Optimal flood control in multireservoir cascade systems with deterministic inflow forecasts

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     

A real-time flood forecasting model based on maximum-entropy spectral analysis: I. Development

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     

Determining Hydraulic Characteristics of Production Wells using Genetic Algorithm

Proper well management requires the determination of characteristic hydraulic parameters of production wells such as well loss coefficient (C) and aquifer loss coefficient (B), which are conventionally determined by the graphical analysis ofstep-drawdowntest data. However, in the present study, the efficacy of a non-conventional optimization technique called Genetic Algorithm (GA), which ensures near-optimal or optimal solutions, is assessedin determining well parameters from step-drawdown test data. Computer programs were developed to optimize the well parametersby GA technique for two cases: (i) optimization of B and C only, and (ii) optimization of B, C and p (exponent) as well as to evaluate the well condition. The reliability and robustness of the developed computer programs were tested usingnine sets of published and unpublished step-drawdown data from varying hydrogeologic conditions. The well parameters obtained by the GA technique were compared with those obtained by the conventional graphical method in terms of root mean square error(RMSE) and visual inspection. It was revealed that the GA technique yielded more reliable well parameters with significantlylow values of RMSE for almost all the datasets, especially in caseof three-variable optimization. The optimal values of the parametersB, C and p for the nine datasets were found to range from 0.382 to 2.292 min m^-2, 0.091 to 3.262, and 1.8 to 3.6, respectively. Because of a wide variation of p, the GA techniqueresulted in considerably different but dependable and robust well parameters as well as well specific capacity and well efficiency compared to the graphical method. The condition of three wells was found to be good, one well bad and that of the remaining five wells satisfactory. The performance evaluation of the developed GA code indicated that a proper selection of generation number and population size is essential to ensure efficient optimization. Furthermore,a sensitivity analysis of the obtained optimal parameters demonstrated that the GA technique resulted in a unique set ofthe parameters for all the nine datasets. It is concluded thatthe GA technique is an effective and reliable numerical tool for determining the characteristic hydraulic parameters of production wells.     

Water table fluctuation with a random initial condition

The nonlinear Boussinesq equation is used to understand water table fluctuations in various ditch drainage problems. An approximate solution of this equation with a random initial condition and deterministic boundary conditions, recharge rate and aquifer parameters has been developed to predict a transient water table in a ditch-drainage system. The effects of uncertainty in the initial condition on the water table are illustrated with the help of a synthetic example. These results would find applications in ditch-drainage design.Notation A / tanh t - a lower value of the random variable representing the initial water table height at the mid point - a+b Upper value of the random variable representing the initial water table height at the midpoint - B tanh t - C 4/ - h variable water table height - h mean of the variable water table height - h _m variable water table height at the mid point - h _m mean of the variable water table height at the mid point - K hydraulic conductivity - L half spacing between the ditches - m ₀ initial water table height at the mid point - N Uniform rate of recharge - S specific yield - t time of observation - x distance measured from the ditch boundary - (4/SL)(NK)^1/2 - (L/4)(N/K)^1/2 - dummy integral variable     

Time-dependent soil-water distribution under a circular trickle source

Soil-water distribution in homogeneous soil profiles of Yolo clay loam and Yolo sand (Typic xerorthents) irrigated from a circular source of water, was measured several times after the initiation of irrigation. The effect of trickle discharge rates and soil type on the locations of the wetting front and soil-water distribution was considered. Soil-water tension and hydraulic conductivity, as functions of soil-water content, were also measured. The theories of time-dependent, linearized infiltration from a circular source and a finite-element solution of the two-dimensional transient soil-water equation were compared with the experimental results. In general, for both soils the computer horizontal and vertical advances of the wetting front were closely related to those observed. With both theories, a better prediction of the wetting front position for the clay loam soil than for the sandy soil is shown. The calculated and measured horizontal vertical advances did not agree over long periods of time. With the linearized solution, overestimated and underestimated vertical advances for the clay and sandy soils, respectively, were shown. The finite-element model approximate in a better way the vertical advances than the linearized solution, while an opposite tendency for the horizontal advances indicated, especially in sandy soil.Notation k constant (dK/d) - K hydraulic conductivity - K ₀ saturated hydraulic conductivity - J ₀,J ₁ Bessel functions of the first kind - h soil water tension - q Q/r ₀ ² - Q discharge rate - r cylindrical coordinate; also horizontal distance in soil surface - R dimensionless quantity forr - r ₀ constant pond radius - R ₀ dimensionless quantity forr ₀ - t time - T dimensionless quantity fort - x, y Cartesian coordinates - z vertical coordinate; also vertical distance along thez axis chosen positively downward - Z dimensionless quantity forz - empirical soil characteristic constant - dummy variable of integration - volumetric soil water content - matrix flux potential - dimensionless quantity for      

On the predictability of the water-table variation in a ditch-drainage system

The irrigation in regions of brackish groundwater in many parts of the world results in the rise of the water-table very close to the groundsurface. The salinity of the productive soils is therefore increased. A proper layout of the ditch-drainage system and the prediction of the spatio-temporal variation of the water table under such conditions are of crucial importance in order to control the undesirable growth of the water-table. In this paper, an approximate solution of the nonlinear Boussinesq equation has been derived to describe the water-table variations in a ditch-drainage system with a random initial condition and transient recharge. The applications of the solution is discussed with the help of a synthetic example.Notations a lower value of the random variable representing the initial water-table height at the groundwater divide - a+b upper value of the random variable representing the initial water-table height at the groundwater divide - h variable water-table height measured from the base of the aquifer - K hydraulic conductivity - L half width between ditches - m ₀ initial water-table height at the groundwater divide - N(t) rate of transient recharge at time t - N ₀ initial rate of transient recharge - P N ₀/K - S Specific yield - t time of observation - t ₀ logarithmic decrement of the recharge function - T Kt/SL - x distance measured from the ditch boundary - X x/L - Y h/L - Y mean of Y - Y Variance of Y     

Determination of optimal unit hydrographs by linear programming

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     

Chemical Evaluation of Ma’an Sewage Effluents and its Reuse in Irrigation Purposes

The present study investigates the chemical composition of Ma’an Wastewater Treatment Plant in south Jordan. Samples of effluent of this plant were collected over 1 year period. All samples were analyzed for pH, conductivity, major ions (Cl⁻, , , , , Na⁺, K⁺, Ca^2 + and Mg^2 +) and trace metals B, Fe, Cu, Zn, Cd and Pb. The pH value ranges from 6.79 to 8.15 with a median value of 7.39 ± 0.32. The water quality was characterized by its high salinity hazard (C3) and low sodium hazard (S1) which can be considered as marginal for human consumption. Moreover, concentrations of trace metals in treated wastewater were found to be low and within guidelines for irrigation water due to low level of industrialization activities in the study area. Generally, the result of this study suggests that the treated wastewater is suitable for irrigational purposes, while these effluents can be considered as possible additional resources for irrigation in Jordan.     

Correspondence of the scheme and method of calculating the stability factor

The use of the main methods of calculating K_c in practice guarantees an accuracy of the calculation within

Evaluation of the SCS-CN-Based Model Incorporating Antecedent Moisture

Using a large set of rainfall-runoff data from 234 small to large watersheds from USA, this paper evaluates the modified version of the [Mishra, S. K. and Singh, V. P., 2002a, SCS-CN-based hydrologic simulation package, in V. P. Singh and D. K. Frevert (eds), Mathematical Models in Small Watershed Hydrology, Water Resources Publications, Chap. 13, pp. 391–464] (MS) model which is based on the Soil Conservation Service Curve Number (SCS-CN) methodology and incorporates the antecedent moisture in direct surface runoff computations. Comparison with the existing SCS-CN method using the t-test and the ranking-based grading shows that the modified MS model performs far better than the existing SCS-CN model.     

Comparative Evaluation of Analytic Solutions in Predicting Soil Moisture Profiles in Vertical One-Dimensional Infiltration under Ponded and Constant Flux Boundary Conditions

The problem of vertical one-dimensional infiltration for both ponded and constant flux boundary conditions was studied through the use of existing analytic solutions. Main objective was to compare the soil moisture profile developed under constant flux boundary condition at the time of ponding , with that moisture profile developed under ponded conditions at an earlier time . Time t _C denotes the time when the decreasing infiltration rate for the ponded conditions becomes equal to the constant flux _q , applied for the constant flux case. One might state that the analytical solutions, for both cases do not give identical profiles. An approximate coincidence might be brought about through a modification in the diffusivity which, in many respects, seems justified. Practical outcome of the above analysis is the determination of the time of ponding T, after which surface runoff starts, for the constant flux case. This is of practical significance either under natural conditions of rainfall or under conditions of sprinkle-irrigation, since surface runoff is directly related with soil erosion and waste of irrigation water. Therefore any attempt to determine the time of ponding T is well merited.     

Forecast model of water consumption for Naples

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     

Two new characteristic parameters for runoff computation

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.     

Effect of the outlet angle β2 on the characteristics of low specific-speed centrifugal pumps

Conclusions An experimental check of various low specific-speed pumps showed that on changing the geometry of the vaning of the impeller, in conformity with the recommendations [1], the main feature of which is an increase of the outlet angles of the vanes to values 90°₂130° and installation of short intermediate vanes in the passages between the main vanes, the head increases considerably without a decrease of pump efficiency.For centrifugal pumps with a volute and diffuser whose impellers have a different number of main vanes, the most rational is the installation of three intermediate vanes in each vane passage with a change in the direction of the flow by the main vane toward an increase of the outlet angle on diameter D₂=0.5–0.65D₃ for fecal pumps and pumps with short vanes.The use of the new impeller design is especially effective for high rotative-speed pumps. For multistage submersible pumps with a guide-vane apparatus at the inlet and axial outlet, the optimal variant is the installation of one intermediate vane with a start of the change in the flow direction D₃=0.63D₂.In conclusion, it is necessary to note that all tests were conducted without modifying the casing and branch pipes, which leaves room for a further increase of the head developed by pumps with the new type of impeller.Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 5, pp. 21–25, May, 1982.     

An economic input-output analysis for urban stormwater quality planning

A new approach using input-output techniques is proposed for the analysis of urban stormwater pollution caused by urban land development. The input-output model provides projections of sectoral outputs within an urban region. By defining land as an input to production, these output projections may be translated into projections of commercial and industrial land development. Furthermore, the closed version of the input-output model is used to project residential land development as a function of projected wage income. The pollutant generation in urban stormwater is related to the quantity of each category of land development by a pollutant coefficient matrix. Thus, the model can be used to predict the impact of various economic growth scenarios on pollution loadings in runoff water. This will help planners in assessing the environmental costs of various scenarios, and in preparing for remedial actions. A numerical example is provided to illustrate the applications of the model.     

Spectral Reflectance,Growth and Chlorophyll Relationships for Rice Crop in a Semi-Arid Region of India

Relations among spectral reflectance, chlorophyll a, and growth of rice plants grown on irrigated light textured soil in a semi arid region are presented here. There was a linear relation between spectral reflectance and rice plant height (r = 0.97), for band 1 (0.45–0.52 m) reflectance values. On the other hand, in bands 2 (0.52–0.60 m) and 3 (0.63–0.69 m), reflectance values decreased until 70 days after planting (DAP) and then increased during the reproductive phase of the crop. The near infrared band 4 (0.76–0.90 m) showed a maximum reflectance at 59 DAP (panicle initiation stage) and a decline in reflectance thereafter through maturity. The peak value of IR/R ratio was 16.39 at 62 DAP during the early reproductive phase; thereafter, it declines gradually with the maturity of the crop. Chlorophyll a concentration was high during early growth (vegetative and early reproductive stages) and decreased during the flowering and maturity stages. The rice plant canopy show a high chlorophyll a concentration at 64 and 59 DAP for sites A and B, respectively. Chlorophyll a concentration is higher in site A plant canopies than it is in site B during the entire crop cycle. A good inverse correlation (r = 0.91) has been found between chlorophyll a and band 1, while the IR/R ratio and the normalised difference vegetation index (NDVI) showed a relationship (r = 0.78) with the chlorophyll a concentration during the crop cycle. Band 2, 3 and 4 radiance values show a biphasic linear relationship with chlorophyll a concentrations, negative for early growth and positive for flowering and maturity stages. Results indicate that the period between 66 to 70 DAP is most suitable for the assessment of rice crop yield, based on chlorophyll a concentration.     

Vladimirskaya hydroelectric pumped storage power plant as the first pilot power project of the 21st century

The authors describe the Vladimirskaya hydroelectric pumped storage power plant that is currently being designed as the pilot Russian power project for the 21st century.Translated from Gidrotekhnicheskoe Stroitelstvo, No. 8, August 2004, pp. 15 – 20.     

Optimization-simulation models for the design of an Irrigation project

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] - Omin_t Lower bound onO _t in timet [L³] - Omax_t 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 - Tr_t 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 (Ymax_t –Ymin_t) of the reservoir in timet [L³] - Yd Dead storage of the reservoir [L³] - Ymax_t Capacity up to the normal pool level of the reservoir in timet [L³] - Ymax_t Live capacity up to the normal pool level of the reservoir in timet [L³] - Ymin_t Capacity up to the minimum pool level of the reservoir in timet [L³] - Ymin_t Live capacity up to the minimum pool level of the reservoir in timet [L³]