Analytic solution to swing equations in power grids with ZIP load models.

OBJECTIVE: This research pioneers a novel approach to obtain a closed-form analytic solution to the nonlinear second order differential swing equation that models power system dynamics. The distinctive element of this study is the integration of a generalized load model known as a ZIP load model (constant impedance Z, constant current I, and constant power P loads). METHODS: Building on previous work where an analytic solution for the swing equation was derived in a linear system with limited load types, this study introduces two fundamental novelties: 1) the innovative examination and modeling of the ZIP load model, successfully integrating constant current loads to augment constant impedance and constant power loads; 2) the unique derivation of voltage variables in relation to rotor angles employing the holomorphic embedding (HE) method and the Padé approximation. These innovations are incorporated into the swing equations to achieve an unprecedented analytical solution, thereby enhancing system dynamics. Simulations on a model system were performed to evaluate transient stability. RESULTS: The ZIP load model is ingeniously utilized to generate a linear model. A comparison of the developed load model and analytical solution with those obtained through time-domain simulation demonstrated the remarkable precision and efficacy of the proposed model across a range of IEEE model systems. CONCLUSION: The study addresses the key challenges in power system dynamics, namely the diverse load characteristics and the time-consuming nature of time-domain simulation. Breaking new ground, this research proposes an analytical solution to the swing equation using a comprehensive ZIP model, without resorting to unphysical assumptions. The close-form solution not only assures computational efficiency but also preserves accuracy. This solution effectively estimates system dynamics following a disturbance, representing a significant advancement in the field.

Distributed optimal power flow.

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.

Analytical solution to swing equations in power grids.

OBJECTIVE: To derive a closed-form analytical solution to the swing equation describing the power system dynamics, which is a nonlinear second order differential equation. EXISTING CHALLENGES: No analytical solution to the swing equation has been identified, due to the complex nature of power systems. Two major approaches are pursued for stability assessments on systems: (1) computationally simple models based on physically unacceptable assumptions, and (2) digital simulations with high computational costs. MOTIVATION: The motion of the rotor angle that the swing equation describes is a vector function. Often, a simple form of the physical laws is revealed by coordinate transformation. METHODS: The study included the formulation of the swing equation in the Cartesian coordinate system, which is different from conventional approaches that describe the equation in the polar coordinate system. Based on the properties and operational conditions of electric power grids referred to in the literature, we identified the swing equation in the Cartesian coordinate system and derived an analytical solution within a validity region. RESULTS: The estimated results from the analytical solution derived in this study agree with the results using conventional methods, which indicates the derived analytical solution is correct. CONCLUSION: An analytical solution to the swing equation is derived without unphysical assumptions, and the closed-form solution correctly estimates the dynamics after a fault occurs.

Multiple-rank modification of symmetric eigenvalue problem.

Rank-1 modifications applied k-times (k > 1) often are performed to achieve a rank-k modification. We propose a rank- k modification for enhancing computational efficiency. As the first step toward a rank- k modification, an algorithm to perform a rank-2 modification is proposed and tested. The computation cost of our proposed algorithm is in O ( n 2 ) where n is the cardinality of the matrix of interest. We also propose a general rank-k update algorithm based on the Sturm Theorem, and compare our results to those of the direct eigenvalue decomposition and of a perturbation method.