ITOM: an interval-parameter two-stage optimization model for stochastic planning of water resources systems

Abstract

Planning of water resources systems is often associated with many uncertain parameters and their interrelationships are complicated. Stochastic planning of water resources systems is vital under changing climate and increasing water scarcity. This study proposes an interval-parameter two-stage optimization model (ITOM) for water resources planning in an agricultural system under uncertainty. Compared with other optimization techniques, the proposed modeling approach offers two advantages: first, it provides a linkage to pre-defined water policies, and; second, it reflects uncertainties expressed as probability distributions and discrete intervals. The ITOM is applied to a case study of irrigation planning. Reasonable solutions are obtained, and a variety of decision alternatives are generated under different combinations of water shortages. It provides desired water-allocation patterns with respect to maximum system benefits and highest feasibility. Moreover, the modeling results indicate that an optimistic water policy corresponding to higher agricultural income may be subject to a higher risk of system-failure penalties; while, a too conservative policy may lead to wastage of irrigation supplies.

Appendix

List of symbols

The following symbols are used in this paper:

f ^± :

Net system benefit ($/m³)

f ^±_opt :

Optimized net system benefit ($/m³)

B ^± _i :

Net benefit to farm i per m³ of water allocated ($)

W ^± _i :

Water policy in terms of fixed allocation amount of water that is promised to farm i (m³)

W ^±_imax :

Maximum allowable allocation amount for farm i (m³)

C ^± _i :

Loss to farm i per m³ of water not delivered, C _i > B _i ($)

S ^± _ij :

Decision variable representing shortage of water, which is the amount by which W _i is not met when the seasonal flow is q _j (m³)

S ^±_{ij opt} :

Optimized solution of S ^± _ij decision variable

q ^± _j :

Reservoir flow quantity with probability p _j of occurrence under flow level j (m³)

p _j :

Probability of occurrence of flow level j (%)

i :

Cropping farm index; i=1, 2, 3 where i=1 represents alfalfa farm, 2 represents wheat farm, and 3 represents potato farm

j :

Flow level index, j = 1, 2, ..., 5 where j = 1 for very low flows, 2 for low flows, 3 for medium flows, 4 for high flows, and 5 for very high flows

m :

Total number of farms

n :

Total number of flow levels

z _i :

Binary decision variable