Stochastic Programming

The canonical structure splits decisions into two temporal stages. First-stage decisions (e.g., generator commitment) are made before the random vector ξ is observed. After uncertainty realization, second-stage recourse actions (e.g., dispatch adjustments, load shedding) compensate for constraint violations.

• Objective: Minimize first-stage cost + expected value of second-stage cost
• Recourse matrix W defines how corrective actions map to constraints
• Penalizes infeasibility through explicit cost functions rather than hard constraints
• Example: Committing thermal units day-ahead (stage 1), then dispatching based on actual wind output (stage 2)

Continuous uncertainty distributions are discretized into a finite set of scenarios ω ∈ Ω, each with probability p_ω. This transforms the stochastic program into a large-scale deterministic equivalent that can be solved with standard optimization solvers.

• Each scenario represents one possible realization of all uncertain parameters
• Scenario tree captures multi-stage temporal dependencies through branching
• Scenario reduction techniques (e.g., fast forward selection) prune similar scenarios to maintain tractability
• The deterministic equivalent grows linearly with |Ω|, motivating efficient sampling strategies

A fundamental structural requirement ensuring decisions cannot exploit perfect foresight of future outcomes. Non-anticipativity mandates that decisions sharing the same information history must be identical across scenarios.

• Enforced through explicit equality constraints in the deterministic equivalent
• In multi-stage formulations, constraints apply at each node of the scenario tree
• Violating non-anticipativity produces artificially optimistic solutions that assume clairvoyance
• Lagrangian decomposition methods exploit this structure by dualizing these constraints

Standard stochastic programming minimizes expected cost, which is risk-neutral. Risk-averse formulations incorporate higher moments or tail measures to penalize costly extreme outcomes.

• Mean-risk models add a weighted risk term: min E[cost] + λ · Risk_measure
• Conditional Value at Risk (CVaR) minimizes expected cost in the α-tail of the distribution
• Second-order stochastic dominance constraints ensure the solution distribution is preferred by all risk-averse decision makers
• Critical for grid applications where low-probability blackouts carry catastrophic consequences

The deterministic equivalent quickly becomes computationally intractable as scenarios multiply. Benders decomposition (L-shaped method) exploits the block-diagonal structure of the recourse problem.

• Master problem optimizes first-stage variables with an approximation of recourse costs
• Subproblems (one per scenario) evaluate recourse costs and generate optimality/feasibility cuts
• Cuts iteratively refine the master problem's approximation until convergence
• Progressive hedging offers an alternative by decomposing across scenarios and penalizing non-anticipativity violations

A quantitative metric justifying the computational expense of stochastic programming over simpler deterministic approaches. VSS measures the expected cost improvement from explicitly modeling uncertainty.

• EEV: Expected result of using the Expected Value solution (solve deterministic mean problem, then evaluate under uncertainty)
• RP: Recourse Problem solution (true stochastic optimum)
• VSS = EEV − RP, representing the value of incorporating distributional information
• A positive VSS demonstrates that uncertainty matters; near-zero VSS suggests deterministic approximations suffice

The foundational structure of stochastic programming where first-stage decisions (here-and-now) are made before uncertainty is realized, and second-stage recourse actions (wait-and-see) correct any infeasibility after the random event occurs. In grid planning, a first-stage decision might be committing a baseload generator, while the recourse action is dispatching a fast-ramping battery to compensate for a wind forecast error. The objective minimizes first-stage costs plus the expected value of recourse costs over all scenarios.

A discrete representation of how uncertainty unfolds over time, structured as a rooted tree where nodes represent possible states of the world at each decision stage. Each path from root to leaf is a scenario with an associated probability. In power systems, a scenario tree might branch on wind speed realizations at each hour. The challenge is scenario generation and reduction—too many branches cause computational intractability, while too few lose statistical fidelity. Techniques like Kantorovich distance are used to merge similar nodes.

A fundamental principle requiring that decisions at any stage depend only on information available up to that point—you cannot anticipate the future. Mathematically, if two scenarios are indistinguishable up to stage t, the decisions must be identical. These constraints are enforced explicitly in scenario formulation or implicitly in nodal formulation. Violating non-anticipativity is equivalent to assuming perfect foresight, which produces optimistically biased solutions that underestimate the true cost of uncertainty.

A solution algorithm that exploits the block-diagonal structure of two-stage stochastic programs. The master problem solves for first-stage variables, while subproblems evaluate each scenario independently. Optimality cuts and feasibility cuts are iteratively added to the master problem based on dual information from subproblems. This decomposition is particularly effective for stochastic unit commitment where the commitment decisions couple all scenarios, but the economic dispatch can be solved per scenario in parallel.

A Monte Carlo-based method that replaces the true probability distribution with an empirical distribution derived from a finite set of N i.i.d. samples. The stochastic program is then solved as a deterministic equivalent problem. Key considerations include in-sample bias (the optimal value is optimistically biased downward for minimization) and out-of-sample validation on a fresh test set. SAA is widely used in probabilistic power flow when the underlying distributions of renewable generation are known only through historical data.

A metric quantifying the benefit of explicitly modeling uncertainty versus using a deterministic expected-value approach. VSS is computed as the difference between the expected result of using the expected value solution (EEV) and the recourse problem solution (RP). A high VSS indicates that ignoring uncertainty leads to significantly suboptimal decisions. In grid contexts, VSS can justify investment in stochastic optimization software by demonstrating avoided costs from better reserve procurement.