Optimizing Operation of Water Supply Reservoir: The Role of Constraints

Table 1

Dual values of constraints in reservoir operation optimization.

Dual value	Meaning of dual value
𝜆WA	Marginal value of WA, for example, marginal utility from one unit increases in WA.
𝜆𝑟,𝑡 (𝑡=1, 2, …, 𝐻)	Marginal value of  𝑟𝑡⋅𝜆𝑟,𝑡 is positive when 𝑟𝑡=𝑟𝑡, and the value indicates utility gain from using one more unit of water in period 𝑡.
𝜆−𝑟,𝑡 (𝑡=1, 2, …, 𝐻)	Marginal value of −𝑟𝑡⋅𝜆−𝑟,𝑡  is positive when  −𝑟𝑡=−𝑟𝑡, and the value indicates utility gain from saving one more unit of water in period 𝑡.
𝜆𝑠,𝑡 (𝑡=1, 2, …, 𝐻−1)	Marginal value of   𝑠𝑡+1⋅𝜆𝑠,𝑡 is positive when 𝑠𝑡+1=𝑠𝑡+1, and the value indicates utility gain from refilling reservoir storage by one more unit at the beginning of period 𝑡+1.
𝜆−𝑠,𝑡 (𝑡=1, 2, …, 𝐻−1)	Marginal value of −𝑠𝑡+1⋅𝜆−𝑠,𝑡 is positive when −𝑠𝑡+1=−𝑠𝑡+1, and the value indicates utility gain from drawing down reservoir storage by one more unit at the beginning of period 𝑡+1.