Security-Constrained Economic Dispatch

• Security-Constrained Economic Dispatch is a mathematical framework that optimally dispatches generators by enforcing reliability and contingency constraints.
• It employs convex optimization with chance constraints to meet power balance and reserve limits while integrating renewables and mitigating operational uncertainties.
• Scenario compression techniques like convex-hull and box methods reduce computational complexity, enabling real-time applications without compromising risk guarantees.

A Security-Constrained Economic Dispatch (SCED) framework is a mathematical and algorithmic paradigm used by system operators to determine the least-cost generation dispatch while rigorously enforcing reliability constraints that ensure continued safe system operation under forecast variability and credible contingencies. SCED serves as the principal real-time operational tool for Transmission System Operators (TSOs) and Independent System Operators (ISOs), enabling efficient market clearing and reliability management in the presence of increasing renewable integration and operational uncertainties.

1. Formal Structure and Problem Formulation

The canonical SCED problem is structured as a constrained convex (often linear or quadratic) optimization problem. Decision variables include generator set-points $g\in\mathbb{R}^{n_g}$ and, where applicable, Automatic Generation Control (AGC) participation factors $\eta\in\mathbb{R}^{n_g}$ with $\sum_i\eta_i=1$ (Zhang et al., 2024). The system is subject to deterministic constraints that enforce:

• Power balance: $1^{\top}g=1^{\top}(d-\hat{w})$
• Generator and reserve capacity limits: $g^{\min}\le g\le g^{\max}$

Security constraints, most commonly in the form of $N-1$ contingency criteria, mandate that the post-contingency system (after removal of one line, generator, or other component) remains within operational limits. To address variability (especially due to wind/solar uncertainty), a joint chance constraint is imposed: $$\Pr_{\tilde{w}}\left[\begin{array}{l} \underline{f} \leq f(\tilde{w}) \leq \overline{f}\;\ \underline{g} \leq g - (1^{\top}\tilde{w})\,\eta \leq \overline{g}\ R_d \leq -(1^{\top}\tilde{w})\,\eta\leq R_u \end{array}\right] \geq 1-\epsilon$$ where $$f(\tilde{w}) = H_g [g - (1^{\top}\tilde{w})\,\eta] - H_d d + H_w (\hat{w}+\tilde{w})$$ encodes network flows via PTDF matrices $H_g$, $H_d$, $H_w$.

2. Scenario-Based Chance-Constrained and Distributionally Robust Extensions

To operationalize the joint chance constraint, scenario-based (sample-average) and distributionally robust formulations are prevalent. In scenario-based SCED, $N$ i.i.d. samples $\tilde{w}^{1},\ldots,\tilde{w}^{N}$ are generated from empirical or modeled uncertainty distributions. The stochastic constraint is replaced by $N$ deterministic constraints—one for each scenario—yielding the scenario program $\text{SP}(\mathcal{N})$ (Zhang et al., 2024). In distributionally robust approaches, chance constraints are required to hold uniformly over all distributions within a Wasserstein ball of specified radius against the empirical law, yielding tractable polyhedral approximations of the uncertainty set and robust LP formulations (Maghami et al., 2022).

The primary computational challenge is the rapid growth in model size with $N$ and the inclusion of multiple contingencies. High-dimensional scenario sets can be replaced by much smaller compressed representations to maintain tractability without compromising risk guarantees.

3. Scenario Compression: Convex-Hull and Box Methods

Two principal scenario-compression techniques have been advanced:

• Convex-hull compression: The convex hull of the $N$ scenarios is computed, and only its $v$ vertex scenarios are retained for the SCED formulation. Theoretical results prove that convex-hull compressed programs yield identical feasible sets and optimal solutions to the full scenario program: risk is not increased, and all non-vertex scenarios can be safely dropped (Zhang et al., 2024).
• Box compression: Compute axis-aligned minimum and maximum values to form an enclosing hyper-rectangle ("box"), and include only its $2n_w$ corners as scenarios. This method relaxes the feasible set (since Box$\supseteq$ConvHull), possibly reducing cost but at the price of conservatism and increased solution risk.

Both methods can reduce model size by $5\times$ to $10\times$ and make chance-constrained SCED viable in real-time operation. The risk due to compression is quantified either as "solution risk" (the actual violation probability at the dispatch solution) or as "compression risk" (the probability that adding one more scenario would expand the compressed set), with tight scenario-theoretic bounds computable for each.

4. Algorithmic Steps and Practical Implementation

The compressed scenario-based SCED is executed as follows (Zhang et al., 2024):

1. Sampling: Generate $N$ i.i.d. wind-error scenarios.
2. Compression: Apply convex hull extraction (e.g., Quickhull) or box compression.
3. Model construction: Build the reduced SCED model using compressed scenarios.
4. Risk validation: Calculate scenario-theory bounds (Campi-Garatti) for solution or compression risk, depending on method.
5. Dispatch: Solve the resulting convex program (e.g., with Gurobi/CPLEX).

On a 118-bus system with 500 wind scenarios, convex-hull compression (to $v\approx32$ scenarios) reduced formulation and solve time from 6.9s to $<0.5$s, with negligible impact on risk or cost; box compression (6 corners) halved time again but slightly increased cost/risk.

5. Economic and Reliability Implications

Incorporation of security constraints, especially $N-1$ reliability, can significantly increase operational cost; this effect is quantified by the "price of security" metric—the ratio of SCED to economic dispatch cost (Hajiesmaili et al., 2017). Worst-case inefficiency is realized when cheap generation is plentiful but cheap–to–expensive-region transmission is bottlenecked. Transmission upgrades or targeted demand-shifting can mitigate these costs. Convex-hull compression facilitates maintaining reliability (rigorous risk guarantees) without incurring unnecessary inefficiency due to excessive scenario size. Operators must carefully trade off risk tightness (favoring convex hull) against computational efficiency (favoring box compression).

6. Extensions and Related Models

The scenario compression and scenario-theory risk validation framework generalizes to other SCED settings:

• Distributionally robust chance constraints: Wasserstein-ambiguous DRO approaches admit adjustable conservatism via ambiguity radii and robustify against distributional misspecification (Maghami et al., 2022).
• Risk-sensitive (CVaR) SCED: Weighted cost–CVaR objective enables explicit optimization of the trade-off between nominal cost and the risk of high-consequence events (e.g., large load shed), with established revenue adequacy under security-constrained pricing (Madavan et al., 19 Feb 2025).
• Network flexibility integration: Chance-constrained dispatch can incorporate continuous line-susceptance control (e.g., TCSC devices) to alleviate congestion and further reduce risk/cost under uncertainty (Song et al., 2021).
• Iterative decomposition and distributed algorithms: Algorithms leveraging Lagrangian or augmented Lagrangian routines can implement privacy-preserving, distributed, or parallelized solution methods for geographically partitioned grids (Amini et al., 2015).

7. Practical Insights and Guidelines

Convex-hull scenario compression is recommended whenever the number of extreme points is much smaller than the original scenario ensemble and risk tightness is paramount. Box compression is advantageous for extremely large-scale or time-constrained applications, albeit at increased solution risk. In both cases, scenario-theoretic bounds provide rigorous a posteriori guarantees, ensuring the risk acceptance criteria of system operators are met (Zhang et al., 2024).

For operational deployment, total computation time for model building and solution must be less than real-time intervals (e.g., 5 min), which is achieved in both 118-bus and 2000-bus grid benchmarks. These techniques are thus enabling for practical chance-constrained SCED implementation in modern, uncertainty-dominated power systems.