Nonlinear chance-constrained problems with applications to hydro scheduling Enrico Malaguti DEI - University of Bologna Joint work with Andrea Lodi ◦ , Giacomo Nannicini ∗ , Dimitri Thomopulos ◦ ◦ DEI - University of Bologna ∗ Singapore University of Technology and Design Aussois, January 7th, 2015 (slides stolen from Giacomo) Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 1 / 29
The mid-term hydro scheduling problem 1 Problem decomposition and branch-and-cut 2 A separation routine 3 Finiteness 4 Computational experiments 5 Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 2 / 29
Scheduling for a hydro network Motivating problem: Hydro power generation requires scheduling the operation of multiple power plants in advance. The decision maker must determine how to produce electrical power to maximize profit taking into account the demand. All decisions affect the future state of the network, because water released from upstream reservoirs flows into downstream reservoirs. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 3 / 29
Scheduling for a hydro network Motivating problem: Hydro power generation requires scheduling the operation of multiple power plants in advance. The decision maker must determine how to produce electrical power to maximize profit taking into account the demand. All decisions affect the future state of the network, because water released from upstream reservoirs flows into downstream reservoirs. Uncertainty is a complicating factor: Rainfall cannot be accurately predicted beyond a few days. Aggregated demand can be partially inferred from historical data, but exogenous factors (e.g. weather) make it uncertain. Energy prices fluctuate over time. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 3 / 29
The uncertainty model What we know: [Baslis and Bakirtzis, 2011] assume that forecasts for aggregated demand and precipitations are available as random parameters with discrete support and discrete probabilities. To optimize over a long enough time period (i.e. six months), they assume that random parameters in some time period depend on the realization in the previous period. This gives rise to a scenario tree, where a scenario is a realization of the random parameters. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 4 / 29
The scenario tree Stage 1: month 1 Stage 2: months 2−4 Stage 3: months 5−7 5 inflows, 3 demands 3 inflows, 2 demands inflow: very low demand: low In total, there are 90 scenarios. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 5 / 29
Main characteristics of the problem Problem structure: Multi-stage: at each stage we must decide how much water to release from the reservoirs (determines how much energy we can produce). Underlying flow network structure: water flows between reservoirs. The constraints impose lower and upper bounds on water levels, maximum water released, and minimum production requirements. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 6 / 29
Main characteristics of the problem Problem structure: Multi-stage: at each stage we must decide how much water to release from the reservoirs (determines how much energy we can produce). Underlying flow network structure: water flows between reservoirs. The constraints impose lower and upper bounds on water levels, maximum water released, and minimum production requirements. The difficult part: The production function is nonlinear: at a first approximation concave, but in fact nonconvex. The inflows and the demands are random parameters: the profit over the entire time horizon is therefore nondeterministic. Prices are given by a step function depending on the amount produced and the energy market: modeling this requires discrete variables. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 6 / 29
Dealing with uncertainty The decision-making model: As time moves forward, new information comes in, and the decision-maker may re-evaluate her options. At the time of solving the problem, we should be concerned with current decisions: future decisions can be determined at a later stage. Since profit is uncertain, what do we optimize? Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 7 / 29
Dealing with uncertainty The decision-making model: As time moves forward, new information comes in, and the decision-maker may re-evaluate her options. At the time of solving the problem, we should be concerned with current decisions: future decisions can be determined at a later stage. Since profit is uncertain, what do we optimize? Why not worst case: We could optimize a lower bound on the profit in the worst-case scenario. This is typically a very conservative bound. We would like a more flexible model where the trade-off between robustness and profit can be controlled. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 7 / 29
Optimizing quantiles Interpreting the profit: Assume there are k possible future scenarios ξ 1 , . . . , ξ k . Let ϕ ( x = ¯ x, ξ = ξ i ) be the profit obtained in the i -th scenario with decision variables ¯ x . If ¯ x is given, we can compute the distribution of ϕ . Our formulation: choose x in order to maximize the α -quantile of ϕ . The parameter α controls the level of risk. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 8 / 29
The mid-term hydro scheduling problem 1 Problem decomposition and branch-and-cut 2 A separation routine 3 Finiteness 4 Computational experiments 5 Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 9 / 29
Formulation as chance-constrained problem Feasible region and scenarios: Some constraints of the problem are deterministic: scenario-independent, x ∈ X . Other constraints are nondeterministic: scenario-dependent. Let C ( ξ ) be the feasible region of the problem given scenario ξ . We want to take decisions in such a way that the probability associated with satisfied scenarios is large. max { lower bound on profit : Pr( x ∈ C ( ξ )) ≥ 1 − α, x ∈ X } Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 10 / 29
Basic idea for the formulation Linear case (for simplicity): Variables: general-variables x , scenario-variables y . Introduce binary variables z to determine which scenarios are satisfied. A possible but impractical formulation is: min cx s.t.: Ax ≥ b b 1 − Mz 1 A 1 x H 1 y 1 + ≥ b 2 − Mz 2 A 2 x H 2 y 2 + ≥ . . . . . . b k − Mz k A k x H k y K + ≥ p 1 z 1 + p 2 z 2 + . . . + p k z k ≤ α y 1 , y 2 , y k x, . . . ≥ 0 z 1 , z 2 , . . . z k ∈ { 0 , 1 } . Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 11 / 29
Decomposing the problem Master problem: First and second stage decisions: x (because of nonanticipativity). A variable x ϕ to indicate the α -quantile of the profit, which is maximized. Constraints that are independent of the realization of the random parameters: x ∈ X . Scenario subproblems, one for each scenario: First and second stage decisions x , third stage decisions: y . A constraint: x ϕ ≤ profit for current scenario. Scenario-dependent constraints. Let C x,y ( ξ i ) be the feasible region of a scenario, and define C x ( ξ i ) := Proj x C x,y ( ξ i ) . So ˆ x is feasible for scenario i if ˆ x ∈ C x ( ξ i ) . Basic idea: Generate solutions for the master. If they are not feasible for enough scenarios to satisfy the joint chance-constraint, cut them off. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 12 / 29
Benders decomposition and Branch-and-Cut An exact solution approach: A Branch-and-Cut was proposed by [Luedtke, 2013] for this type of problems with finite support – some conditions must be satisfied. It uses a separation routine for the scenario subproblems, combined with the variables z i . We can enforce the constraint z i = 0 ⇒ x ∈ C x ( ξ i ) using an approach similar to [Codato and Fischetti, 2006]. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 13 / 29
A Branch-and-Cut approach: overview Assumptions: The scenario subproblems are polyhedra with the same recession cone. The objective function of the master problem is “well-behaved”, e.g. linear or convex over a compact. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 14 / 29
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