Multistage scenario-based interval-stochastic programming for planning water resources allocation

Abstract

In this study, a multistage scenario-based interval-stochastic programming (MSISP) method is developed for water-resources allocation under uncertainty. MSISP improves upon the existing multistage optimization methods with advantages in uncertainty reflection, dynamics facilitation, and risk analysis. It can directly handle uncertainties presented as both interval numbers and probability distributions, and can support the assessment of the reliability of satisfying (or the risk of violating) system constraints within a multistage context. It can also reflect the dynamics of system uncertainties and decision processes under a representative set of scenarios. The developed MSISP method is then applied to a case of water resources management planning within a multi-reservoir system associated with joint probabilities. A range of violation levels for capacity and environment constraints are analyzed under uncertainty. Solutions associated different risk levels of constraint violation have been obtained. They can be used for generating decision alternatives and thus help water managers to identify desired policies under various economic, environmental and system-reliability conditions. Besides, sensitivity analyses demonstrate that the violation of the environmental constraint has a significant effect on the system benefit.

Appendix 1 Nomenclatures for variables and parameters

f ^± :

net system benefit over the planning horizon ($)

t :

time period, and t = 1, 2,…,T

$A_{1}^{0}$ :

storage-area coefficient for reservoir 1

$A_{2}^{0}$ :

storage-area coefficient for reservoir 2

$A_{1}^{a}$ :

area (per unit of active storage volume) above $A_{1}^{0} ;$

$A_{2}^{a}$ :

area (per unit of active storage volume) above $A_{2}^{0} ;$

$De_{t}^{ \min }$ :

minimum amount of water demand for the municipality in period t (m³)

$De_{t}^{ \max }$ :

maximum water demand for the municipality in period t (m³)

e_1t :

average evaporation rate for reservoir 1 in period t

e_2t :

average evaporation rate for reservoir 2 in period t

$E_{1t}^{ \pm }$ :

evaporation loss of reservoir 1 in period t (m³)

$E_{2t}^{ \pm }$ :

evaporation loss of reservoir 2 in period t (m³)

$K_{1}^{t}$ :

number of possible scenarios for stream 1 in period t

$K_{2}^{t}$ :

number of possible scenarios for stream 2 in period t

$NB_{t}^{ \pm }$ :

net benefit per unit of water allocated in period t ($/m³)

$PE_{t}^{ \pm }$ :

penalty per unit of water not delivered in period t ($/m³), and PE _t  > NB _t

$p_{{tk_{1} }}$ :

probability of occurrence of scenario k ₁ (for stream 1) in period t, with $p_{{tk_{1} }} > 0$ and $\sum\limits_{{k_{1} = 1}}^{{K_{1}^{t} }} {p_{{tk_{1} }} = 1}$

$p_{{tk_{2} }}$ :

probability of occurrence of scenario k ₂ (for stream 2) in period t, with $p_{{tk_{2} }} > 0$ and $\sum\limits_{{k_{2} = 1}}^{{K_{2}^{t} }} {p_{{tk_{2} }} = 1}$

$Q_{{tk_{1} }}^{ \pm }$ :

random inflow into stream 1 in period t under scenario k ₁ (m³)

$Q_{{tk_{2} }}^{ \pm }$ :

random inflow into stream 2 in period t under scenario k ₂ (m³)

$R_{{tk_{1} }}^{ \pm }$ :

release flow from reservoir 1 in period t under scenario k ₁ (m³)

$R_{{tk_{1} k_{2} }}^{ \pm }$ :

release flow from reservoir 2 in period t under scenarios k ₁ and k ₂ associated with joint probabilities of $p_{{tk_{1} }} p_{{tk_{2} }}$ (m³)

$RSC_{1}^{ \pm }$ :

storage capacity of reservoir 1 (m³)

$RSC_{2}^{ \pm }$ :

storage capacity of reservoir 2 (m³)

$RSV_{1}^{ \pm }$ :

reserved storage level for reservoir 1 (m³)

$RSV_{2}^{ \pm }$ :

reserved storage level for reservoir 2 (m³)

$S_{{tk_{1} }}$ :

storage level in reservoir 1 in period t under scenario k ₁ (m³)

$S_{{tk_{1} k_{2} }}$ :

storage level in reservoir 2 in period t under scenarios k ₁ and k ₂ (m³)

X _t :

water allocation target that is promised to the municipality in period t (m³)

$Y_{{tk_{1} k_{2} }}^{ \pm }$ :

shortage level by which the water-allocation target is not met under scenarios k ₁ and k ₂ which is associated with joint probabilities of $p_{{tk_{1} }} p_{{tk_{2} }}$(m³)