Risk Measures

Why risk matters in power system planning

Standard SDDP minimizes the expected total cost across all scenarios. This produces policies that perform well on average but can lead to extremely high costs in adverse scenarios – for example, a prolonged drought that depletes reservoirs and forces expensive thermal generation or load shedding.

Risk-averse SDDP addresses this by incorporating a risk measure into the optimization objective. Instead of minimizing only the average cost, the algorithm also penalizes outcomes in the tail of the cost distribution. The result is a policy that sacrifices a small amount of average-case performance for substantially better worst-case behavior.

Risk-neutral vs risk-averse

In risk-neutral SDDP, each backward pass aggregates per-scenario cut coefficients using the scenario probabilities $p(\omega )$. All scenarios contribute equally (weighted by probability) to the cut that approximates the future cost function.

In risk-averse SDDP, the scenario probabilities are replaced by risk-adjusted weights ${\mu }_{\omega }^{\ast }$ that place more emphasis on expensive (adverse) scenarios. The cut generation mechanics are otherwise identical – only the aggregation weights change.

The convex combination risk measure

Cobre uses a convex combination of expectation and Conditional Value-at-Risk (CVaR):

$${\rho }^{\lambda ,\alpha }[Z]=(1-\lambda )\mathbb{E}[Z]+\lambda \cdot {\text{CVaR}}_{\alpha }[Z]$$

This formulation has two parameters:

• $\lambda$ (risk aversion weight): Controls the blend between expected value and CVaR. Setting $\lambda =0$ recovers risk-neutral SDDP; setting $\lambda =1$ uses pure CVaR.
• $\alpha$ (confidence level): Controls how deep into the tail CVaR looks. Smaller $\alpha$ means focusing on more extreme adverse scenarios.

Both parameters can vary by stage, allowing operators to apply stronger risk aversion to near-term decisions (where consequences are immediate) and weaker risk aversion to distant stages.

Implications for convergence monitoring

A critical subtlety of risk-averse SDDP is that the first-stage LP objective is not a valid lower bound on the true risk-averse optimal cost. It remains a useful convergence indicator (it increases monotonically and plateaus), but it cannot be interpreted as a bound in the same way as in risk-neutral SDDP. Valid upper bounds require the inner approximation (SIDP) method.

Further reading

• CVaR – Intuitive explanation of Conditional Value-at-Risk
• Risk Measures (spec) – Complete formal specification: dual representations, subgradient theorem, risk-averse Bellman equation, cut generation algorithm, and lower bound validity warning
• Cut Management – How risk-adjusted weights integrate into cut aggregation
• Convergence – Bound computation and stopping criteria