application of stochastic differential equations in risk assessment for flood releases
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Application of stochastic differentialequations in risk assessment for floodreleasesSHUHAI JIANG aa Nanjing Hydraulic Research Institute , 223 Guangzhou Road,210029, Nanjing, ChinaPublished online: 25 Dec 2009.
To cite this article: SHUHAI JIANG (1998) Application of stochastic differential equationsin risk assessment for flood releases, Hydrological Sciences Journal, 43:3, 349-360, DOI:10.1080/02626669809492131
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HydrologicalSciences—Journal—des Sciences Hydrologiques, 43(3) June 1998 ->Aq
Application of stochastic differential equations in risk assessment for flood releases
SHUHAI JIANG Nanjing Hydraulic Research Institute, 223 Guangzhou Road, 210029 Nanjing, China
Abstract After analysing the Wiener process characteristics of reservoir storage capacity in flood regulation, an Ito stochastic differential equation including a random input term is proposed to describe and estimate the stochastic reservoir level hydrograph in a flood routing process. An expression of risk for flood release is given and the probability density functions of the reservoir level hydrograph, which are related to the risk of overtopping failure, are computed using a Fokker-Planck equation. Analysed and calculated results indicate that, by using the stochastic differential equations, the uncertainty influence of various random factors on the reservoir level hydrograph can be taken into account in the flood routing process. Therefore, the procedure of evaluating the risk of dam overtopping failure developed in this paper provides an improvement over existing methods.
Application des équations différentielles stochastiques à l'évaluation du risque de lâchure de crue Résumé Après avoir analysé les caractéristiques d'un processus de Wiener pour la capacité de stockage d'un réservoir de laminage de crue, l'auteur propose une équation différentielle d'Ito incluant un terme aléatoire pour décrire et estimer de façon stochastique le limnigramme du niveau du réservoir au cours du routage des crues. Une expression du risque de lâchure de crue est proposée. Cette expression de même que la fonction densité de probabilité du niveau du réservoir, qui sont liées au risque de débordement, sont calculées en utilisant une équation de Fokker-Planck. Les résultats analysés et calculés montrent que l'influence des incertitudes de différents facteurs aléatoires sur le niveau du réservoir peut être prise en compte dans le routage des crues grâce aux équations différentielles stochastiques. La procédure d'évaluation du risque de débordement du réservoir présentée ici est donc particulièrement fondée.
INTRODUCTION
The risk of dam overtopping failure has received increasing attention in recent years. This risk assessment is important in the determination of the dimensions of spillways, their satisfactory operation, and in the assessment of dam safety and investment.
Traditionally, to size the spillways, the inflow design flood (IDF) is routed through the reservoir storage, spillways and any other outlet structures that release water downstream. The results of the routing procedure are a reservoir level hydrograph and an outflow hydrograph representing the attenuation of the IDF hydrograph. The spillway capacity can then be judged. Obviously, the risk of flood release associated with the routing process is identified as the probability that the reservoir water level would be at an elevation that would lead to overtopping. This would be caused by inadequate spillway capacity during extreme flood events. In this
Open for discussion until 1 December 1998
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350 Shuhai Jiang
process uncertainties exist in: the available information on hydrological conditions that determine the IDF hydrograph; the hydraulic conditions that determine the spillway capabilities; the accumulated storage volumes; and the initial condition of the flood control level of the storage. These uncertainties cause randomness of the reservoir level hydrograph and the
corresponding outflow discharges. Finally, they lead to a random maximum reservoir surface level, which is associated with a risk of dam overtopping failure (Ganoulis, 1991).
To prevent the hazards of dam failures due to overtopping during extreme floods, engineers have to evaluate the likelihood of occurrence of overtopping. However, the influence of stochastic hydraulic and hydrological processes during extreme floods on the risk of overtopping is not taken into account in their interpretations. Usually a limited description of overtopping is defined as the peak of the inflow design flood exceeding the spillway capacity. This will lead to a mistake since the effect of available storage on the attenuation of the IDF is not considered. Therefore a method to analyse the problem has yet to be developed.
Using stochastic differential equations (SDE) based on the theories of probability and differential equations provides an approach to handling the uncertain effects of stochastic input processes and random initial conditions on the risk of flood release, the reservoir level hydrograph, the flood routing process, the sizing of spillways and their operating rales.
STOCHASTIC DIFFERENTIAL EQUATION FOR THE FLOOD ROUTING PROCESS
Deterministic flood routing procedures utilize some form of the volumetric conservation equation:
dw(h)/dt = Q(t) - q(h, c) (1)
Letting dw/dh = GQi), and defining the initial condition, the differential equation for the reservoir routing process can be obtained from equation (1) as:
\dhldt = ÏQ(t) - q(h,c)}G{h)
\ ( \ ( 2 )
[h{t0) = h0
where h{t) is the reservoir level hydrograph at time t, h0 is the initial reservoir level at time zero, w(h) is the volume of water stored in the reservoir, which is a function of h, Q(t) is the inflow design flood hydrograph at time t and q(h, c) is the outflow discharge hydrograph which is a function of h and a hydraulic parameter c (discharge coefficient). All these variables are deterministic, so the stochastic input and output processes cannot be allowed for in equation (2). The calculated results of such a routing procedure yield only a deterministic level hydrograph h(t).
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Application of stochastic differential equations in risk assessment for flood releases 351
To analyse a stochastic reservoir routing process, a stochastic differential equation with a stochastic input term and a random initial condition must be established.
Firstly, the probability characteristics of a flood routing process have to be discussed. In this procedure, it is emphasized that the stochastic variations of the storage volume temporarily occupied by a flood, control the stochastic reservoir level hydrograph and are controlled by the stochastic inflow and outflow hydrographs. The process can be investigated as follows:
The stochastic IDF hydrograph Q(t)
The uncertainties due to the scarcity of observational hydrological data and a limited modelling capability for selecting the IDF hydrograph result in the inflow flood series being a stochastic process. Usually, it may be assumed that the stochastic process Q{t) is normal, with mean given by the IDF hydrograph determined by standard design methods.
The stochastic outflow hydrograph q(h, c)
For a given size of spillway the releases are affected by the uncertainties of h and c. Therefore, the outflow is to be regarded as a stochastic process. However, in this procedure, there is only one independent variable, c, to cause the random variations of q(h, c).
The stochastic available storage curve w(h)
The inadequacy of topographic surveys and the inherent randomness of the sedimentation process cause uncertainties in the relationship between reservoir level (h) and available storage (w(h)). Suppose that the stochastic w(h) is normal, with mean given by the w - h curve from the standard design process.
A stochastic hydrograph of the storage volume occupied by the IDF in the routing process is mainly determined by the effect of the above-mentioned stochastic processes. It seems to satisfy the definition conditions of the Wiener process viz., the probability density of w(t) - w(tiA) depends only on (?,• - tiA), the variance of w(t,) -w(?M) is proportional to {t, - tjA) and the process known as a random walk is normal.
Therefore, a Wiener process with zero mean can be found from the expected hydrograph of the storage volume w(t) :
Mt) = j(0 - q)àt + B(t) (3) 0
In which Q and q are inflow and outflow mean hydrographs, respectively, as defined above. The derivative of equation (3) divided by G(h) can be defined as:
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àH{t) Q{t)-q(h,c) âB{t)/àt àt G{h) G{h) (4)
H(t0) = H0
Equation (4) is similar to the general equation (2) except for the term (âB(t)/dt)/G(h), through which the effects of the random factors are accounted for. Therefore, H(t) is a stochastic process and is unlike the deterministic function hit). Equation (4) can be simplified to:
IdH(t) = (p(t,H(t))dt + g(t,H{t))dB(t)
{H(t0) = H0
where cp and g substitute respective components of equation (4). Equation (5) is a typical Ito equation with a stochastic input term and a
random initial condition as shown by Soong (1973). Its solutions are Markov processes of Hit).
Bit) in equation (4) is that of Brownian motion and hence dB/dt represents a white noise. The one-dimensional density of Bit) can be simply found as:
f(B) = , exp V2itfcr
B2
2a2t (6)
where a is the standard deviation of the wit) process and is determined from the variances of inflow, outflow and storage contents hydrographs. The standard deviations aQif), aqit) and aw(f) can be determined from the observed data. In most cases the three stochastic input processes are considered independent, so the variance of the output process, D[Bit)], can be written as:
D[B(t)] = G2(tl) = [aQ(t,)2 +oq(tl)2]At2 +aw(/,)2 (7)
where At = (/,• - ?,„,). The variance a2 it) synthesises the effects of Q, q and w on Bit). The initial condition H0 in equation (4) can be assumed as either deterministic or
random. Usually, the reservoir level at the beginning of a routing process has uncertainty. Its mean value H0 can be determined from a flood control level given by the standard design process.
DENSITY FUNCTION OF RESERVOIR LEVEL HYDROGRAPH
For estimating the risk of a flood release, the probability density functions of the solution process, i.e. the stochastic distributions of the reservoir level at any time within the flood routing process, which are associated with overtopping risk, have been always of primary concern for engineers.
The following stochastic differential equation can be applied to obtain the first order density functions, f(/z, t), of the SDE solution process:
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Application of stochastic differential equations in risk assessment for flood releases 353
df(h,t) ^ ( - 1 ) " d" r / \ / M
in which a„ can be expressed as:
<x„(h, t) = Um^E§H(t + At)- H(t)]"\ H(l)__h} (9)
The initiai condition and boundary condition for equation (8) are shown in the following equations, respectively:
f(/z, t0) = f0(h)
f(hmx, 0 = 0 (10)
Wma, 0 = 0
A property of the density function f(/z, t) is: *„,„
]f(h,t)dh = l (11) '»,„»,
where hmm and hmia sire the maximum and minimum reservoir level, respectively. The Fokker-Planck equation can be derived from equation (8) by applying the additional condition an(h, t) = 0 when n > 3 for a Markov process.
Based on the properties of the Wiener process, the following equations can be obtained:
E{AB(tj\ = 0
E{AB(t)AB(t)} = 2a2At
where AB is the increment of B. To compute the density functions of the reservoir level hydrograph f(h, t) the Fokker-Planck equation and equation (5) may be combined using equation (12) to form the expression:
\df(h,t) d r 1 \ 1 \i 9 l / \ 1 \-> 1 l ' =— \î\h,j\àt,H) \ + -~Y \fih,t)g{t,H)~a
rih L v 7 x /1 ah \ • } v '
2
3t a A L - v " / T v - y j - a ^ L - v ' / " v ' / j ( B )
f(h,l0) = f0(h)
Equation (13) is a deterministic partial differential equation for a nonlinear Ito equation problem for which it is difficult to obtain an exact theoretical solution. However, a finite difference method may be used to solve equation (13). In the solution process, f(h, t) and the corresponding mean discharge q{t) can be calculated by using an iterative method. The mean values of discharge q weighted by the density functions of reservoir levels can be expressed as:
' 'max
q(t)= ]q{h,c)î(h,t)àh (14)
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354 Shuhai Jiang
OVERTOPPING RISK IN THE FLOOD ROUTING PROCESS
Overtopping occurs when the maximum reservoir level exceeds the level of the dam crest. In the present procedure, it is convenient to define the difference M between the maximum reservoir level H and the level of dam crest Z, and consider risk as the exceedance of M:
M = H-Z (15)
The level of risk of overtopping is considered as the probability of the exceedance of M = 0:
¥f = VjH > Z] (16)
where P is the probability of occurrence of overtopping. The levels of risk ~Pf can be conveniently expressed by the reliability index, P,
defined as:
P = <D-'(l-P /) (17)
where (3 is the reliability index, and €>"' (•) is the standard normal inverse cumulative distribution function.
The risk assessments are significantly affected by the stochastic factors Z and H as has been shown in equation (15). Before beginning the risk analysis for overtopping failure, a review should be made of any available design records for the dam and spillway. The uncertainties of data on stochastic variables such as Z and H should be examined.
The uncertainties of the elevation of the dam crest are easily estimated. The random characteristics of Z are influenced by the scatter of elevations of the dam crest during construction. It is often assumed that the random variable Z is normal, with mean \xz given by the design elevation of the dam crest and a small standard deviation <rz.
As has been discussed, the stochastic process of the reservoir level, with a series of density functions and corresponding \iH and aH obtained by solving equation (13), depends on the combined effect of various random factors taken into consideration in the procedure for analysing the uncertainty of H{f). The density functions of H(t), which is one of the basic variables in equation (15), cannot be assumed normal. To improve the accuracy of the risk assessment a second moment method as outlined by Jiang (1993) may be employed.
VERIFICATION AND DISCUSSION
The stochastic mathematical model outlined above was verified with an example of the flood procedure developed by Zhu & Wan (1988) for a reservoir in China.
A regression expression for the storage can be established empirically from the storage-elevation curve given by site observed data:
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Application of stochastic differential equations in risk assessment for flood releases 355
w = [1.4594 + 0.0475/z + 0.0175/*2] x 108 (18)
in which h is the reservoir elevation (m) based on a datum at the level of the spillway crest.
The outlet facilities were designed to cope with the 1000-year flood inflow to the reservoir with a peak IDF of gmax = 12 400 m3 s"1. The IDF hydrograph is shown in Fig. 1. The crest of the dam is at elevation Z = 19.5 m (above the datum level of the spillway crest). The outlet facilities consist of (a) a spillway with an ogee-shaped crest structure controlled by six radial gates each 9.4 m wide and with a discharge coefficient of tnl = 0.49; and (b) a bottom outlet tunnel with a rectangular cross section 6 m high by 4 m wide with its axis 5.5 m below the spillway crest and a discharge coefficient of m2 = 0.90. The flood control level, which is the reservoir level at the beginning of routing processes, is at an elevation of 12.5 m. According to the operating rule for flood regulation, when the inflow discharge Q < 3000 m3 s"1
no outlet facilities operate, when 3000 m3 s ' < Q < 6300 m3 s"1 (a peak inflow for a 20-year flood) the outflow discharge q is 3000 m3 s"1, (both through the tunnel and over the spillway crest) and when Q > 6300 m3 s4 all outlet facilities operate fully and the outflow discharge q changes with the reservoir level.
By applying the deterministic flood routing procedure a hydrograph of design outflow discharge and a corresponding hydrograph of the reservoir level can be obtained as shown in Figs 1 and 2. The peak outflow of 10 200 m3 s"1 with maximum reservoir level hmax of 18.6 m for the 1000-year flood is shown. It indicates that the reservoir levels h(t) in the routing process have not exceeded the level Z = 19.5 m of the dam crest throughout and a freeboard just less than 1 m would be provided. Therefore, it is thought from the traditional deterministic routing procedure that the outlet facilities have adequate capacity to avoid overtopping. However, the question of quantitative assessment of the possible margin of the freeboard has yet to be
10000
500C
a~i
—
, /
; - 1
1
/ x
•
- \ ~ t
\ \ \ \ \ \ \ \
0 20 40 60
t t h ) Fig. 1 Hydrograph of Q ~ t and q ~ t.
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356 Shuhai Jiang
h cm)
14 -
12
)
a ^cT
I
\6[
o <r-o -, M
û g = 30000m ' s / 2
3 44 a 0" =90000m s / 2
.. . \ -V, (
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D (7 = 150000m s ^
r a d i t i o n a l
a c o n t r o l l e v é l
NS
0 20 40 Fig. 2 Hydrograph ofh~t.
60 t ( h )
answered. A method to provide a better basis for estimating the magnitude of the freeboard needs to be used.
To consider the uncertainty effects of the input on the routing process a stochastic procedure may be carried out. Assume that the coefficient of variation 8G = aQ{f)l]xQ{t) = 0.1, Ô, = 0.01 and bjh) = 0.05. The probability density
Fig. 3 Probability distribution profiles of f(/z, t) f c h . t )
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Application of stochastic differential equations in risk assessment for flood releases 357
€1
•
Ô
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Ô
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t>
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a £
ê
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A ff = 30000 m3 s
• (J = 90000 m 3 s ' I
« o" = -- 150000m s
• %
•y±—
à 1 I , l_ 0 20 40 60
fth) Fig. 4 Relationship between a and cxw.
fonctions f(/z, /) and corresponding u.H, CTH are calculated and shown in Figs 2, 3 and 4 by selecting input standard deviations a of 0, 30 000, 90 000 and 150 000 m3 s~0'5
and incorporating an initial condition of H0 with a discrete distribution, i.e.:
. . / s f 1 if h = } \ fJh) = d[h~hJ = in , . (19)
0 v 0 / [0 otherwise {iy)
The calculated results show that the reservoir level hydrograph is no longer a deterministic curve, and becomes a stochastic process with some density distributions changing gradually with time. A series of mean values \iH(f) agrees well with the one given by the deterministic procedure as shown in Fig. 2. The value of CJ continues to increase with increasing a and reaches a maximum value uHnm when all outlet facilities start to operate fully, then uH decreases sharply as shown in Fig. 4. Obviously, a balanced relationship between H and q causes the variance of H(t) to decrease; H controls q, and it is also controlled by q when all outlet facilities are open in the reservoir routing process.
The characteristics of H(t) are significantly affected by the randomness of the initial condition. The density distributions f(h, t) at some points of the routing process are shown in Fig. 5 by assuming the initial conditions H0 with a discrete distribution given by equation (19) and two normal distributions: \xH = 12.5 m,
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358 Shuhai Jiang
h
cm
20
18
16
14
12
10 o I f ( h , t )
Fig. 5 Probability distribution profiles of f(h, t) with different initial conditions, H0.
a Ha = 1.00 m or 0.57 m. The influence of the initial H0 on the density functions
f(/z, t) decreases gradually with the developing process and can be ultimately neglected for a long routing process .
The selection of schemes for flood release from the reservoir is facilitated by an assessment of the risk of overtopping failure. A random variable Z is assumed to have a normal distribution with \xz = 19.5 m and az = 0.01 m. Taking a discrete distribution with h0 = 12.5 m for the initial condition H0 and standard deviation of the Wiener process <r = 30 000 m3 s"05, the Scheme 1 selected has dimensions of the outlet facilities and operating rules outlined above. The distributions of (3(0 and \i^(t) are calculated as shown in Figs 6 and 7. Also, it is shown from the calculated results for Scheme 1 that fjmin of 0.765 and the corresponding P ^ of 22 .27% occur at t ime t = 60 h with the occurrence of j % m a x , i .e. 6 -7 h after the inflow flood peak. The mean \xH and standard deviation uH are relatively big at time t = 42 h when all outlet facilities begin to operate fully. Accordingly, the value p of 0.829, with corresponding Pf of 2 0 . 3 6 % , would be another small peak. It is evident that the probability of overtopping varies throughout the routing process of Scheme 1, which is difficult to demonstrate quantitatively using the deterministic flood routing procedure.
) 0 0.2
\
è ^V
/
*^s
t-= 6 h
0.4 0.6 0-8
5^—— v.
ï = 2
if
U~*^
4h
/
to f
\ \
i=48h
\ i l
~J V
t=60h
Aln=l2.5m discrete
dis tr ibut ion
normal g-^oSIm distr ibution
normal , , _ , „ . dis tr ibution -
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Application of stochastic differential equations in risk assessment for flood releases 359
6
3
u
sell erne scheme
1 \ 2
/s*^
\ \
\
\ /
x W \
20 40 t c f l )
60
Fig. 6 Variation of p(f) with different operating schemes.
HK
18
16
14
\?
sch erne
s c h e m e
s c h e m e
! i
/ ^ /
1
2
3 /
. ^ - '
/r* /
/ '
/ / / / ~7 / /*
1 /
/ /
/ )
•s,
^
20 40 60 -t c h>
Fig. 7 Variation of nH(f)with different operating schemes.
A spillway capacity diminished to 5/6 of that for the original design is selected as Scheme 2. For this condition the maximum mean of reservoir levels (%max exceeds Z, and pmm of -0.435 with corresponding P ^ of 66.83% can be calculated as shown in Figs 6 and 7. An inadequate spillway capacity in Scheme 2 causes a bigger P7, thus
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creating potential hazards of overtopping failure. To diminish the P ^ Scheme 3 is proposed by changing the operating rule and
keeping the dimensions of the outlet facilities of Scheme 2. According to the property of the hydrograph of the IDF the peak outflow discharge is reduced from 6300 m3 s"1
to 5500 m3 s"1, and the beginning of full operation of all outlet facilities is advanced 15 h in Scheme 3. The values of \xH and <JH decrease, and the risk levels of overtopping decrease sharply when all outlet facilities are operating fully open. For this scheme pmin = 1.50 and the corresponding P/max is 6.71% (see Figs 6 and 7). Also the maximum reservoir mean level u^^ of 18.34 m and the maximum mean outflow discharge u.çmax of 7960 m3 s"1 are markedly smaller than those of Schemes 1 and 2. Especially, it demonstrates that the operating rale of flood regulation has a dramatic influence on the reservoir level hydrograph and on the level of flood release.
CONCLUSION
A stochastic differential equation appears to provide a useful mathematical approach to analysing the stochastic phenomena in the flood routing process and to summarising the effects of various random factors on the stochastic reservoir level hydrograph. Through an analysis of the characteristics of the Wiener process for the reservoir storage contents in the flood routing process, an Ito equation with a stochastic input term and a random initial condition is derived. Also, by using the Fokker-Planck equation, the probability density distributions of the reservoir level hydrograph in flood routing are solved.
Risk assessment for dam overtopping should take into account the influence of major random factors. The procedure described for doing this produces a stochastic reservoir level hydrograph and can be used as a powerful tool in conjunction with engineering judgement to evaluate and improve dam safety.
Acknowledgements The research was supported by the Nanjing Hydraulic Research Institute, Nanjing, China. The writer wishes to acknowledge with gratitude the assistance of Professor Eric M. Laurenson in the preparation of the paper.
REFERENCES
Ganoulis, J. (1991) Water resources engineering risk assessment. NATO ASI, Series 358-394. Jiang, S. H. (1993) A stochastic mathematical model for reservoir flood routing. Adv. Wat. Sci. Nanjing, China (in
Chinese) 4(4), 294-300. Soong, T. T. (1973) Random Differential Equations in Science and Engineering. McGraw-Hill, New York. Zhu, C. C. & Wan, C. O. (1988) Handbook of Water Resources Design (in Chinese), 41-44. Water Resources and
Hydro Power Press of China, Beijing, China.
Received 15 October 1996; accepted 26 September 1997
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