Planning hydroelectric resources with recourse-based multistage interval-stochastic programming

Abstract

Optimization models play an important role in long-term hydroelectric resources planning. The effectiveness of an optimization model, however, depends on its capability of dealing with uncertainties. This study presents a multistage interval-stochastic programming model for long-term hydropower planning, in which uncertainties are reflected as randomness and intervals. The model is developed based on interval programming technique and recourse-based multistage stochastic programming and using the expected value of long-term hydroelectric profit as the objective function. A solution method of the developed model is also presented, which is based on a decomposition method by partitioning the multistage interval-stochastic program into two-stage stochastic programming sub-problems in each scenario-tree node. A hypothetical case study is used to demonstrate the developed model and its solution method. Modeling results demonstrates the computationally effectiveness of the solution method and reveal the applicability of the developed model for long term planning of hydroelectric resources.

Appendix: Linear features in a conventional TSP

A conventional TSP often couples a linear master program and a subproblem with an objective function presented as convex polyhedrom. In each iteration of the basic algorithm for solving TSP, the later is used to recursively update a piecewise linear approximation of the recourse function, while the former is used as a means to generate successive iterates (Wets 1974). Such a linear feature in the conventional TSP is important to solve the MSP model presented in this study. Consider a simple TSP in which model variables have same definitions as model (7):

$$ \hbox{Max} \,f = cx - \max E_{{\tilde{\xi}}} [\beta y({\tilde{\xi}})] $$

(18a)

$$ \hbox{s.t}.\ 0 \leq x - y({\tilde{\xi}}) \leq {\tilde{\xi}} $$

(18b)

$$ x_{\max} \geq x \geq 0,\, y({\tilde{\xi}}) \geq 0 $$

(18c)

After piecewise linearization of ${\tilde{\xi}},$ program (18) can be solved with the method introduced by Birge and Louveaux (1988). Assuming its solutions as {f _opt, x _opt, y _opt } and ${\hat{x}}$ the value of x when f reaches its extreme $f({\hat{x}}).$ Here, x _opt = ${\hat{x}}$ when ${\hat{x}} < x_{\max}$ and equals x _max when ${\hat{x}} \geq x_{\max}.$ When ${\hat{x}} < x_{\max}$ exists, assuming all values of ${\tilde{\xi}}$ an increase of ${\bar{\xi}} ({\bar{\xi}} \geq 0).$ Thus, ${\tilde{\xi}}$ becomes ${\tilde{\xi}}+{\bar{\xi}}$ and a new TSP can be formulated as:

$$ \hbox{Max}\,f^{(1)} = cx^{(1)} - \max E_{{\tilde{\xi}}} [\beta y^{(1)} ({\tilde{\xi}})] $$

(19a)

$$ \hbox{s.t}. \,0 \leq x^{(1)} - y^{(1)} ({\tilde{\xi}}) \leq {\tilde{\xi}} + {\bar{\xi}} $$

(19b)

$$ x_{\max} \geq x^{(1)} \geq 0,\, y^{(1)} ({\tilde{\xi}}) \geq 0 $$

(19c)

Let $x^{(1)} = x^{(2)} + {\bar{\xi}}.$ Problem (19) can be reformulated as:

$$ \hbox{Max} f^{(1)} = f^{(2)} + {\bar{\xi}} $$

(20a)

and,

$$ f^{(2)} = \{cx^{(2)} - \max E_{{\tilde{\xi}}} [\beta y^{(2)}, {\tilde{\xi}}]\} $$

(20b)

$$ \hbox{s.t}.\ 0 \leq x^{(2)} - y^{(2)} ({\tilde{\xi}}) \leq {\tilde{\xi}} $$

(20c)

$$ x_{\max} \geq x^{(2)} + {\bar{\xi}} \geq 0, y^{(2)} ({\tilde{\xi}}) \geq 0 $$

(20d)

According to constraint (18c), 0 ≤ x ≤ x _max holds. Obviously, program f ⁽²⁾ and (18) have the same solutions. The solutions of program (20) can be expressed as:

$$ f_{\rm opt}^{(1)} = f_{\rm opt}^{(2)} + c \cdot {\bar{\xi}} = f_{\rm opt} + c \cdot {\bar{\xi}},x_{\rm opt}^{(1)} = {\hat{x}} + {\bar{\xi}}, \,\hbox{and}\,y_{\rm opt}^{(1)} = y_{\rm opt}^{(2)} = y_{\rm opt} $$

(21)

If all values of ${\tilde{\xi}}$ has a decrease in ${\bar{\xi}} ({\bar{\xi}}\geq 0),$ a new TSP can be formulated, with a similar structure as problem (20). The solutions of this new TSP model also have linear relations as problem (18) has, which is expressed in (21). When ${\bar{\xi}}$ has a change, a corresponding TSP with a similar structure as problem (18) can be formulated and their solutions can be obtained based on the linear relation between ${\bar{\xi}}$ and f _opt. When relation ${\hat{x}} \geq x_{\max}$ exists, a series of new TSP models can be formulated with a similar structure as problem (19). A general type of such new TSP problems is expressed as:

$$ \hbox{Max}\, f^{(3)} = cx^{(3)} - \max E_{{\tilde{\xi}}} [\beta y^{(3)} ({\tilde{\xi}})] $$

(22a)

$$ \hbox{s.t}.\ 0 \leq x^{(3)} - y^{(3)} ({\tilde{\xi}}) \leq {\tilde{\xi}} - {\bar{\xi}} $$

(22b)

$$ x_{\max} \geq x^{(3)} \geq 0, y^{(3)} ({\tilde{\xi}}) \geq 0 $$

(22c)

Based on (18)–(21), the solutions of model (22) can be obtained, which can be expressed as:

$$ f_{\rm opt}^{(3)} = f_{\rm opt} - c \cdot {\bar{\xi}}, x_{\rm opt}^{(3)}= {\hat{x}} - {\bar{\xi}}, \,\hbox{and}\,y_{\rm opt}^{(3)} = y_{\rm opt} $$

(23)

Model (22) can have its maximum objective-function value $f_{\rm opt} = c \cdot ({\hat{x}} - x_{\max})$ when ${\bar{\xi}} = {\hat{x}} - x_{\max}.$ The above analyses reveal linear relations among ${\bar{\xi}},f_{\rm opt},$ and x _opt. When ${\bar{\xi}}$ changes, the convex polyhedrom $E[\beta y,{\bar{\xi}})$ in problem (18) has no influences on f _opt and x _opt; meanwhile, y _opt remains unchanged. When all values of ${\tilde{\xi}}$ increases to ${\tilde{\xi}}+{\bar{\xi}},$ the value of f _opt will increase to $f_{\rm opt} + c \cdot {\bar{\xi}};$ at the same time, the value of x _opt will increase to $x_{opt}+ {\bar{\xi}}.$ Such linear relations can facilitate a simple solution method for model (5).