Distributed optimal power flow

Abstract

Objective: The objectives of this paper are to 1) construct a new network model compatible with distributed computation, 2) construct the full optimal power flow (OPF) in a distributed fashion so that an effective, non-inferior solution can be found, and 3) develop a scalable algorithm that guarantees the convergence to a local minimum.

Existing challenges: Due to the nonconvexity of the problem, the search for a solution to OPF problems is not scalable, which makes the OPF highly limited for the system operation of large-scale real-world power grids-"the curse of dimensionality". The recent attempts at distributed computation aim for a scalable and efficient algorithm by reducing the computational cost per iteration in exchange of increased communication costs.

Motivation: A new network model allows for efficient computation without increasing communication costs. With the network model, recent advancements in distributed computation make it possible to develop an efficient and scalable algorithm suitable for large-scale OPF optimizations.

Methods: We propose a new network model in which all nodes are directly connected to the center node to keep the communication costs manageable. Based on the network model, we suggest a nodal distributed algorithm and direct communication to all nodes through the center node. We demonstrate that the suggested algorithm converges to a local minimum rather than a point, satisfying the first optimality condition.

Results: The proposed algorithm identifies solutions to OPF problems in various IEEE model systems. The solutions are identical to those using a centrally optimized and heuristic approach. The computation time at each node does not depend on the system size, and Niter does not increase significantly with the system size.

Conclusion: Our proposed network model is a star network for maintaining the shortest node-to-node distances to allow a linear information exchange. The proposed algorithm guarantees the convergence to a local minimum rather than a maximum or a saddle point, and it maintains computational efficiency for a large-scale OPF, scalable algorithm.

Fig 1. The largest number of nodes in the partitioned subsystems of various power system test cases reported in the literature [, –19].

Fig 2. A star and linear network for a modified IEEE-4 bus system.

The traditional network model is shown on the left (A) and the proposed network model is presented on the right (B). Red lines indicate the power channel, and green lines are the voltage channel.

Fig 3. x-update from the result of optimization with 1) a feasible solution x~jn+1, 2) an infeasible solution, but the distance between the solution and the projection onto the feasible region is close enough ‖ζ~jk+1-x~jk+1‖≤εjk, and 3) an infeasible solution in that the projection onto the feasible region is too far ‖ζ~jm+1-x~jm+1‖≤εjm.

Fig 4. The path length, the maximum number of nodes in a subsystem, and the number of subsystems for the IEEE 3-, 4-, 9-, 14-, 24-, 30-, 39-, 57-, 85-, 118-, 300-, and 2,000-bus systems.

The lines indicate the positive correlations on the size of the system. The red-dotted line, the blue broken line, and the green solid line represent the best-fit curves for the path length, the maximum number of nodes in a subsystem, and the number of subsystems, respectively. Two algorithms are applied for clustering nodes: (A) unnormalized algorithm, and (B) normalized based on the algorithm in [28].

Fig 5. Cardinalities of nodal variables in x-optimization in terms of system size.

Fig 6. Convergence of proposed algorithm for selected systems (i.e., IEEE 3-, 4-, 9-, 14-, 24-, 39-, and 85-bus systems).

Fig 7. Convergence of proposed algorithm for selected small systems (i.e., IEEE 9-, 14-, 30-, 57-, 118-, and 300-bus systems).

The convergences of the ADMM approaches [, –19], are compared.

Fig 8. Convergence of proposed algorithm for synthetic 2,000-bus system on a footprint of Texas.

Fig 9. Cardinalities of central OPF and proposed algorithm for the test systems.

Fig 10. N_iter for central OPF and of the proposed algorithm on the tested systems.

Dotted lines are averages, and solid lines are best-fit curves.

Fig 11. Computation times in seconds for nodal OPF of the maximum cardinality of the nodal variables.

Dotted line is the best-fit line with the slope of 2.64.