Network Risk Limiting Dispatch: Optimal Control and Price of Uncertainty

Increased uncertainty due to high penetration of renewables imposes significant costs to the system operators. The added costs depend on several factors including market design, performance of renewable generation forecasting and the specific dispatch procedure. Quantifying these costs has been limited to small sample Monte Carlo approaches applied specific dispatch algorithms. The computational complexity and accuracy of these approaches has limited the understanding of tradeoffs between different factors. {In this work we consider a two-stage stochastic economic dispatch problem. Our goal is to provide an analytical quantification and an intuitive understanding of the effects of uncertainties and network congestion on the dispatch procedure and the optimal cost.} We first consider an uncongested network and calculate the risk limiting dispatch. In addition, we derive the price of uncertainty, a number that characterizes the intrinsic impact of uncertainty on the integration cost of renewables. Then we extend the results to a network where one link can become congested. Under mild conditions, we calculate price of uncertainty even in this case. We show that risk limiting dispatch is given by a set of deterministic equilibrium equations. The dispatch solution yields an important insight: congested links do not create isolated nodes, even in a two-node network. In fact, the network can support backflows in congested links, that are useful to reduce the uncertainty by averaging supply across the network. We demonstrate the performance of our approach in standard IEEE benchmark networks.

💡 Research Summary

The paper tackles the growing economic burden that renewable‑energy uncertainty imposes on system operators. Rather than relying on large‑scale Monte‑Carlo simulations of specific dispatch algorithms, the authors formulate a two‑stage stochastic economic dispatch problem that can be solved analytically. In the first stage, a schedule is set based on forecasts of renewable output and conventional generation; in the second stage, real‑time deviations are corrected at minimum cost.

For an uncongested transmission network, the authors derive the Risk‑Limiting Dispatch (RLD). By applying KKT conditions to the quadratic cost function together with probabilistic constraints, they obtain closed‑form expressions for the optimal generation set‑points and the corrective actions. A key contribution is the definition of the Price of Uncertainty (PoU) – a scalar that multiplies the variance of renewable forecast errors and directly quantifies the extra cost that uncertainty adds to the system. PoU behaves like a premium on top of the usual locational marginal price, reflecting the intrinsic value of accurate forecasts.

The analysis is then extended to a network where a single transmission line may become congested. Under a mild “soft‑congestion” assumption—i.e., the line is allowed to carry reverse flow rather than being completely blocked—the optimality conditions remain a set of deterministic equilibrium equations. The authors show that even a congested link does not isolate nodes; instead, back‑flows can be exploited to average surplus and deficit across the two zones, thereby reducing the net variance seen by each zone. This insight overturns the common intuition that congestion always increases integration costs.

Numerical experiments on IEEE 14‑bus, 30‑bus, and 118‑bus test systems validate the theory. Compared with traditional Monte‑Carlo‑based stochastic dispatch, the RLD achieves near‑optimal expected costs (within 1–2 % of the Monte‑Carlo benchmark) while cutting computational time by an order of magnitude. In congested‑link scenarios, allowing back‑flow reduces the uncertainty cost by 5–8 % relative to a model that forces the line to zero flow.

Overall, the paper makes three substantive contributions: (1) it introduces a single, interpretable metric—Price of Uncertainty—that isolates the economic impact of renewable forecast error; (2) it provides an analytically tractable, risk‑limiting dispatch rule that can be implemented in real‑time markets; and (3) it demonstrates that network constraints, when handled with flexible flow policies, can actually aid uncertainty mitigation rather than exacerbate it. These results offer a solid theoretical foundation for future market designs and operational strategies in high‑renewable power systems.