FLWL of Huanglongtan reservoir could rise up to 247.64 m from 247.00 m without increasing flood control risk and Huanglongtan reservoir could generate 9 and 7 billion kW.h extra hydropower energy during flood season in wet year 2000 and in dry year 1994, respectively.

* Xu Wang [email protected]

1

College of Water Resources and Hydropower, Sichuan University, Chengdu 610065, China

2

State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

3

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

4

Key Laboratory of Beijing for Water Quality Science and Water Environment Recovery Engineering, College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China

Tan Q.-f. et al.

Keywords Flood limited water level . Dynamic control bound . Capacity compensation . Flood spatial pattern . Uncertainty

1 Introduction With economic and social development, the contradiction between water supply and demand becomes increasingly acute. Reservoirs play an important role in solving the shortage of water resources by altering the spatial and temporal distribution of runoff. A multi-purpose reservoir serves for various purposes, such as hydropower generation, flood control, navigation, recreation and so on (Ahmed and Sarma 2005; Labadie 2005;Eum and Simonovic 2010; Ostadrahimi et al. 2012; Zhou and Guo 2013). Some purposes may conflict with one another, such as flood control and hydropower generation, especially in the flood season (Xu et al. 1997;Long et al. 2008). Usually, the higher the water level is, the more generation will be produced while the more flood control capacity will be sacrificed. In order to balance conflicts between flood control and water conservation, the Chinese Flood Control Act prescribes that the water level of reservoir should be kept below the FLWL during flood season to provide a storage enough for flood prevention (the Standing Committee of the Eighth National People’s Congress of the People’s Republic of China 1997). To improve the economic effect of reservoirs in the flood season without lowering the flood prevention standard, many advanced measures are proposed to manage the FLWL. In short, the development of the FLWL management can be divided into three stages. In the first stage, the FLWL is a fixed value in the entire flood season which is so called the static control of FLWL (SC-FLWL). It is simple to determine a FLWL in this stage, but it neglects the annual and seasonal variation of inflow and wastes a lot of water resources, often resulting in the reservoir being unable to be refilled to the normal water level by the year end(Yun and Singh 2008; Zhou et al. 2014). In the second stage, the flood season is divided into several subseasons, and each sub-season can obtain an independent FLWL (Liu et al. 2015). It improves the utilization of water resources to some extent, but still belongs to static control method without using the hydrological forecast information. In the last stage, the FLWL can fluctuate within a given range based on the hydrological forecast information and operation rules in the flood season, which is so called the dynamic control of FLWL (DC-FLWL) (Zhou et al. 2009; Li et al. 2010;Chen et al. 2013; Zhou et al. 2014). It is an effective way to improve the utilization of flood resources and increase hydropower generation without decreasing the flood prevention standard. The bound of DC-FLWL is generated in the planning phase prior to its application in the real-time operation. In order to control the flood risk in the real-time operation, the uncertainty of inflow should be considered in the design stage. Li et al.(2010) proposed a dynamic control bound calculation model that considered inflow forecasting error and uncertainty of the flood hydrograph shape. Zhang et al. (2011) used the maximum entropy principle to analyze the distribution pattern of flood forecast error and studied the risk of dynamic control of reservoir FLWL within different flood forecast error bounds. In order to improve the total benefits of cascade reservoirs, researchers in China began to study the dynamic control of FLWL for cascade reservoirs. For example, Chen et al. (2013) developed a joint operation and dynamic control model of FLWLs for cascade reservoirs. Zhou et al. (2014) developed a joint operation and dynamic control model of FLWLs for the Three Gorges and Qingjiang cascade reservoirs, which extended the dynamic control of FLWL from a single reservoir or cascade reservoirs to

The Dynamic Control Bound of Flood Limited Water Level

a mixed reservoir system. In the recent years, more and more reservoirs have been built in China, and using the compensation regulation between the cascade reservoirs is an effective way to raise water resources efficiency and flood control ability of the basin. However, due to the complicated hydraulic connection, the DC-FLWL is complicated for cascade reservoirs. The uncertainty of inflow becomes more and more complex as the dimension of reservoirs increases. Literatures involving such problems are very limited. In order to control flood risk, the bound of the DC-FLWL is always determined in the design stage by carrying flood operation calculation on the design flood according to the given operation rules. The specification for calculating design flood in China prescribes that the spatial pattern of the design flood should be determined when there is another hydropower engineering with a high regulation performance upon the design section. The flood control risk may be underestimated if we do not take full consideration about the spatial pattern uncertainty or overestimated if the unreasonable FSPs are considered. The coming of the Copula function, which is widely used in the hydrology and water resources field (Favre et al. 2004; Singh and Zhang 2006; Jenq–Tzong et al. 2007; Chowdhary et al. 2007; Karmakar and Simonovic 2009), thanks to its advantage in setting up the joint distribution of multivariable, makes it possible to acquire experiential information from the historical flood data to describe the uncertainty of the FSP. In this paper, Twodimensional Gumbel-Hougaard Copulas function was used to describe the uncertainty of the FSP, thus a new model considering capacity compensation regulation and the uncertainty of FSP for a serial multipurpose reservoir system is developed to calculate the dynamic control bound of FLWL. The Pankou-Huanglongtan cascade reservoirs, which lie in Du River, a branch of Han River basin was selected as a case study. The next section will introduce the proposed methods. Study area is introduced in Section 3 and the application results are presented in Section 4. The conclusions are drawn in Section 5.

2 Methods The proposed method mainly consists of three modules: a compensation regulation module to analyze the feasibility to raise the FLWL and calculate the probable maximum upper bound of the FLWL; a risk analysis module to control flood risk in the design stage and a simulation operation module to simulate the flood control operation for cascade reservoirs.

2.1 Compensation Regulation Module When the upstream reservoir reserves a part of flood control capacity for the downstream reservoir, it can share the flood control pressure for the downstream reservoir. Assume that: the release and the highest water level of the downstream reservoir B during flood operation is unchanged. If the FLWL is raised from the original FLWL ZB , 0 to Z'B , 0 by using the reserved capacity of upstream reservoir A, the correspondingly elevated FLWL of B is:

  Z 0 B;0 ¼ Z V Z B;0 þ ΔV

ð1Þ

where V(*) is the function to convert the water level to storage; Z(*)is the function to convert the storage to water level; ΔVis the reserved capacity of upstream reservoir A for the downstream reservoir B.

Tan Q.-f. et al.

Considering that the FLWL cannot increase unlimitedly, this paper defines the probable maximum FLWL as: 

   Z max B;0 ¼ min Z V Z B;0 þ ΔV ; Z lit max ð2Þ where ZmaxB , 0 is the probable maximum FLWL of reservoir B only considering the capacity compensation of reservoir A and the water level constraint of the reservoir itself with no reference to the uncertainty of the inflow, and Zlitmax is the limited highest initial dispatching water level of the reservoir B. Usually, Zlitmax can be calculated by the following expression:
 Z lit max ¼ min Z flood ; Z design ; Z check

ð3Þ

where Zflood, Zdesign and Zcheck are three characteristic water levels of reservoirs in China. Zflood called the upper water level for flood control, is the highest water level during operation when the reservoir meets a flood which is the downstream flood control standard; Zdesign called design flood water level, is the highest water level during operation when the reservoir confronts a design standard flood; Zcheck called check flood water level, is the highest water level during operation when the reservoir confronts a check standard flood. Taking the minimum of those three characteristic water levels as the value of Zlitmax, the safety of the downstream flood control points and the reservoir itself can be ensured, while the reservoir submergence can be avoided.

2.2 Risk Control Module 2.2.1 Risk Constraint Description The general sketch of cascade reservoirs is illustrated in Fig.1, where A and B are the upstream and downstream reservoirs respectively. X and Y are the natural inflows of the reservoir A and interval area between reservoir A and B. Z is the natural inflow of the reservoir B not considering the regulation function of the reservoir A. To ensure the safety of flood control, when the downstream encounters a flood zp with frequency p, the highest water level of the downstream reservoir during flood operation should be lower than the limited highest water level, no matter how zp is allocated between the upstream and the interval area. Therefore, the risk constraint is: Fig. 1 The sketch of cascade reservoirs

X

Y

Z

The Dynamic Control Bound of Flood Limited Water Level

8 <

zp ¼ xi þ yi ; i ¼ 1; 2:::n; ∀X ¼ xi ; xi ∈½xl ; xu ; Z i;max ≤Z p lit;max; : xi > 0; yi > 0;

ð4Þ

where xi and yi are the floods of the upstream and the interval area whenzpis allocated by ith FSP, and Zi , max is the corresponding highest water level of the downstream reservoir during flood operation; n is the number of discrete ; Zplit , max is the limited highest water level of the downstream reservoir when it encounters a flood with frequency p; xl and xu are the minimum and the maximum floods the upstream area may meet when the downstream reservoir encounters a flood zp. The method based on Copula function to determine the [xl, xu] will be described in the following chapter.

2.2.2 Confidence Interval Computation Gumbel-Hougaard Copula is proved to be effective to establish the joint distribution FX , Y(x, y) of the floods (Karmakar and Simonovic 2009; Yan et al. 2010). Two-dimensional GumbelHougaard Copulas function expression is:  h i1=θ  ð5Þ F X ;Y ðx; yÞ ¼ PðX ≤x; Y ≤yÞ ¼ C ðu; vÞ ¼ exp − ð−lnuÞθ þ ð−lnvÞθ where u = FX(x),v = FY(y) are the marginal distributions of the stochastic variable X and Y; θ is the structural parameter of the Copula function. The corresponding density function is: cθ ðu; vÞ ¼ C θ ðu; vÞ

o ðlnulnvÞθ−1 n ðθ−1Þ* ½−lnC θ ðu; vÞ1−2θ þ ½−lnC θ ðu; vÞ2−2θ uv

ð6Þ

In China, the Pearson Type III (PTIII) distribution is commonly employed to construct the distributions of the hydrologic variables (Yun and Singh 2008). The probability density functions (PDFs) of X and Y are: β x αx ðx−γ x Þαx −1 e−βx ðx−γx Þ Γ ðαx Þ

ð7Þ

αy −1 β Y αy 
 y−γ y e−βy ðy−γy Þ Γ αy

ð8Þ

f X ð xÞ ¼

f Y ðyÞ ¼

where αx and αy are the shaper parameters of the PTIII distribution of X and Y; βx and βy are the scale parameters; γx and γy are the location parameters; and Γ(*)is a gamma function. The joint density function fx , y(x, y) of the stochastic variable X and Y is:

*

 f X ;Y ðx; yÞ ¼ f X ;Y x; zp −x ¼ f X ðxÞ* f Y zp −x cθ F X ðxÞ; F y zp −x

ð9Þ

The normalized joint density function is then obtained, as shown in Fig.2:

gðxÞ ¼

Zzp



*
 cθ F X ðxÞ; F y zp −x f X ðxÞ* f Y zp −x





cθ F X ðxÞ; F y zp −x 0

 *


 f X ðxÞ* f Y zp −x dx

ð10Þ

Tan Q.-f. et al.

1−α

g ( x)

Fig. 2 The sketch of the normalized joint density function g(x)

... α1 ( x1) xl x 2 x3 x 4

... α2 xi xi + 1 xn − 1 xu ( xn) x

Assuming that the probability that the upstream flood volume falls into the range [0, xl] is α1, and that for the range [xu, zp] is α2, then α = α1 + α2 can be called significant level. That is to say: Zxl gðxÞdx ¼ α1 ð11Þ 0

Zzp g ðxÞdx ¼ α2

ð12Þ

xu

Pðxl ≤X ≤xu Þ ¼ 1−ðα1 þ α2 Þ ¼ 1−α

ð13Þ

α is a balance index of risk and benefit. The smaller value theαis, the more risk is taken into consideration while the benefit of the reservoir may be sacrificed. Local operators can set the value of α by analyzing the characteristic of the historical floods or the physical reasons of local floods. The reasonable range of the upstream flood [xl, xu] can be calculated under a given significant level α, which can be described as [xl, xu]α.

2.3 Simulation Operation Module for Cascade Reservoirs The general framework of joint operation for cascade reservoirs is shown in Fig.3. The steps are: (1) Calculate the PDFs for upstream flood X (fx(x)) and the interval area flood Y (fx(x)) respectively. Then construct the normalized joint distribution function g(x) of two by using a Copula function. (2) Calculate the confidence interval [xl, xu]α under a given significant levelαand a certain design floodzpwith frequency p. (3) Disperse the confidence interval [xl, xu]α into a total of n valuesx1 , x2 , . . . , xn. Thus derive a total of n FSPs, where X = xi,Y = zp − xi, and i = 1 , 2 . . . n (4) Simulate the operation of the reservoir A according to the given flood operation rules and obtain the outflow of the reservoir A. Assuming that the range of DC-FLWL for reservoir B is [Zlow, Zup]. The initial upper bound is calculated by the compensation regulation module, and the lower bound remains unchanged from that of original design. (5) Add the outflow of the reservoir A to the interval flood as the inflow of the reservoir B considering the time difference of the flood routing, then simulate the operation of the reservoir B based on the assumed bound of DC-FLWL. (6) If the simulation result satisfies the risk constraint, let Zup = Zup ‐ 0.01and go back Step (4). If the risk constraint is not satisfied, stop and output the final range [Zlow, Zup].

The Dynamic Control Bound of Flood Limited Water Level Fig. 3 The framework of the simulation operation for cascade reservoirs

In China, the FSP is usually set by typical year spatial pattern method or homogeneous frequency spatial pattern method. In order to compare with various FSPs, the study set four kinds of FSP: I Homogeneous frequency spatial pattern I: flood in upstream area is with a same frequency as the design section, and corresponding interval flood is calculated by water balance; II Homogeneous frequency spatial pattern II: flood in interval area is with a same frequency as the design section and corresponding upstream flood is calculated by water balance; III Typical year spatial pattern: set the FSP according to the chosen typical flood; IV Given a significance level and obtaining a confidence interval, the confidence interval was dispersed into many sub-ranges, then obtained different spatial patterns. Scheme IV can get the upper FLWL by following the step (1) to step (6), while the other schemes just need to follow the step (4) to step (6).

3 Study Area 3.1 Study Area and Data Pankou reservoir (PK) and Huanglongtan reservoir (HLT) are cascade reservoirs on the Du River (see Fig.4), an important branch of the Han River in China. The downstream HLT was built in 1976, which controls a drainage area of 11,892 km2, about 95.1 % of the Du River basin area. The upstream PK began to store water in 2009, with a drainage area of 8950 km2, about 71.6 % of the Du River basin area. The distance between the two reservoirs are about

Tan Q.-f. et al.

Fig. 4 Location of PK and HLT in China

108 km. The interval area between Pankou and Huanglongtan reservoir (PK-HLT) is over 3008 km2, which is 24.9 % of the catchment area controlled by HLT. Reservoirs’ characteristic parameters are listed in Table 1. The adopted inflow series in flood season of PK and HLT were from 1959 to 2015, and the data was collected on three-hour basis. The flood season of Du River basin is from 1st May to 30th September totally of 153 days (1244 periods of time). Zhushan is the nearest hydrologic station downstream the PK. As the inflow of PK-HLT was not measured, the interval inflow was calculated by subtracting the routed quantity of outflow of PK from the inflow of HLT. Figure 5 shows the inflow of Zhushan, HLT and corresponding derived interval inflow in 2000 wet year and 1994 dry year. The storms affecting PK and HLT can be divided into two major categories: frontal rain and typhoon rain. The former can cover the whole river basin upon HLT, while the latter only has an effect on the downstream river basin. This study chose two typical floods: a flood happening in 1964 and formed by frontal rain; a flood happening in 1975 and formed by typhoon rain. The measured maximum three-day flood volume of HLT occurred in 1964 typical flood and the maximum three-day flood volume of HLT, PK and PK-HLT are 13.76, 11.10 and 2.68 hundred million m3 respectively. The largest interval flood occurred in the Table 1 The characteristic water levels of PK and HLT reservoirs Reservoir Dead pool level (m)

Original FLWL (m)

Design flood level(m)

Check flood level (m)

Crest elevation (m)

PK

330.00

347.60

357.14

360.82

362.00

HLT

226.00

[240.00,247.00]

248.20

251.70

252.00

The Dynamic Control Bound of Flood Limited Water Level 1994 dry year

4000

HLT

Inflow(m3/s)

Zhushan

3000

PK-HLT

2000 1000 0

0

200

400

Inflow(m3/s)

4000

(A)

600 Time(3h)

800

1000

2000 wet year

1200

HLT

3000

Zhushan PK-HLT

2000 1000 0

0

200

400

(B)

600 Time(3h)

800

1000

1200

Fig. 5 The inflow of HLT, Zhushan and PK-HLT (a) in 1994 dry year, (b) in 2000 wet year

typical flood in 1975 and the maximum three-day flood volume of HLT, PK and PK-HLT are 6.35, 3.28 and 3.14 hundred million m3 respectively. Based on the chosen typical floods, the study set four kinds of FSP to calculate the design flood data: I Homogeneous frequency spatial pattern I: PK and HLT were with the same frequency; II Homogeneous frequency spatial pattern II: PK ~ HLT and HLT were with the same frequency; III Typical year spatial pattern: set the FSP according to 1964 typical flood and 1975 typical flood; IV Given a significance level of 0.05, the confidence interval was dispersed into 200 subranges, then obtained two hundred spatial patterns.

3.2 Flood Operation Rules The HLT uses flood forecasting and pre-discharge operation rule. The flood operation rule for HLT is given as follows: (1) (2) (3) (4)

The initial dispatch water level is 247.00 m; If Qin , hlt ≤ 6000m3/s, Qout , hlt = Qin , hlt; If 6000 < Qin , hlt < 1100m3/s and Zhlt > 240.00m, Qout , hlt = min (85 % * Qfore , max, Qin , hlt); If Qin , hlt > 11000m3/s, ungated operation should be carried out to protect the safety of the dam itself; (5) After the flood peak, close the gates gradually and control water level at 247.00 m. where Qin , hlt is the inflow of the HLT; Qout , hlt is the outflow; Zhlt is the water level; and Qfore , max is the maximum forecasting flow in the 24 h. PK undertakes a task to improve the flood control standard of HLT. The 3.11 hundred million m3 flood control capacity between water level 353.20 m and 358.40 m is used to share the flood control pressure with HLT. The flood operation rule for PK is given as follows:

Tan Q.-f. et al.

(1) The initial dispatch water level is 353.20 m; (2) If Qin , pk < Qmax , 353.2, Qout , pk = Qin , pk; (3) If Qin , pk > Qmax , 353.2, Qout , pk < min (Qmax, 10700)m3/s to protect the safety of the downstream area; (4) If Zpk > 358.40m, ungated operation should be carried out to protect the safety of the dam itself; (5) After the flood peak, close the gates gradually and keep the water level fixed at 353.20 m. where Qin , pk is the inflow of PK; Qout , pk is the outflow; Zpk is the water level ; Qmax is the maximum drainage ability; and Qmax , 353.2 is the maximum drainage ability at the water level of 353.20 m.

4 Results and Discussions With the building of PK, HLT can use the reserved flood control storage to raise the upper bound of the FLWL over 247.00 m. The main purpose of HLT is hydropower generation without flood control task for the downstream area. Therefore, the flood control risk constraints mainly lie in the design standard flood to ensure the safety of itself and the check standard flood to avoid submergence. The design and the check frequencies of HLT are 1 % and 0.2 % respectively.

4.1 Marginal Distribution and Normalized Joint Probability Density Function HLT is a seasonal regulation reservoir, while PK is an annual regulation reservoir. For big reservoirs, the maximum three-day flood volume is chosen to be a flood control indicator. The PTIII distribution statistical parameters of flood variables at HLT, PK and their interval area as well as the corresponding goodness-of-fit statistics are shown in Table 2, where Qm is the flood x ,Cv and Cs are the means, variation peak and W3d is the maximum three-day flood volume; ̅ coefficients and skewness coefficients of the flood variables; R is correlation coefficient of empirical and theoretic cumulative distribution function (CDF). The Chi-square test is used to detect whether PTIII distribution could be used as the marginal distributions of the flood variables. The critical value of Chi-square test at 5 % significance level is expressed as χ0.05. χ2 is an index to measure the difference between the actual frequency and theoretical frequency, and the smaller the value is ,the better of the fitting result. We can see form Table 2 that χ2 < χ0.05, thus the PTIII distributions are satisfactory. Meanwhile, Fig.6 illustrates the fitted marginal distributions for flood variables of HLT, PK and PK-HLT. Combined with Table 2 The statistical parameters of flood variables and corresponding goodness-of-fit statistics

Qm/(m3/s) W3d/ (hundred million m3)

Site or subarea

̅

x

Cv

Cs/Cv

R

χ2

χ0.05

HLT HLT

4920 6.00

0.55 0.61

2.5 3.0

0.9952 0.9948

3.24 2.23

7.81 5.99

PK-HLT

1.03

0.91

2.5

0.9920

2.65

5.99

PK

4.92

0.61

3.0

0.9917

2.78

5.99

The Dynamic Control Bound of Flood Limited Water Level

Empirical frequency

y = 1.01*x - 0.00394

0.8

Theoretic frequency(P3)

Theoretic CDF

Peak flow of HLT(m3/s)

1 15000

10000

5000

0.6 0.4 0.2

0

0 0.1

1 2

5 10

30 50 70 Probability(%)

90 95

99

99.9

Theoretic frequency(P3)

0.8

1

0.8

1

y = 0.9837*x + 0.033

0.8

25

Theoretic CDF

3-day flood v olume of HLT(10 8 m3)

0.4 0.6 Empirical CDF

Empirical frequency

30

20 15 10

0.6 0.4 0.2

5 0 0.1

1 2

5

10

30 50 70 Probability(%)

90 95

99

0

99.9

15

0

0.2

0.4 0.6 Empirical CDF

1 Empirical frequency

y = 0.9803*x + 0.01588

0.8

Theoretic frequency(P3)

10

Theoretic CDF

3-day flood v olume of PK(10 8 m3)

0.2

1

35

5

0.6 0.4 0.2

0

0.1

1 2

5

10

30 50 70 Probability(%)

90 95

99

0

99.9

7

0

0.2

0.4 0.6 Empirical CDF

0.8

1

1

6

Empirical frequency

y = 1.041*x - 0.02766

0.8

Theoretic frequency(P3)

5

Theoretic CDF

3-day flood v olume of PK-HLT(108 m3)

0

4 3 2

0.6 0.4 0.2

1 0

0 0.1

1 2

5 10

30 50 70 Probability(%)

90 95

99

99.9

0

0.2

0.4 0.6 Empirical CDF

0.8

1

Fig. 6 Illustration of fitted marginal distributions for flood characteristics: peak flow of HLT (first row), 3day flood volume of HLT (second row), 3-day flood volume of PK (third row) and 3-day flood volume of PK-HLT (forth row). First column shows the exceeding probability of fitted distributions. Second column shows the fitting results

correlation coefficients and Chi-square test results, we can find that the PTIII distribution is validated as our marginal distribution. Combined with Eq. (7) and Eq. (8), we can get the PDFs of the upstream flood X and interval area flood Y. The two-dimensional joint distribution of X and Y were established by using Gumbel-Hougaard Copula function. The structural parameter θ of the Copula

Tan Q.-f. et al.

function is 1.91 estimated based on inversion of Kendall’s τ (Reddy and Ganguli 2012). The joint distribution is:  h i1=1:91  F X ;Y ðx; yÞ ¼ PðX ≤x; Y ≤ yÞ ¼ C ðu; vÞ ¼ exp − ð−lnuÞ1:91 þ ð−lnvÞ1:91 The Kolmogorov-Smirnov (KS) test is used to detect whether the Gumbel-Hougaard Copula function can be used to establish the joint distribution of X and Y. The critical value of KS test statistic at 5 % significance level is 0.1801. The maximum deviations between observed data and the corresponding distribution is 0.1246, which indicates that the joint distribution is satisfactory. Combining with Eq. (6) to (10), the normalized joint distribution g(x) can be obtained.

4.2 Confidence Interval Computation The paper focused on the uncertainty of flood volume allocation, while the uncertainty of flood hygrograph is out of the scope of our research. So we just chose the 1964 typical flood to obtain the design flood by homogeneous multiple enlargement. The maximum three-day flood volumes of HLT with a 100-year return period and a 500-year return period are 18.80 hundred million m3 and 24.40 hundred million m3, respectively. The confidence intervals in two cases are [12.32, 16.04] and [16.48, 20.23] respectively, when the significant level αis 0.05, which means the maximum three-day flood volume of PK is 95 % likely to fall into these intervals when HLT encounters these two kinds of flood.

4.3 Calculation Result of DC- FLWL 4.3.1 Dynamic Control Bound of FLWL The initial upper bound calculated by the compensation regulation module was 248.20 m, which was actually the design flood level. Comprehensive consideration of the safety of the reservoir itself and submergence constraint, the upper FLWLs of the four kinds of FSP are shown in Table 3. The upper FLWLs calculated by Scheme I and Scheme II were higher than that of Scheme IV, which were 248.11 m, 247.82 m, and 247.64 m, respectively. The reason was that these two spatial pattern schemes could not include all the FSPs that were against flood control. Scheme IV set the significance level at 0.05 and obtained a confidence interval. By dispersing the confidence interval into 200 tiny sub-ranges, two hundred FSPs were Table 3 The upper FLWLs of the four kinds of FSP Spatial pattern schemes

Flood volume allocation

Upper FLWL (m)

Five-hundred-year design flood (hundred million m3)

A-hundred-year design flood (hundred million m3)

I

x = 20.00,y = 4.40

x = 15.50,y = 3.30

248.11

II III

x = 18.21,y = 6.19 1964:x = 19.68,y = 4.75

x = 14.26,y = 4.54 1964:x = 15.17,y = 3.66

247.82 247.93

1975:x = 12.60,y = 12.07

1975:x = 9.71,y = 9.30

247.00

IV

x∈[16.48,20.23], y = 24.40-x

x∈[12.32, 16.04], y = 18.80-x

247.64

The Dynamic Control Bound of Flood Limited Water Level

considered. Therefore, Scheme IV considered the uncertainty of the FSP, and thus was safer than the others. The upper FLWL calculated by typical year spatial pattern was much related to the choice of the typical year. If the flood volume was allocated according to the typical year of 1964, the upper FLWL was 247.93 m. Otherwise, the upper FLWL should remain unchanged at 247.00 m. The study found that the improvement of the FLWL for HLT is mainly restrained by the design standard flood. For the check standard flood, even if the upper FLWL is 248.2 m, the highest water level of HLT during flood operation is lower than the check flood level (251.7 m). Figure 7 shows the changing process of the highest water level of HLT during flood operation with the maximum three-day flood volume of PK when a design standard flood happens in HLT. The four horizontal dotted lines represent the locations of FSP Scheme I to Scheme III. The dash area represents the location of the confidence interval when a design standard flood occurs and the significance level is 0.05. The vertical dotted line represents the location of the design flood level. By iterative computation, the final upper FLWL is 247.64 m based on the simulation operation of design standard flood. We can find following results from Fig.7: (1) When the initial dispatch water level is set to be 247.80 m, the reservoir is safe and there is no submergence for the homogeneous frequency spatial pattern and the 1964 typical year spatial pattern. But for some spatial patterns included in the confidence interval, the highest water level of HLT during operation is over 248.20 m, which will result in reservoir submergence. If the uncertainty of spatial pattern is not taken into full consideration, the flood control risk will be underestimated and will result in economic loss. (2) For a single reservoir, the highest water level during operation is increased with inflow and initial dispatch water level. However, for the cascade reservoirs, the inflow of downstream reservoir is not only controlled by the release of the upstream reservoirs but the interval inflow, leading to the size of the inflow has no obvious change rule. As a result, the highest water level of the downstream reservoir HLT during flood operation has a phenomenon of irregular change. Therefore, all possible spatial patterns should be considered in the design phase to reduce the flood control risk in the real-time operation.

Fig. 7 The changing process of the highest water level of HLT during flood operation with the maximum threeday flood volume of PK when a design standard flood happens in HLT

Tan Q.-f. et al.

(3) The confidence interval can include the homogeneous frequency spatial pattern and 1964 typical year spatial pattern as well as other spatial patterns which are against the flood control. Given the fact that the design standard flood usually cover the whole river basin, the flood volume is almost proportional to the drainage area. The 1975 typical year spatial pattern is not very reasonable. When this situation is taken into consideration, the flood risk will be overestimated and the utilizable benefit will be limited. The confidence interval which is calculated by using empirical information from the historical flood data can exclude such an unreasonable spatial pattern in advance. As a result, the reserved flood control capacity offered by PK can be fully used to increase power generation benefit of HLT.

4.3.2 Allocation of Different Spatial Patterns on g(x) Figure 8 shows the allocation of the different schemes on the normalized joint density function when the HLT encounters a 100-year design flood. Following conclusions can be get: ① Homogeneous frequency spatial pattern is not the most likely case, while it is not the most unsafe FSP that is against flood control from the combination analysis with Fig. 7. ② Typical year spatial pattern is greatly influenced by the choice of the typical year. The volume allocation is highly random. Sometimes, the allocation result may be outside of the confidence interval (such as the 1975 typical year spatial pattern) unless we have a rich experience.

4.3.3 Benefit Calculation Using the additional flood control storage capacity of the PK, HLT can raise the upper bound of the FLWL to 247.64 m. The study chose a wet year in 2000 and a dry year in 1994 to calculate the hydropower generation benefit. In 1994, the power output of HLT in flood season increased from 469 million kW.h to 478 million kW.h by 9 million kW.h. In 2000, the power output of HLT in flood season increased from 661 million kW.h to 668 million kW.h by 7 million kW.h.

X

Fig. 8 The allocation of the different spatial patterns on the normalized joint density function

The Dynamic Control Bound of Flood Limited Water Level

5 Conclusions The study chose Gumbel-Hougaard Copula function to establish the joint distribution of the upstream and interval floods. A confidence interval of upstream flood volume was calculated under a given significant level. By dispersing the confidence interval, we can obtain various spatial patterns, thus obtaining the uncertain inflow of the downstream reservoir. By simulating the operation of the cascade reservoirs under a risk constraint, the dynamic control bound of the FLWL can be calculated. Combining the application results in PK-HLT cascade reservoirs, the following results can be obtained: 1. By digging empirical information from the historical flood data, a reasonable confidence interval can be obtained under a given significant level, which provides reference information to set the FSPs. This improvement avoids the blindness to allocate the volume of the research section to the sub-areas and including the unreasonable spatial patterns. 2. The proposed model can give a sufficient consideration about the uncertainty of the FSP, which can significantly reduce the flood control risk in the real-time operation. Comparing with the typical year spatial pattern and the homogeneous frequency spatial pattern, the derived dynamic control bound of FLWL is much safer and more reasonable. 3. For the HLT, the dynamic control bound of FLWL could be widened from [240.00, 247.00] m to [240.00, 247.64] m, which would increase 9 and 7 million kW.h extra hydropower generation during a wet year in 2000 and a dry year in 1994, respectively. This paper chose the PTIII distribution and Gumbel-Hougaard Copula as our marginal distribution and joint probability density function respectively. Hypothesis testing was made to justify the selections. What kinds of distributions are more suitable needs further research. Meanwhile, owing to the increasing anthropogenic activities, flood frequency analysis becomes more and more complicated because of the non-stationarity of flood series. This paper focused on the uncertainty of design flood due to the FSP uncertainty. Thus stationarity assumption was made for the marginal distributions. For the uncertainty coming from the non-stationarity of flood series need further research. On the other hand, given the uncertainty of the design flood, one should not only consider the uncertainty of the flood volume allocation, but also consider the uncertainty of flood hygrograph in the near future. Acknowledgments This paper was jointly supported by National 973 project (2013CB036406), Ministry of Environmental Protection’s Special Funds for Scientific Research on Public Causes (2013467042) and Beijing Municipal Science and Technology Project (Z141100006014049).

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