Branch Flow Model: Relaxations and Convexification (Parts I, II) (1204.4865v4)

Abstract: We propose a branch flow model for the anal- ysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

• The paper introduces a branch flow model that employs two relaxation steps to solve the optimal power flow (OPF) problem precisely in radial networks.
• It demonstrates that while the conic relaxation is exact for mesh networks, additional phase shifters enable mapping relaxed solutions to global optima.
• The research offers practical methods and strong theoretical foundations for efficient power system optimization and further advances in grid convexification.

Overview of "Branch Flow Model: Relaxations and Convexification (Part I)"

The paper by Farivar and Low introduces a Branch Flow Model (BFM) for the analysis and optimization of both radial and mesh networks in power systems. It presents a novel approach to solving the Optimal Power Flow (OPF) problem that involves two relaxation steps. The authors demonstrate that, for radial networks, these relaxation steps are always exact when no upper bounds on loads are imposed. For mesh networks, while the conic relaxation remains exact, the angle relaxation may not be. The paper also introduces methods to determine global optimality of relaxed solutions and proposes the use of phase shifters for efficient OPF resolution in mesh networks.

Introduction

The research focuses on the BFM as an alternative to the traditional bus injection model. The BFM emphasizes the currents and powers on individual branches rather than nodal variables. This approach provides new insights and methods for the OPF problem, which seeks to optimize objectives like power loss and generation cost subject to physical and operational constraints.

Key Contributions

1. Branch Flow Model Formulation:
   • The authors present the mathematical formulation of the BFM, emphasizing branch-specific variables such as currents and power flows.
2. Two Relaxation Steps:
   • Angle Relaxation: Eliminates voltage and current angles, resulting in a simplified problem (OPF-ar) that remains non-convex.
   • Conic Relaxation: Further relaxes the problem to a convex form (OPF-cr) by approximating a quadratic equality constraint with a conic inequality.
3. Exactness in Radial Networks:
   • For radial networks, both relaxation steps are proven to be exact, ensuring that solutions of the relaxed problems correspond to solutions of the original problem.
4. Angle Recovery and Global Optimality in Mesh Networks:
   • The paper discusses conditions under which the relaxed solutions for mesh networks can be mapped back to globally optimal solutions of the original problem.
5. Phase Shifters for Convexification:
   • The authors propose using phase shifters outside a spanning tree of the network to ensure that any relaxed solution can be efficiently mapped back to an optimal solution, improving computational efficiency.

Theoretical Foundations

The BFM relies on mathematical formulations to represent the power flow equations:

• Ohm’s Law: Connects voltage differences across branches to the branch currents.
• Power Balance: Ensures that the sum of the power flows into and out of each node equals the net power injection at that node.
• Relaxed Branch Flow Equations: These equations, obtained by relaxing the phase angles, result in a set of linear and quadratic equality constraints.

Strong Numerical Results

• The authors prove that the conic relaxation (OPF-cr) is always exact for radial networks without load upper bounds.
• They provide characterizations (angle recovery conditions) to determine when relaxed solutions are globally optimal for mesh networks.

Implications and Future Work

1. Practical Implications:
   • Systems designers can use the BFM to formulate and solve OPF problems more efficiently, particularly in distribution networks that are predominantly radial.
2. Theoretical Implications:
   • The paper extends theoretical understanding of power flow optimization, demonstrating that non-convex problems can often be tackled effectively using convex relaxations.
3. Future Developments:
   • Future work, discussed in Part II of the paper, will address the placement of phase shifters in mesh networks and further investigate numerical aspects and simulations.

Conclusion

The BFM and the proposed relaxation techniques represent a significant advancement in the optimization of power systems. By proving the exactness in radial networks and providing tools for global optimality in mesh networks, the research paves the way for more efficient and theoretically sound methods in power flow optimization. This work lays a robust foundation for further advancements, particularly in the convexification of mesh networks and applications in real-world power distribution and transmission systems.