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1.
A unit hydrograph (UH) obtained from past storms can be used to predict a direct runoff hydrograph (DRH) based on the effective rainfall hyetograph (ERH) of a new storm. The objective functions in commonly used linear programming (LP) formulations for obtaining an optimal UH are (1) minimizing the sum of absolute deviations (MSAD) and (2) minimizing the largest absolute deviation (MLAD). This paper proposes two alternative LP formulations for obtaining an optimal UH, namely, (1) minimizing the weighted sum of absolute deviations (MWSAD) and (2) minimizing the range of deviations (MRNG). In this paper the predicted DRHs as well as the regenerated DRHs by using the UHs obtained from different LP formulations were compared using a statistical cross-validation technique. The golden section search method was used to determine the optimal weights for the model of MWSAD. The numerical results show that the UH by MRNG is better than that by MLAD in regenerating and predicting DRHs. It is also found that the model MWSAD with a properly selected weighing function would produce a UH that is better in predicting the DRHs than the commonly used MSAD.Notations
M
number of effective rainfall increments
-
N
number of direct runoff hydrograph ordinates
-
R
number of storms
- MSAD
minimize sum of absolute deviation
- MWSAD
minimize weighted sum of absolute deviation
- MLAD
minimize the largest absolute deviation
- MRNG
minimize the range of deviation
- RMSE
root mean square error
-
P
m
effective rainfall in time interval [(m–1)t,mt]
-
Q
n
direct runoff at discrete timent
-
U
k
unit hydrograph ordinate at discrete timekt
-
W
n
weight assigned to error associated with estimatingQ
n
-
n
+
error associated with over-estimation ofQ
n
-
n
–
error associated with under-estimation ofQ
n
-
max
+
maximum positive error in fitting direct runoff hydrograph
-
max
–
maximum negative error in fitting direct runoff hydrograph
-
max
largest absolute error in fitting obtained direct runoff
-
E
r,1
thelth error criterion measuring the fit between the observed DRHs and the predicted (or reproduced) DRHs for therth storm
-
E
1
averaged value of error criterion overR storms 相似文献
2.
Si-Hui Dong 《Water Resources Management》2008,22(4):525-533
Unit hydrograph method is usually used to resolve surface runoff concentration process. Empirical unit hydrograph is sometimes
jagged and not smooth because of estimation errors. Owing to the randomicity of correction method, there is some localization
in its application. The application of instantaneous unit hydrograph is relatively wider. Generally, moment method is used
to estimate the parameters of instantaneous unit hydrograph. However, the error is obvious between the observed flood process
and the predicted flood process with moment method, especially near the flood peak. The genetic algorithm toolbox of matlab software is used to optimize the parameters of instantaneous unit hydrograph. The statistical function gamcdf(x, α, β) in matlab toolbox is used to calculate S(t) curve, which can avoid the errors caused by approximate formula method. The case study indicates that the weighted sum of
absolute error applying the method in this paper is 25, the result applying moment method is 63, and the result of approximate
formula method is 49. The results show that the method in this paper is more effectual than the other two methods. 相似文献
3.
This paper is concerned with the estimation of aggregated direct runoff from small watersheds during a time interval (0,t), homogeneous with respect to rainfall characteristics. The storm events are simulated by a Poisson process, whereas direct runoff is estimated by the SCS method or a linear regression model. The probability of the occurrence of direct runoff is incorporated in the proposed method by examining the possibility of each storm exceeding the watershed losses index. A closed form solution is derived for the expected total direct runoff in the interval (0,t). Finally, the proposed method is applied to a particular set of conditions.Notation
Q
direct runoff
-
P
rainfall depth
-
S
index of watershed storage
- CN
Curve Number of SCS method
-
t
time
-
T
i
time interval between successive storm events (i andi+1)
-
X
i
storm depth of theith event (case a) excess storm depth of theith event (case b)
-
Y(t)
total direct runoff in (0,t)
-
N(t)
number of storm events in (0,t)
-
F(t)
distribution function of the time between storm events
-
G(x)
distribution function of the storm depth
-
F
n(t),F
n+1(t)
n-fold and (n+1)-fold convolution ofF(t), respectively
-
G
n(x),G
n+1(x)
n-fold and (n+1)-fold convolution ofG(x), respectively
-
E[X]
expected mean value
-
p
probability of exceeding the thresholde,p+q=1
- *
convolution operation 相似文献
4.
This paper, the first of two, develops a real-time flood forecasting model using Burg's maximum-entropy spectral analysis (MESA). Fundamental to MESA is the extension of autocovariance and cross-covariance matrices describing the correlations within and between rainfall and runoff series. These matrices are used to derive the model forecasting equations (with and without feedback). The model may be potentially applicable to any pair of correlated hydrologic processes.Notation
a
k
extension coefficient of the model atkth step
-
B
k
backward extension matrix forkth step
-
B
ijk
element of the matrixB
k
(i,j=1, 2)
-
c
k
coefficient of the entropy model atkth step in the LB algorithm
- e
k
(e
x
,e
y
)k = forecast error vector atkth step
-
E
k
error matrix atkth step
-
E
ijk
element of theE
k
(i,j=1, 2)
-
f
frequency
-
F
k
forward extension matrix atkth step
-
F
ijk
element of theF
k
matrix (i,j=1, 2)
-
H(f)
entropy expressed in terms of frequency
-
H
X
entropy of the rainfall process (X)
-
H
Y
entropy of the runoff process (Y)
-
H
XY
entropy of the rainfall-runoff process
-
I
identity matrix
-
forecast lead time
-
m
model order, number of autocorrelations
-
R
correlation matrix
-
S
x
standard deviation of the rainfall data
-
S
y
standard deviation of the runoff data
-
t
time
-
T
1
rainfall record
-
T
2
runoff record
-
T
rainfall-runoff record (T=T
1
T
2)
-
x
t
rainfall data (depth)
-
X
X() = rainfall process
-
mean of the rainfall data
-
y
t
direct runoff data (discharge)
-
Y
Y() = runoff process
-
mean of the runoff data
- (x, y)
t
rainfall-runoff data (att T)
- (x, y, z)
t
rainfall-runoff-sediment yield data (att T)
-
z
complex number (in spectral analysis)
-
k
coefficient of the LB algorithm atkth step
-
nj
Lagrange multiplier atjth location in the
n
matrix
-
n
n
= matrix of the Lagrange multiplier atkth step
-
X
(k),
Y
(k)
autocorrelation function of rainfall and runoff processes atkth lag
-
XY
(k)
cross-correlation function of rainfall and runoff processes atkth lag
-
W
1(f)
power spectrum of rainfall or runoff
-
W
2(f)
cross-spectrum of rainfall or runoff
Abbreviations acf
autocorrelation function
- ARMA
autoregressive moving average (model)
- ARMAX
ARMA with exogenous input
- ccf
cross-correlation function
- det()
determinant of the (...) matrix
- E[...]
expectation of [...]
- FLT
forecast lead time
- KF
Kalman filter
- LB
Levinson-Burg (algorithm)
- MESA
maximum entropy spectral analysis
- MSE
mean square error
- SS
state-space (model)
- STI
sampling time interval
-
forecast ofx
-
forecast ofx -step ahead
-
x
F
feedback ofx-value (real value)
- |x|
module (absolute value) ofx
-
X
–1
inverse of the matrixX
-
X*
transpose of the matrixX 相似文献
5.
Kumar Rakesh Chatterjee C. Lohani A. K. Kumar Sanjay Singh R. D. 《Water Resources Management》2002,16(4):263-278
For estimation of runoff response of an ungauged catchment resulting from a rainfall event, geomorphologicalinstantaneous unit hydrograph (GIUH) approach is getting popularbecause of its direct application to an ungauged catchment. Itavoids adoption of tedious methods of regionalization of unithydrograph; wherein, the historical rainfall-runoff data of anumber of gauged catchments are required to be analyzed. In thisstudy, the GIUH derived from geomorphological characteristics ofa catchment has been related to the parameters of Clark IUH modelfor deriving its complete shape. The DSRO hydrographs estimatedby the GIUH based Clark model have been compared with the DSROhydrographs computed by the Clark IUH model option of the HEC-1package and the Nash IUH model by employing some of the commonlyused error functions. Sensitivity analysis of the GIUH basedClark model has been conducted with the objective to identify thegeomorphological and other model parameters which are moresensitive in estimation of peak of unit hydrographs computed bythe GIUH based Clark model. So that these parameters may beevaluated with more precision for accurate estimation of floodhydrographs for the ungauged catchments. 相似文献
6.
The MESA-based model, developed in the first paper, for real-time flood forecasting was verified on five watersheds from different regions of the world. The sampling time interval and forecast lead time varied from several minutes to one day. The model was found to be superior to a state-space model for all events where it was difficult to obtain prior information about model parameters. The mathematical form of the model was found to be similar to a bivariate autoregressive (AR) model, and under certain conditions, these two models became equivalent.Notation
A
k
parameter matrix of the bivariate AR model
-
B
backshift operator in time series analysis
-
eT
forecast error (vector) at timet = T
-
t
uncorrelated random series (white noise)
-
F
k
forward extension matrix of the entropy model forkth lag
-
I
identity matrix
-
m
order of the entropy model
-
N
number of observations
-
P
order of the AR model
-
Q
p
peak of the direct runoff hydrograph
-
R
correlation matrix
-
t
p
time to peak of the direct runoff hydrograph
-
1
coefficient of variation
-
2
ratio of absolute error to the mean
-
forecasted runoff
-
x
i
observed runoff
-
mean of the observed runoff
-
X
–1
inverse ofX matrix
-
X*
transpose of theX matrix
Abbreviations AIC
Akaike information criterion
- AR
autoregressive (model)
- AR(p)
autoregressive process of thepth order
- ARIMA
autoregressive integrated moving average (model)
- acf
autocorrelation function
- ccf
cross-correlation function
- FLT
forecast lead time
- MESA
maximum entropy spectral analysis
- MSE
mean square error
- STI
sampling time interval 相似文献
7.
Estimation of Clark’s Instantaneous Unit Hydrograph Parameters and Development of Direct Surface Runoff Hydrograph 总被引:2,自引:1,他引:1
Muhammad Masood Ahmad Abdul Razzaq Ghumman Sajjad Ahmad 《Water Resources Management》2009,23(12):2417-2435
We present a method to estimate Time of Concentration (T
c) and Storage Coefficient (R) to develop Clark’s Instantaneous Unit Hydrograph (CIUH). T
c is estimated from Time Area Diagram of the catchment and R is determined using optimization approach based on Downhill Simplex technique (code written in FORTRAN). Four different objective
functions are used in optimization to determine R. The sum of least squares objective function is used in a novel way by relating it to slope of a linear regression best fit
line drawn between observed and simulated peak discharge values to find R. Physical parameters (delineation, land slope, stream lengths and associated drainage areas) of the catchment are derived
from SPOT satellite imageries of the basin using ERDAS: Arc GIS is used for geographic data processing. Ten randomly selected
rainfall–runoff events are used for calibration and five for validation. Using CIUH, a Direct surface runoff hydrograph (DSRH)
is developed. Kaha catchment (5,598 km2), part of Indus river system, located in semi-arid region of Pakistan and dominated by hill torrent flows is used to demonstrate
the applicability of proposed approach. Model results during validation are very good with model efficiency of more than 95%
and root mean square error of less than 6%. Impact of variation in model parameters T
c and R on DSRH is investigated. It is identified that DSRH is more sensitive to R compared to T
c. Relatively equal values of R and T
c reveal that shape of DSRH for a large catchment depends on both runoff diffusion and translation flow effects. The runoff
diffusion effect is found to be dominant. 相似文献
8.
A. Sarangi C. A. Madramootoo P. Enright S. O. Prasher 《Water Resources Management》2007,21(7):1127-1143
The predictability of unit hydrograph (UH) models that are based on the concepts of land morphology and isochrones to generate direct runoff hydrograph (DRH) were evaluated in this paper. The intention of this study was to evaluate the models for accurate runoff prediction from ungauged watershed using the ArcGIS® tool. Three models such as exponential distributed geomorphologic instantaneous unit hydrograph (ED-GIUH) model, GIUH based Clark model, and spatially distributed unit hydrograph (SDUH) model, were used to generate the DRHs for the St. Esprit watershed, Quebec, Canada. Predictability of these models was evaluated by comparing the generated DRHs versus the observed DRH at the watershed outlet. The model input data, including natural drainage network and Horton's morphological parameters (e.g. isochrone and instantaneous unit hydrograph), were prepared using a watershed morphological estimation tool (WMET) on ArcGIS® platform. The isochrone feature class was generated in ArcGIS® using the time of concentration concepts for overland and channel flow and the instantaneous unit hydrograph was generated using the Clark's reservoir routing and S-hydrograph methods. An accounting procedure was used to estimate UH and DRHs from rainfall events of the watershed. The variable slope method and phi-index method were used for base flow separation and rainfall excess estimation, respectively. It was revealed that the ED-GIUH models performed better for prediction of DRHs for short duration (≤6 h) storm events more accurately (prediction error as low as 4.6–22.8%) for the study watershed, than the GIUH and SDUH models. Thus, facilitated by using ArcGIS®, the ED-GIUH model could be used as a potential tool to predict DRHs for ungauged watersheds that have similar geomorphology as that of the St. Esprit watershed. 相似文献
9.
10.
A method capable of estimating the hydrograph from a prescribed storm for a practical mild slope upstream catchment is proposed. This method makes use of two new characteristic parameters, andS, in conjunction with the kinematic wave equation to compute lateral inflows of the main stream of the catchment. The depth profile of overland flow at any instant within the catchment and hydrograph at any location can be easily found. Lag times for individual lateral inflows are then considered and are linearly combined to obtain the hydrograph at the outlet of the catchment or depth profile of the main stream at any instant. The validity of the excess rainfall-surface runoff linear relationship in this study has also been verified with Tatsunokuchiyama catchment, and it shows good results for this computed runoff. 相似文献
11.
Forecast model of water consumption for Naples 总被引:1,自引:1,他引:0
The data refer to the monthly water consumption in the Neapolitan area over more than a 30 year period. The model proposed makes it possible to separate the trend in the water consumption time series from the seasonal fluctuation characterized by monthly peak coefficients with residual component. An ARMA (1,1) model has been used to fit the residual component process. Furthermore, the availability of daily water consumption data for a three-year period allows the calculation of the daily peak coefficients for each month, and makes it possible to determine future water demand on the day of peak water consumption.Notation
j
numerical order of the month in the year
-
i
numerical order of the year in the time series
-
t
numerical order of the month in the time series
-
h
numerical order of the month in the sequence of measured and predicted consumption values after the final stage t of the observation period
-
Z
ji
effective monthly water consumption in the month j in the year i (expressed as m3/day)
-
T
ji
predicted monthly water consumption in the month j in the year i minus the seasonal and stochastic component (expressed as m3/day)
-
C
ji
monthly peak coefficient
-
E
ji
stochastic component of the monthly water consumption in the month of j in the year i
-
Z
i
water consumption in the year i (expressed as m3/year)
-
Z
j
(t)
water consumption in the month j during the observation period (expressed as m3/day)
-
evaluation of the correlation coefficient
-
Z
j
(t)
water consumption in the month j during the observation period minus the trend
-
Y
t
transformed stochastic component from E
t
: Y
t
=ln Et
-
Y
t+h
measured value of stochastic component for t+h period after the final stage t of the observation period
-
Y
t
(h)
predicted value of stochastic component for t+h period after the final stage t of the observation period
- j
transformation coefficients from the ARMA process (m, n) to the MA () process 相似文献
12.
基于信息熵理论的流域瞬时单位线 总被引:1,自引:0,他引:1
基于流域汇流是一种受确定性和随机性因素影响的随机现象,从流域的瞬时单位线等价于水质点在流域内流达时间的概率密度函数入手,引入信息熵理论,提出了一种流域瞬时单位线。这种模型的数学菜同于Nash瞬时单位线,具有概念不禁,计算简单等优点,并且计算精度等同ash时单位线,以圩缺乏水文资料地区的汇流计算也是可行的。 相似文献
13.
On relationship between curve numbers and phi indices 总被引:1,自引:0,他引:1
David A. Chin 《水科学与水工程》2018,11(3):187-195
The curve number and phi (φ)-index models each provide a simple one-parameter relationship between storm-event rainfall and runoff. It is shown that the curve number and φ-index models can both be used to segregate the rainfall hyetograph into initial abstraction, retention, and runoff amounts. However, the principal advantages of the φ-index model are that both rainfall distribution and duration can be explicitly taken into account in calculating runoff, and the φ index is more physically based than the curve number. The quantitative relationship between the curve number and the φ index is presented and validated with field measurements. Knowing the relationship between the curve number and the φ index is useful in that it facilitates using the extensive database of curve numbers in the more realistic φ-index model in calculating a runoff hydrograph from a given rainfall hyetograph. It is demonstrated that conventional adjustments to curve numbers can be largely explained by variations in storm duration, which suggests that variable rainfall duration can possibly be an essential factor in accounting for deviations from the median curve number of a catchment. 相似文献
14.
Synthetic generation of daily streamflow sequences is one of the most critical issues in stochastic hydrology. In this study,
a new wavelet transform method is developed for synthetic generation of daily streamflow sequences. Firstly, daily streamflow
sequences with different frequency components are decomposed into the series of wavelet coefficients W
1(t), W
2(t),...,W
P
(t) and scale coefficients (the residual) C
P
(t) at a resolution level P using wavelet decomposition algorithm. Secondly, the series of W
1(t), W
2(t),...,W
P
(t) and C
P
(t) are divided into a number of sub-series based on a yearly period. Thirdly, random sampling is performed from sub-series
of W
1(t), W
2(t),...,W
P
(t) and C
P
(t), respectively. Based on these sampled sub-series, a large number of synthetic daily streamflow sequences are obtained using
wavelet reconstruction algorithm. The advantages of this newly developed method include: (1) it is a nonparametric approach;
(2) it is able to avoid assumptions of probability distribution types (Normal or Pearson Type III) and of dependence structure
(linear or nonlinear); (3) it is not sensitive to the original data length and suitable for any hydrological sequences; and
(4) the generated sequences from this method could capture the dependence structure and statistical properties presented in
the data. Finally, a case study in Jinsha River, China, indicates that the new method is valid and efficient in generating
daily streamflow sequences based on historical data. 相似文献
15.
In drainage of agricultural lands, the upward vertical recharge from a semi-confined aquifer depends on the difference of the piezometric heads on the two sides of the semi-impermeable layer through which this recharge takes place. This means that the recharge through the semi-impermeable base depends on the unknown height of the unsteady water table. In the nonhomogeneous Boussinesq equation, which describes the drainage problems, the downward uniform rate of the recharge from rain or irrigation and the recharge from the semiconfined aquifer are expressed by two terms. By solving the Boussinesq equation expressions for the nondimensional height of the water table and the nondimensional discharge of the drains per unit drained area are obtained for three different initial conditions. Some known solutions are shown as special cases of the present solutions. Variation of nondimensional water table heights at half distance of the drain spacing and the nondimensional discharge of the drains with nondimensional time have been graphically illustrated with the help of synthetic examples.Notation
B
s
thickness of the semi-impervious layer [L]
-
c
hydraulic resistance of the semi-impervious layer [T]
-
D
depth of the drains from the base [L]
-
d
e
equivalent depth [L]
-
h=h(x, t)
height of the water table [L]
-
h
0
initial height of the water table [L]
-
h
t
water table height at mid-distance of drains att [L]
-
h
j
,h
k
water table height at mid-distance of drains at timej andfk, respectively [L]
-
H
0
piezometric head in the semi-confined aquifer [L]
-
K
hydraulic conductivity of the soil [LT–1]
-
K
s
hydraulic conductivity of the semi-impervious layer [LT–1]
-
k
0,k
1,k
2
nondimensional constants
-
L
distance between the drains [L]
-
q
0
upward recharge per unit surface area through the semi-impervious layer [LT–1]
-
q
t
discharge per unit drainable area of drains at timet [LT–1]
-
R,R
0
recharge per unit surface area from rain or irrigation during the unsteady and steady-state, respectively, [LT–1]
-
S
specific yield of the soil
-
t
time of observation [T]
-
x
distance measured from the drain [L]
-
leakage factor [L]
-
nondimensional distance
-
nondimensional time 相似文献
16.
R. K. Rai M. K. Jain S. K. Mishra C. S. P. Ojha V. P. Singh 《Water Resources Management》2007,21(11):1829-1848
This paper presents a technique to derive the unit impulse response functions (UIRF) used for determination of unit hydrograph
by employing the Z-transform technique to the response function derived from the Auto Regressive Moving Average (ARMA) process
of order (p, q). The proposed approach was applied to reproduce direct surface runoff for single storm event data registered over four watersheds
of area ranging from 0.42 to 295 km2. It is observed that the UIRF based on ARMA (1, 2) and ARMA (2, 2) provides a better representation of the watershed response.
Further, to test the superiority of the developed impulse response function form ARMA process, the direct runoff hydrographs
were computed using the simple ARMA process and optimized Nash’s two parameter model and compared with the results obtained
from UIRF’s of ARMA model. The performance of the models based on the graphical presentation as well as from the test statistics
viz. RMSE and MAPE indicates that UIRF-ARMA (p, q) performs better than optimized Nash Model and mostly similar to simple ARMA (p,q) model. Further more, the ARMA process of order p ≤ 2 and q ≤ 2 is generally sufficient and less cumbersome than the Argand diagram based approach for UIRF derivation. 相似文献
17.
Abstract Synthetic unit hydrographs are frequently used to estimate hydrograph characteristics when observed data are not available. A number of synthetic unit hydrograph approaches are available, but the ones that found widespread use are those based on models of Snyder, Clark, and the U.S. Soil Conservation Services (SCS). The major goal of the study is to develop a synthetic unit hydrograph for Wadi Al Fara'a Catchment, which is un-gauged and considered one of the West Bank's most important catchments. Unfortunately, none of the wadis in the West Bank are gauged and flow records are not available; therefore, it is hoped that this method will be applied successfully for Wadi Al Fara'a catchment and the results can hopefully to be applied to all West Bank catchments, which will facilitate estimation of potential runoff in the whole West Bank. 相似文献
18.
Optimization-simulation models were used for the systems analysis of a water resources system. The Karjan Irrigation reservoir project in India was taken as the system. Two types of optimization models, i.e., linear programming, and dynamic programming (continuous and discontinuous) were used for preliminary design purposes. The simulation technique was used for further screening. The linear programming model is most suitable for finding reservoir capacity. Dynamic programming (continuous and discontinuous models) may be used for further refining the output targets and finding the possible reservoir carry-over storages, respectively. The simulation should then be used to obtain the near optimum values of the design variables.Notations
a
1
Unit irrigation benefit [Rs.105 L–3]
-
B
1
Gross annual irrigation benefit [Rs.105]
-
B
1,t
Gross irrigation benefit in periodt [Rs.105]
-
C
1
Annual capital cost of irrigation [Rs.105]
-
C
1
Annual capital cost function for irrigation [Rs.105 L–3]
-
C
1,t
Fraction of annual capital cost for irrigation in periodt [Rs.105]
-
C
2
Annual capital cost of reservoir [Rs.105]
-
C
2
Annual capital cost function for reservoir [Rs.105 L–3]
-
C
2,t
Fraction of annual capital cost for reservoir in periodt [Rs.105]
-
El
t
Reservoir evaporation in timet [L3]
-
f
t
Optimal return from staget [Rs.105]
-
g
t
The return function for periodt [Rs.105]
-
I
t
Catchment inflow into the reservoir in periodt [L3]
-
I
t
Water that joins the main stem just above the irrigation diversion canal in timet [L3]
-
t
Local inflow to the reservoir from the surrounding area in timet [L3]
-
Ir
Annual irrigation target [L3]
-
K
t
Proportion of annual irrigation targetIr to be diverted for irrigation in timet
-
K
t
Amount by whichK
t exceeds unity is the fraction of the end storage which is assigned to reservoir evaporation losses
-
L
Loss in irrigation benefits per unit deficit in the supply [Rs.105 L–3]
-
L
1
Lower bound on annual irrigation target,Ir [L3]
-
L
2
Lower bound on reservoir capacity,Y [L3]
-
N
Number of time periods in the planning horizon
-
O
t
Total water release from the reservoir in periodt [L3]
-
O
t
*
The optimal total water release from the reservoir in timet [L3]
-
t
Secondary water release from the reservoir in timet [L3]
-
O
t
Reservoir release to the natural channel in timet [L3]
-
Od
t
Irrigation demand in timet [L3]
-
Om
1
Annual OM cost of irrigation [Rs.105]
-
Om
1
Annual OM cost function for irrigation [Rs.105 L–3]
-
Om
1,t
Fraction of annual OM cost for irrigation in periodt [Rs.105]
-
Om
2
Annual OM cost of reservoir [Rs.105]
-
Om
2
Annual OM cost function for reservoir [Rs.105 L–3]
-
Om
2,t
Fraction of annual OM cost for reservoir in periodt [L3]
-
Omint
Lower bound onO
t in timet [L3]
-
Omaxt
Upper bound onO
t in timet [L3]
-
P
t
Precipitation directly upon reservoir in timet [L3]
-
S
t
Gross/live reservoir storage at the end of timet (gross storage in the linear program and live storage in the dynamic program) [L3]
-
S
t–1
Gross/live reservoir storage at the beginning of timet [L3]
-
t
Any time period
- Trt
Transformation function
-
U
1
Upper bound onIr [L3]
-
U
2
Upper bound onY [L3]
-
Y
Total capacity of reservoir at maximum pool level [L3]
-
Ya
Fixed active (live) capacity of the reservoir (Y-Yd) [L3]
-
Ya
t
Active (live) capacity (Ymaxt –Ymint) of the reservoir in timet [L3]
-
Yd
Dead storage of the reservoir [L3]
-
Ymaxt
Capacity up to the normal pool level of the reservoir in timet [L3]
-
Ymaxt
Live capacity up to the normal pool level of the reservoir in timet [L3]
-
Ymint
Capacity up to the minimum pool level of the reservoir in timet [L3]
-
Ymint
Live capacity up to the minimum pool level of the reservoir in timet [L3] 相似文献
19.
20.
Andrzej Karbowski 《Water Resources Management》1993,7(3):207-223
This article presents the formal analysis of a problem of the optimal flood control in systems of serially connected multiple water reservoirs. It is assumed, that the basic goal is minimization of the peak flow measured at a point (cross-section) located downstream from all reservoirs and that inflows to the system are deterministic. A theorem expressing sufficient conditions of optimality for combinations of releases from the reservoirs is presented together with the relevant proof. The main features of the optimal combinations of controls are thoroughly explained. Afterwards, two methods of determining the optimal releases are presented. Finally, the results of the application of the proposed methodology to a small, four reservoir system are presented.Notations
c
i
contribution of theith,i=1, ...,m, reservoir to the total storage capacity of the multireservoir system
-
d
i
(t)
one of the uncontrolled inflows to the cascade at timet (fori=1 main inflow to the cascade, fori=2, ...,m, side inflow to theith reservoir, fori=m+1 side inflow at pointP)
-
total inflow to theith reservoir,i=2, ...,m, at timet (i.e., inflowd
i
augmented with properly delayed releaser
i–1 from the previous reservoir) (used only in figures)
-
d(t),d
S
(t)
(the first term is used in text, the second one in figures) aggregated inflow to the cascade (natural flow at pointP) at timet
-
time derivative of the aggregated inflow at timet
-
i
reservoir index
-
m
number of reservoirs in cascade
-
P
control point, flood damage center
-
minimal peak of the flow at pointP (cutting level)
-
Q
p
(t)
flow measured at pointP at timet
-
flow measured at pointP at timet, corresponding to the optimal control of the cascade
-
r
i
(t)
release from theith reservoir at timet, i=1, ...,m
-
optimal release from theith reservoir at timet, i=1, ...,m
-
r
1
*
(t)
a certain release from theith reservoir at timet, different than
,i=1, ...,m, (used only in the proof of Theorem 1)
-
a piece of the optimal release from themth reservoir outside period
at timet
-
assumed storage of theith reservoir at time
(used only in the proof of Theorem 1)
-
s
i
(t)
storage of theith reservoir at timet, i=1, ...,m
-
time derivative of the storage of theith reservoir at timet, i=1, ...,m
-
storage capacity of theith reservoir,i=1, ...,m
-
(the first term is used in text, the second one in figures) total storage capacity of the cascade of reservoirs
-
S*
sum of storages, caused by implementingr
i
*
,i=1, ...,m, of all reservoirs measured at
(used only in the proof of Theorem 1)
-
t
time variable (continuous)
-
t
0
initial time of the control horizon
-
t
a
initial time of the period of constant flow equal
at pointP
-
initial time of the period of the essential filling of theith reservoir,i=1, ...,m (used only in the proof of Theorem 1)
-
t
b
final time of the period of constant flow equal
at pointP
-
final time of the period of the essential filling of theith reservoir,i=1, ...,m (used only in the proof of Theorem 1)
-
time of filling up of theith reservoir while applying method with switching of the active reservoir
-
t
f
final time of the control horizon
-
fori=1, ...,m–1, time lag betweenith andi+1th reservoir; fori=m time lag between the lowest reservoir of the cascade and the control pointP 相似文献