An Analytical Risk Analysis Method for Reservoir Flood Control Operation Considering Forecast Information

Abstract

Risk analysis is vital for reservoir flood control operation considering forecast because forecast uncertainties may create risks of multiple hazard events, including the single-hazard events, the union and intersection of single-hazard events (UHE and IHE). The probability of UHE and IHE are the major concern of decision-makers. This research proposes an analytical flood risk probability calculation framework for single-hazard event, UHE and IHE caused by uncertainties in flood forecasting and takes Dahuofang Reservoir, located in the Hunhe River basin, Northeast China, as a case study. In the framework, the risk is calculated by the integral of the joint probability density function (Pdf) of the risk sources over the hazard domain, which is the collection of the values of risk sources that cause hazards. The determination method of the hazard domain and the Pdf of risk sources and the procedure of risk analysis are elaborated. Results of single-hazard events show that the proposed methodology is of higher precision compared with the risk analysis method based on the law of total probability. Meanwhile, the flood risks of UHE and IHE are calculated for the reservoir operation under different levels of flood limit water level. The study provides a new method of flood risk analysis for flood control reservoir operation.

Appendices

Appendix 1

1.1 Calculation Methods of f _E(e) and f _Y(y)

• (1) The probability density function of the forecast error of cumulated net rainfall f_E(e)

With real and forecast values of cumulated net rainfall, the probability density function of the forecast error of cumulated net rainfall f_E(e) can be obtained. The common fitting method is a widely used method, which assumes that the forecast error of cumulated net rainfall follows a definite distribution, such as normal distribution, lognormal distribution, or Pearson type III distribution. However, no single distribution mentioned above has been accepted as a global standard (Perreault et al. 1999). Jaynes (1957) thought that the assumption is arbitrary and recommended the maximum entropy model (Siddall 1983). The maximum entropy model can be used to derive more generalized distributions using different constraints and provides a unified approach to derive any desired distribution for hydrometeorological variables (Chen and Singh 2018). Here, the maximum entropy model is applied to derive f_E(e). In this model, the objective is to maximize the entropy of the forecast cumulated net rainfall error E, shown as follows:

$$ {I}_{\mathrm{max}}=\underset{f_E(e)}{\max}\left[-\int {f}_E(e)\ln {f}_E(e)\mathrm{d}e\right] $$

(8)

s.t.

$$ \Big\{{\displaystyle \begin{array}{c}\underset{\varPhi }{\int }{f}_E(e)\mathrm{d}e=1\\ {}\underset{\varPhi }{\int }{e}^i{f}_E(e)\mathrm{d}e={m}_i,\left(i=1,2,\cdots, m\right)\end{array}} $$

(9)

where Φ is the feasible set of the forecast cumulated net rainfall error E; m_i is the ith order origin moment of E; m is the maximum of the order i.

f_E(e) can be obtained by Lagrange multiplier method and variation method (Mishra et al. 2020):

$$ {f}_E(e)=\exp \left({\lambda}_0+{\lambda}_1e+{\lambda}_2{e}^2+{\lambda}_3{e}^3+\cdots \right) $$

(10)

where λ₀, λ₁, λ₂, λ₃… denotes the Language multipliers and can be calculated by optimization algorithm (Siddall 1983).

• (2) The probability density function of the cumulated net rainfall f_Y(y)

For the requirements of flood control, hydrologists will estimate the probability density function or cumulative distribution function of flood volume W, i.e., f_W(w) or F_W(w). However, they rarely derive the probability density function of the cumulated net rainfall f_Y(y). Obviously, Y and W are positively correlated, as shown in Eq. (11).

$$ Y=\frac{10^3}{A}W $$

(11)

where A is the drainage area. The unit of the cumulated net rainfall Y, the drainage area A, and flood volume W are mm, km², and 10⁶ m³, respectively. We define the parameter k in Eq. (12).

$$ k=\frac{10^3}{A} $$

(12)

In Appendix 2, we derive the relationship between f_Y(y) and f_W(w) from Eqs. (11) and (12), i.e., Eq. (23).

The flood volume W follows a Pearson type III distribution, which has been recommended by the Chinese Ministry of Water Resources as a uniform procedure for flood frequency analysis in China (Zhang et al. 2015), with the formulation as follows,

$$ {f}_W(w)=\frac{\beta^{\alpha }}{\varGamma \left(\alpha \right)}{\left(w-{a}_0\right)}^{\alpha -1}\exp \left[-\beta \left(w-{a}_0\right)\right] $$

(13)

From Eqs. (13) and (23) the probability density function of the cumulated net rainfall f_Y(y) can be obtained,

$$ {f}_Y(y)=\frac{1}{k}\frac{\beta^{\alpha }}{\varGamma \left(\alpha \right)}{\left(\frac{1}{k}y-{a}_0\right)}^{\alpha -1}\exp \left[-\beta \left(\frac{1}{k}y-{a}_0\right)\right] $$

(14)

Appendix 2

1.1 Relationship between the Probability Density Function of the Flood Volume and the Cumulated Net Rainfall

According to Eq. (12) on the arbitrary interval [a, b] in the definition domain, we obtain

$$ P\left(a\le W\le b\right)=P\left( ka\le Y\le kb\right) $$

(15)

which means that

$$ {\int}_a^b{f}_W(w)\mathrm{d}w={\int}_{ka}^{kb}{f}_Y(y)\mathrm{d}y $$

(16)

By the mean value theorem (Zorich 2016), there exists w₀ ∈ [a, b], y₀ ∈ [ka, kb]

$$ {\int}_a^b{f}_W(w)\mathrm{d}w=\left(b-a\right){f}_W\left({w}_0\right) $$

(17)

$$ {\int}_{ka}^{kb}{f}_Y(y)\mathrm{d}y=\left( kb- ka\right){f}_Y\left({y}_0\right) $$

(18)

We infer from Eqs. (17) and (18) that

$$ \left(b-a\right){f}_W\left({w}_0\right)=\left( kb- ka\right){f}_Y\left({y}_0\right) $$

(19)

$$ {f}_W\left({w}_0\right)=k\times {f}_Y\left({y}_0\right) $$

(20)

Note that Eq. (20) is also tenable when b → a then

$$ \underset{b\to a}{\lim }{f}_W\left({w}_0\right)=\underset{b\to a}{\lim }k\times {f}_Y\left({y}_0\right) $$

(21)

where $\underset{b\to a}{\lim }{w}_0=a$, $\underset{b\to a}{\lim }{y}_0= ka$

Hence

$$ {f}_W(a)=k\times {f}_Y(ka) $$

(22)

It follows from Eq. (22) that

$$ {f}_Y(y)=\frac{1}{k}{f}_W\left(\frac{1}{k}y\right) $$

(23)

Appendix 3

1.1 Comparison with the traditional method

The fundamental equation of AMIT, Eq. (6), can be derived as follows

$$ {\displaystyle \begin{array}{c}P(H)=\underset{\varOmega }{\iint }{f}_{E,Y}\left(e,y\right)\mathrm{d}e\mathrm{d}y\\ {}=\underset{\sum \limits_{i=1}^{n-1}{\varOmega}^i}{\iint }{f}_{E,Y}\left(e,y\right)\mathrm{d}e\mathrm{d}y\\ {}=\sum \limits_{i=1}^{n-1}\underset{\varOmega^i}{\iint }{f}_{E,Y}\left(e,y\right)\mathrm{d}e\mathrm{d}y\\ {}=\sum \limits_{i=1}^{n-1}P\left(H,e\in \left[{e}_{i-1},{e}_i\right]\right)\\ {}=\sum \limits_{i=1}^{n-1}P\left(H|e\in \left[{e}_{i-1},{e}_i\right]\right)\times P\left(e\in \left[{e}_{i-1},{e}_i\right]\right)\end{array}} $$

(24)

where Ωⁱ is the risk domain when e ∈ [e_i − 1, e_i], and e₀ = e_min; e_n − 1 = e_max; P(e ∈ [e_i − 1, e_i]) denotes the probability that forecast error e is within the interval of [e_i − 1, e_i]; P(H|e ∈ [e_i − 1, e_i]) denotes the conditional probability that the hazard H occurs when forecast error is within the range e ∈ [e_i − 1, e_i]; P(H, e ∈ [e_i − 1, e_i]) denotes the probability that e ∈ [e_i − 1, e_i] and the hazard H coincides.

The last two equations are the basis of the risk analysis method based on the law of total probability (MLTP) (Zhang et al. 2011). That is to say, the MLTP is the discrete or simplified form of the AMIT on calculation formula of risks. In MLTP, f_E(e) also need to be derived out in the same way in Appendix 1. Despite a close connection exists between the two methods, the difference in the calculation formula of risks leads to the difference in the calculation process.

In the MLTP, the P(e ∈ [e_i − 1, e_i]) in Eq. (24) can be calculated by:

$$ P\left(e\in \left[{e}_{i-1},{e}_i\right]\right)=\underset{e_{i-1}}{\overset{e_i}{\int }}{f}_E(e)\mathrm{d}e $$

(25)

P(H| e ∈ [e_i − 1, e_i]) is usually calculated by the approximative method (Zhang et al. 2011), as follows

$$ P\left(H|e\in \left[{e}_{i-1},{e}_i\right]\right)\approx \frac{P\left(H|{e}_{i-1}\right)+P\left(H|{e}_i\right)}{2} $$

(26)

where P(H| e_i) represents the conditional probability that the hazard H occurs on the condition of e = e_i. P(H| e_i) is obtained by interpolation based on the relationship between the maximum water level and design frequency of flood, which is derived through reservoir regulation of the designed floods when forecast error is e_i (Zhang et al. 2011).

From Eq. (26), we can know that the error of MLTP mainly comes from the approximation of P(H| e ∈ [e_i − 1, e_i]). Besides, calculation error also exists in P(H| e_i) because of interpolation (Fenton 1992). MLTP can be applied in the risk analysis of single-hazard event caused by a single and double risk sources. However, subject to the restrictions and limitations of the method structure, MLTP is difficult to deal with the risk analysis of UHE and IHE.

As can be seen from the above analysis that the accuracy of AMIT is higher because there is no approximation in this method. Additionally, the introduction of the hazard domain makes AMIT easy to understand and applicable to deal with the risk analysis of UHE and IHE.

Appendix 4

1.1 Probability density function of cumulated net rainfall Y and its forecast error E

• (1) The probability density function of cumulated net rainfall forecast error f_E(e)

In this section, we use the analysis results of f_E(e) calculated by Zhang et al. (2011). By subtracting the cumulated net rainfall from the forecast value, Zhang et al. (2011) obtained the forecast error of cumulated net rainfall from 1951 to 2005 in the Dahuofang Reservoir basin. It should be noted that all the forecasts of cumulated net rainfall were calculated altogether using same model to obtain a stationary series.

In Eq. (10), the more coefficients used, the higher the accuracy. In this study, 4 parameters (i.e., λ₀, λ₁, λ₂, λ₃) are adopted and calculated by a genetic algorithm, which are −0.3645, −0.017, −0.002, −2.000 × 10⁻⁵, respectively. The pdf function f_E(e) is shown in Fig. 6 (a). When the forecast cumulated net rainfall error E is smaller than −48 mm or bigger than 48 mm, f_E(e) is smaller than 5 × 10⁻⁴. The probability of error locates within the range E∈[−48, 48] is 99.54%. So the range of [−48, 48] is identified as the domain of the forecast cumulated net rainfall error E in the Dahuofang Reservoir.

• (2) The probability density function of the cumulated net rainfall f_Y(y)

Fig. 6

The probability density function of the forecast error of cumulated net rainfall f_E(e) (a) and the 13-day cumulated net rainfall f_Y(y) (b)

Rainfall that leads to flood events during flood season in the Dahuofang River Basin usually lasts for 13 days, and correspondingly, 13-day cumulated net rainfall is selected as the risk source.

The moment method is applied to estimate the parameters in the formulation of f_Y(y) and f_W(w), shown in Eq. (13), i.e., α, β, and a₀, while parameter k can be obtained by Eq. (12). The estimated values of k, α, β, and a₀ are 0.174, 1.384, 0.003 and 0, respectively, with the distribution shown in Fig. 6 (b).

Appendix 5

1.1 The rules of Dahuofang Reservoir operation considering forecast information

The rules of Dahuofang Reservoir operation considering forecast information contain 6 steps which are shown as follows:

• (i) The initial stage

If the water level of the reservoir is between 126.4 m and 127.6 m and heavy rain or torrential rain begin on the whole basin, the conveyance tunnel and spillway should be opened.

• (ii) The stage to avoid the peak of incremental inflow

When the forecast cumulated net rainfall of the catchment between Dahuofang Reservoir and Shenyang City in 12 h is bigger than 35 mm, the equal strength rainfall is lasting, and the rolling rainfall forecast of every 6 h shows that a torrential rainfall which contains about 50 mm to 100 mm rain is coming, then the spillway should be closed to avoid the peak of incremental inflow. If the flood peak of the catchment between Dahuofang Reservoir and Shenyang City is smaller than 3750 m³/s and the flood is declining, the reservoir inflow is bigger than 2120 m³/s, and the forecast information shows that there is rainfall in 6 h to 12 h, then the spillway also should be closed to avoid the peak of incremental inflow.

• (iii) The end of the stage to avoid the peak of incremental inflow

If the forecast information shows that the incremental inflow in 3 h is bigger than 3750 m³/s, or the cumulated net rainfall of the catchment between Dahuofang Reservoir and Shenyang City is bigger than 140 mm, or the water level of the reservoir is bigger than 131.5 m, or the length of period to avoid the peak of incremental inflow is bigger than 15 h, then the stage to avoid the peak of incremental inflow is over and the main spillway should be opened.

• (iv) The stage after the period to avoid the peak of incremental inflow

If the cumulated net rainfall of the catchment between Dahuofang Reservoir and Shenyang City is bigger than 150 mm, or the cumulated net rainfall of Dahuofang Reservoir is bigger than 150 mm and the rainfall is lasting, or the peak of the catchment between Dahuofang Reservoir and Shenyang City is over and the water level of the reservoir is rising, then 5 gates of the main spillway should be opened.

• (v) The stage to regulate the design flood

When the water level of the reservoir is bigger than 136.32 m, or the cumulated net rainfall of the Dahuofang reservoir is bigger than 260 mm and the equal strength rain exists in near future, 5 gates of the main spillway and 3 gates of emergency spillway should be opened.

• (vi) The stage to regulate the check flood

When the water level of the reservoir is bigger than 136.52 m, 5 gates of the main spillway and 7 gates of emergency spillway should be opened.

Appendix 6

1.1 Figures and tables

Fig. 7

Flowchart to obtain the hazard domain Ω₁

Fig. 8

The joint probability density function of the forecast error of cumulated net rainfall and the cumulated net rainfall f_{E, Y}(e, y)

Table 3 Flood control standard and safety streamflow of protected areas by Dahuofang Reservoir

Table 4 The maximum water level of the Dahuofang Reservoir obtained by regulating the design flood of each frequency using the reservoir conventional operation method (FLWL = 126.4 m)