Risk analysis for the downstream control section in the real-time flood control operation of a reservoir

Abstract

Many uncertainty factors are associated with the joint operation of a reservoir and its downstream river, which create risks in flood control decisions. Therefore, this paper proposes an analytical method for the estimation of the uncertainties and their risks in real-time flood control decisions. Three uncertainty factors, including reservoir discharge errors, forecasting errors of lateral inflows and river food routing errors are proposed and modeled as stochastic processes, and their internal transforming formulas are derived based on the theory of routing before combination. The definition and calculation formulas for the risks of each moment and the integrated risk of the entire flood process at the downstream flood control section are proposed by an analytical approach based on the combination theory of stochastic processes. The Dahuofang reservoir in northern China is selected as the study case. The results indicate that the risk of the flood peak is higher than that of other moments under the same controlled flood discharge and that the risk that arises from the uncertainties of the reservoir discharge and lateral inflow is decreased by the river storage function. Compared with the Monte Carlo method, the proposed method is effective and efficient for performing risk analysis of the downstream control section in the real-time flood control operation of a reservoir. The risk analysis results could provide important information regarding flood risks for the operators to implement flood control arrangements.

Appendices

Appendix 1

Subscript t is omitted in the derivation process to facilitate the description.

$$ f[z/g(x),g(y)] = \int\limits_{ - \infty }^{ + \infty } {\frac{1}{{2\pi \sigma_{{\xi_{1} }} \sigma_{{\xi_{2} }} }}\exp \left\{ - \frac{{[z_{1} - g(x)]^{2} }}{{2\sigma_{{\xi_{1} }}^{2} }} - \frac{{[z - z_{1} - g(y)]^{2} }}{{2\sigma_{{\xi_{2} }}^{2} }}\right\} d(z_{1} )} $$

(26)

If $\frac{{z_{1} - g(x)}}{{\sigma_{{\xi_{1} }} }} = m$ and $z - g(x) - g(y) = n$, then Eq. (26) is simplified as:

$$ f[z/g(x),g(y)] = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left[- \frac{{m^{2}}}{2} - \frac{{(n - m\sigma_{{\xi_{1}}})^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}\right]} dm = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left[- \frac{{m^{2} (\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}) - 2mn\sigma_{{\xi_{1}}} + n^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}\right]} dm = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left[- \frac{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}(m^{2} - \frac{{n\sigma_{{\xi_{1}}}}}{{\sigma_{{\xi_{2}}}^{2}}} \cdot m \cdot \frac{{2\sigma_{{\xi_{2}}}^{2}}}{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}) - \frac{{n^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}\right]} dm = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left\{- \frac{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}[m^{2} - \frac{{2n\sigma_{{\xi_{1}}}}}{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}} \cdot m + \frac{{n^{2} \sigma_{{\xi_{1}}}^{2}}}{{(\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2})^{2}}} - \frac{{n^{2} \sigma_{{\xi_{1}}}^{2}}}{{(\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2})^{2}}}] - \frac{{n^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}\right\}} dm = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left[- \frac{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}(m - \frac{{n\sigma_{{\xi_{1}}}}}{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}})^{2} + \frac{{n^{2} \sigma_{{\xi_{1}}}^{2}}}{{2\sigma_{{\xi_{2}}}^{2} (\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2})}} - \frac{{n^{2}}}{{2\sigma_{{\xi_{2}}}^{2}}}\right]} dm = \frac{1}{{2\pi \sigma_{{\xi_{2}}}}}\int_{- \infty}^{+ \infty} {\exp \left[- (\sqrt {\frac{{\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}{2}} \frac{m}{{\sigma_{{\xi_{2}}}}} - \frac{{n\sigma_{{\xi_{1}}}}}{{\sqrt {\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2}}}} \cdot \frac{1}{{\sqrt 2 \sigma_{{\xi_{2}}}}})^{2} - \frac{{n^{2}}}{{2(\sigma_{{\xi_{1}}}^{2} + \sigma_{{\xi_{2}}}^{2})}}\right]} dm $$

(27)

If $\sqrt {\frac{{\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} }}{2}} \frac{m}{{\sigma_{{\xi_{2} }} }} - \frac{{n\sigma_{{\xi_{1} }} }}{{\sqrt {\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} } }} \cdot \frac{1}{{\sqrt 2 \sigma_{{\xi_{2} }} }} = h$, then Eq. (27) is simplified as:

$$ f[z/g(x),g(y)] = \frac{1}{{2\pi \sigma_{{\xi_{2} }} }} \cdot \frac{{\sqrt 2 \sigma_{{\xi_{2} }} }}{{\sqrt {\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} } }} \cdot \exp \left[ - \frac{{n^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}}\right]\int_{ - \infty }^{ + \infty } {e^{{ - h^{2} }} } dh = \frac{1}{{\sqrt 2 \pi \cdot \sqrt {\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} } }}\exp \left[ - \frac{{n^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}}\right] \cdot \sqrt \pi = \frac{1}{{\sqrt {2\pi } \cdot \sqrt {\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} } }}\exp \left\{ - \frac{{\{ z - [g(x) + g(y)]\}^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}}\right\} $$

(28)

Therefore, $f[z/g(x),g(y)] = \frac{1}{{\sqrt {2\pi } \cdot \sqrt {\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} } }}\exp \{ - \frac{{\{ z - [g(x) + g(y)]\}^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}}\}$

Appendix 2

Subscript t is omitted in the derivation process to facilitate the description.

$$P_{t,R} = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{{Q_{R} }} {f[z,g(x),g(y)]d(z)} } } d[g(x)]d[g(y)] = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{{Q_{R} }} {\frac{1}{{2\pi \sigma_{gx} \sigma_{gy} \sqrt {2\pi (\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}\exp \left\{ - \frac{{[z - g(x) - g(y)]^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}} - \frac{{[g(x) - u_{gx} ]^{2} }}{{2\sigma_{gx}^{2} }} - \frac{{[g(y) - u_{gy} ]^{2} }}{{2\sigma_{gy}^{2} }}\right\} d(z)} } } d[g(x)]d[g(y)] = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{2\pi \sigma_{gx} \sigma_{gy} }}} } \exp \left\{ - \frac{{[g(x) - u_{gx} ]^{2} }}{{2\sigma_{gx}^{2} }} - \frac{{[g(y) - u_{gy} ]^{2} }}{{2\sigma_{gy}^{2} }}\right\} \int\limits_{ - \infty }^{{Q_{R} }} {\frac{1}{{\sqrt {2\pi (\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}} \exp \left\{ - \frac{{[z - g(x) - g(y)]^{2} }}{{2(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )}}\right\} d(z)d[g(x)]d[g(y)] $$

(29)

If $z = \sqrt {(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} h + g(x) + g(y),$ then Eq. (29) is simplified as:

$$ P_{t,R} = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{2\pi \sigma_{gx} \sigma_{gy} }}} } \exp \{ - \frac{{[g(x) - u_{gx} ]^{2} }}{{2\sigma_{gx}^{2} }} - \frac{{[g(y) - u_{gy} ]^{2} }}{{2\sigma_{gy}^{2} }}\} \int\limits_{ - \infty }^{{\frac{{Q_{R} - g(x) - g(y)}}{{\sqrt {(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}}} {\frac{1}{{\sqrt {2\pi } }}} \exp ( - \frac{{h^{2} }}{2})d(h)d[g(x)]d[g(y)] $$

(30)

If $\frac{{g(x) - u_{gx} }}{{\sigma_{gx} }} = m$ and$\frac{{g(y) - u_{gy} }}{{\sigma_{gy} }} = n,$ then Eq. (30) is simplified as:

$$ P_{t,R} = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\frac{1}{2\pi }} } \exp \left\{ - \frac{{m^{2} }}{2} - \frac{{n^{2} }}{2}\right\} \int\limits_{ - \infty }^{{\frac{{Q_{R} - m\sigma_{gx} - u_{gx} - n\sigma_{gy} - u_{gy} }}{{\sqrt {(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}}} {\frac{1}{{\sqrt {2\pi } }}} \exp \left( - \frac{{h^{2} }}{2}\right)dhdmdn = \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\frac{1}{2\pi }} } \exp\left \{ - \frac{{m^{2} }}{2} - \frac{{n^{2} }}{2}\right\} \{ \frac{1}{2} + \int\limits_{0}^{{\frac{{Q_{R} - m\sigma_{gx} - u_{gx} - n\sigma_{gy} - u_{gy} }}{{\sqrt {(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}}} {\frac{1}{{\sqrt {2\pi } }}} \exp \left( - \frac{{h^{2} }}{2}\right)dh\} dmdn = 0.5 + \frac{1}{{(\sqrt {2\pi } )^{3} }}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{0}^{{\frac{{Q_{R} - m\sigma_{gx} - u_{gx} - n\sigma_{gy} - u_{gy} }}{{\sqrt {(\sigma_{{\xi_{1} }}^{2} + \sigma_{{\xi_{2} }}^{2} )} }}}} {\exp ( - \frac{{h^{2} + m^{2} + n^{2} }}{2})} } } dhdmdn $$

(31)

If $Q^{\prime}_{R} = \frac{{Q_{R} - m(t)\sigma_{gx} (t) - u_{gx} (t) - n(t)\sigma_{gy} (t) - u_{gy} (t)}}{{\sqrt {\sigma_{{\xi_{1} }}^{2} (t) + \sigma_{{\xi_{2} }}^{2} (t)} }},$ then Eq. (31) is simplified as:

$$ P_{t,R} = 0.5 + \frac{1}{{(\sqrt {2\pi } )^{3} }}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_{0}^{{Q^{\prime}_{R} }} {\exp ( - \frac{{h^{2} + m^{2} + n^{2} }}{2})} } } dhdmdn $$

(32)