Approximating Dispatchable Regions in Three-Phase Radial Networks with Conditions for Exact SDP Relaxation

TL;DR

This work addresses the problem of characterizing dispatchable regions for renewable generation in unbalanced three-phase radial networks by framing it with the bus-injection model and relaxing the nonconvex feasible set to an SDP, yielding a convex region $\mathcal{U}'$ that contains the true region $\mathcal{U}$. It develops an SDP-based projection algorithm to construct a polytopic outer approximation $\mathcal{U}'_{poly}$ of $\mathcal{U}'$ via dual solutions, and introduces a power-loss penalty (PE$(\mathbf{u})$) to guarantee exact SDP relaxation under practical network conditions, producing an inner region $\tilde{\mathcal{U}}$ with $\tilde{\mathcal{U}}\subseteq\mathcal{U}$. The approach is validated on the IEEE 123-bus feeder, showing that the proposed inner approximation is reliable and tighter than linear or CCP-based methods, while the outer polytopes progressively tighten with iterations. This gives operators tractable, certifiable regions for renewable commitment that respect AC power-flow physics and safety constraints, enabling proactive grid management in high-renewables scenarios.

Abstract

The concept of dispatchable region plays a pivotal role in quantifying the capacity of power systems to accommodate renewable generation. In this paper, we extend the previous approximations of the dispatchable regions on direct current (DC), linearized, and nonlinear single-phase alternating current (AC) models to unbalanced three-phase radial (tree) networks and provide improved outer and inner approximations of dispatchable regions. Based on the nonlinear bus injection model (BIM), we relax the non-convex problem that defines the dispatchable region to a solvable semidefinite program (SDP) and derive its strong dual problem (which is also an SDP). Utilizing the special mathematical structure of the dual problem, an SDP-based projection algorithm is developed to construct a convex polytopic outer approximation to the SDP-relaxed dispatchable region. Moreover, we provide sufficient conditions to guarantee the exact SDP relaxation by adding the power loss as a penalty term, thereby providing a theoretical guarantee for determining an inner approximation of the dispatchable region. Through numerical simulations, we validate the accuracy of our approximation of the dispatchable region and verify the conditions for exact SDP relaxation.

Figures (6)

• Figure 1: Relationship between the dispatchable region and our approximations: $\mathcal{U}$ represents the dispatchable region; $\mathcal{U}'$ is its SDP-relaxed region; $\mathcal{U}'_{poly}$ denotes the outer polytopic approximation; and $\tilde{\mathcal{U}}$ depicts an inner approximation of $\mathcal{U}$.
• Figure 2: Visualization of the convergence behavior of the SDP-relaxed region $\mathcal{U}'$: The gray polytope $\mathcal{U}'_{poly}(5)\approx\mathcal{U}'$ is enclosed by $\mathcal{U}'_{poly}(2)$.
• Figure 3: Output of Algorithm \ref{['Algo:cutting']} at different iterations. The polytope $\mathcal{U}'_{poly}(c)$ (solid lines) progressively approximates the SDP-relaxed region $\mathcal{U}'$ (dashed lines).
• Figure 4: Evolution of $\textnormal{dp}'_{\textnormal{max}}$ and polytope volume by Algorithm \ref{['Algo:cutting']} over iterations for six test cases. Solid lines represent relaxed voltage limits (Volt. R), and dotted lines represent tightened voltage limits (Volt. T). Colors indicate different generator capacities.
• Figure 5: Comparison of $\mathcal{U}'_{poly}$ under different scenarios. The solid black line indicates Baseline $\&$ Volt. R, which serves as the reference. The blue dashed and red dotted lines represent Baseline $\&$ Volt. T and Case H $\&$ Volt. R, respectively.

Theorems & Definitions (11)

• Definition 1
• Proposition 1
• Proposition 2
• Theorem 1
• Lemma 1
• Lemma 2
• Definition 2
• Proposition 3
• Proposition 4
• Theorem 2