Researchers from North Carolina A&T State University, Pacific Northwest National Laboratories, and Sandia National Laboratories use quantum computing to solve complex equations in power system dynamics. The team converts high dimensional nonlinear differential-algebraic equations, which model power system dynamics, into ordinary differential equations. These can then be encoded into quantum computers. The results suggest that quantum computing can solve these dynamics with high accuracy and less computational complexity. The team used a symbolic programming framework and the Harrow-Hassidim-Lloyd algorithm.
Quantum Computing and Power System Dynamics
Power system dynamics, typically modeled by many differential-algebraic equations (DAEs), present a significant computational challenge due to the sheer number of generators, loads, and transmission lines that form the network. The complexity of these calculations grows exponentially with the size of the system. This article explores the potential of quantum computing as an alternative approach to solving these equations, focusing on maintaining high accuracy while reducing computational complexity.
The Challenge of Power System Dynamics
Power system dynamics are generally represented by a set of ordinary differential equations (ODEs) that model the dynamics of synchronous generators, along with algebraic nonlinear equations of power flow balances and Kirchhoff voltage laws for individual buses in the network. Transforming these DAEs into standard ODE forms is not a trivial task. Once transformed, these ODEs can be solved by numerical discretization methods such as Euler, the Runge-Kutta, and the backward differentiation formula methods in classical computers. However, the implementation on a classical computer scales exponentially with the size of the problem and the need to linearize the nonlinear terms embedded in the complex power system dynamics, often leading to high computational complexity.
Quantum Computing as an Alternative Approach
Quantum computing offers a different computation method and algorithmically superior scaling for specific problems. For example, the Harrow-Hassidim-Lloyd (HHL) algorithm for linear equations is based on amplitude encoding, where a system with n variables can be represented as an n-level quantum state. Since the number of required qubits equals log2(n), the algorithm provides an exponential memory advantage over the classical computing method. Extending quantum linear equation solvers to tackle high-dimensional systems of linear ODEs offers the prospect of rapidly characterizing the solutions of high-dimensional linear ODEs.
Implementing Quantum Computing for Power System Dynamics
This paper evaluates the potential of quantum computing for solving large-scale power system dynamics using quantum computing algorithms and recent advances in scientific computing frameworks. Unlike linear ODEs, the power systems dynamics pose high-dimensional nonlinear DAEs. To address this, the researchers leverage symbolic programming packages from Julia/SciML (the popular Julia Programming Language), particularly ModelingToolkit, to implement and transform power systems’ DAEs into an equivalent set of ODEs using the Pantelides-based index reduction.
Then, they employ Leyton’s quantum algorithm for nonlinear cases of ODEs. The obtained nonlinear ODEs are linearized by Taylor expansion as a set of polynomial functions of state variables. By storing multiple copies of quantum states, the nonlinearities embedded in polynomial functions are then captured by amplitudes of the tensor of quantum states. Then, state variables, updated in the classical computer according to the Euler method, are now equivalently updated using the HHL algorithm. The complexity, however, is polynomial to the logarithm of the system dimension.
Numerical Studies and Conclusion
The paper concludes with numerical studies on the SMIB and the three-machine nine-bus test systems. The results show that quantum computing can solve the dynamics of the power system with high accuracy while its complexity is polynomial in the logarithm of the system dimension. This suggests that quantum computing could be a viable alternative to classical methods for solving the complex equations involved in power system dynamics.
