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arXiv:2310.19953 · Advances in transdisciplinary engineering

A Hybrid Quantum Algorithm for Load Flow

David A. Neufeld, Sajad Fathi Hafshejani, Daya Ram Gaur, Robert R. Benkoczi

Published Jun 20, 2024• 3 citations
arXivHTMLPDF

Abstract

The goal of the load flow study is to ensure that electrical power is delivered efficiently and reliably to end-users while maintaining the stability and security of the power system. Newton-Raphson is a numerical method used widely for load flow analysis. One of the most computationally expensive steps in this method is an equation-solving step. We propose to replace this step with HHL, a quantum algorithm for solving linear systems of equations. HHL is exponentially faster, but with caveats. In this study, a hybrid quantum algorithm is proposed for solving load flow. The Newton-Raphson method is used as a benchmark to compare the performance of the hybrid quantum algorithm. Although the simulation of the hybrid quantum algorithm takes much time, these preliminary results are encouraging and point to the potential for the use of quantum algorithms to develop hybrid quantum algorithms for load flow analysis and related problems.

DOI
10.3233/atde240359

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The goal of the load flow study is to ensure that electrical power is delivered efficiently and reliably to end-users while maintaining the stability and security of the power system. Newton-Raphson is a numerical method used widely for load flow analysis. One of the most computationally expensive steps in this method is an equation-solving step. We propose to replace this step with HHL, a quantum algorithm for solving linear systems of equations. HHL is exponentially faster, but with caveats. In this study, a hybrid quantum algorithm is proposed for solving load flow. The Newton-Raphson method is used as a benchmark to compare the performance of the hybrid quantum algorithm. Although the simulation of the hybrid quantum algorithm takes much time, these preliminary results are encouraging and point to the potential for the use of quantum algorithms to develop hybrid quantum algorithms for load flow analysis and related problems.
This paper presents Newton-Raphson method using HHL algorithm for power flow quantum computation. The proposed Newton-Raphson HHL Quantum method converges to the solution within four iterations compared to the Newton-Raphson classical solution which converged in five iterations. The Newton-Raphson classical solution voltageV2=0.96674\mathrm{V}_{2}=0.96674V2​=0.96674pu, angleδ2=−0.04816\delta_{2}=-0.04816δ2​=−0.04816rads, angleδ3=0.04096\delta_{3}=0.04096δ3​=0.04096rads. The Newton-Raphson-HHL Quantum method voltageV2=0.98173\mathrm{V}_{2}= 0.98173V2​=0.98173pu, angleδ2=−0.05018\delta_{2}=-0.05018δ2​=−0.05018rads, angleδ3=0.04021\delta_{3}=0.04021δ3​=0.04021rads. This work Newton-Raphson HHL Quantum method for 3-bus time to converge is O.228s228\mathrm{s}228s. The fidelity is calculated to be 0.981436 with circuit width of 4 which is the number of qubits used in the circuit and the circuit depth gate layers is 126.
In this paper, we model and solve a fundamental power system problem, i.e., DC power flow, using a practical quantum computer. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm is used to solve the DC power flow problem. The HHL algorithm for the solution of a system of linear equations (SLE) offers an exponential speedup over the classical computers. The speedup advantage is more significant when the size and the frequency of solving the power flow problem become more substantial. Verified quantum computing applications to power systems are merely nonexistent at this point. Through this paper, we plan to (1) provide a proof-of-concept that practical power system problems can be solved using quantum technology, (2) build a quantum-grid foundation by solving a fundamental power system problem with applications in many other critical grid problems, and (3) apply HHL to solve an SLE which has broad applications in many power system problems. A small 3-bus system is used for testing and demonstration purposes, considering the limitations of the available quantum computing hardware and software. The proposed method's merits and effectiveness are demonstrated using IBM open-source quantum computer and reported through proof-of-concept experimental demonstration using a 4-qubit quantum information processor.
Quantum computers hold promise for solving problems intractable for classical computers, especially those with high time or space complexity. Practicalquantum advantagecan be said to exist for such problems when the end-to-end time for solving such a problem using a classical algorithm exceeds that required by a quantum algorithm. Reducing the power flow (PF) problem into a linear system of equations allows for the formulation of quantum PF (QPF) algorithms, which are based on solving methods for quantum linear systems such as the Harrow-Hassidim-Lloyd (HHL) algorithm. Speedup from using QPF algorithms is often claimed to be exponential when compared to classical PF solved by state-of-the-art algorithms. We investigate the potential for practical quantum advantage in solving QPF compared to classical methods on gate-based quantum computers. Notably, this paper does not present a new QPF solving algorithm but scrutinizes the end-to-end complexity of the QPF approach, providing a nuanced evaluation of the purported quantum speedup in this problem. Our analysis establishes a best-case bound for the HHL-based quantum power flow complexity, conclusively demonstrating that the HHL-based method has higher runtime complexity compared to the classical algorithm for solving the direct current power flow (DCPF) and fast decoupled load flow (FDLF) problem. Notably, our analysis and conclusions can be extended to any quantum linear system solver with rigorous performance guarantees, based on the known complexity lower bounds for this problem. Additionally, we establish that for potential practical quantum advantage (PQA) to exist it is necessary to consider DCPF-type problems with a very narrow range of condition number values and readout requirements.
This paper aims to contextualize the requirements for Quantum Computing (QC) algorithms to achieve a quantum advantage in solving the alternating current power flow (ACPF) problem, with a focus on runtime complexity. First, we establish a benchmark for a QC iterative solver to demonstrate an advantage over the classical Newton-Raphson Load Flow (NRLF) algorithm. Next, we derive a baseline expression for the end-to-end runtime complexity of any Gate-based QC algorithm as Ω(Nκ/ε),Ω(N κ/\varepsilon),Ω(Nκ/ε), reflecting dependence on system size NNN, condition number κκκ, and error tolerance ε\varepsilonε. Finally, we highlight key areas where QC algorithms may offer potential benefits over NRLF in addressing the standard ACPF problem.
This paper explores the use of quantum computing, specifically the use of HHL and VQLS algorithms, to solve optimal power flow problem in electrical grids. We investigate the effectiveness of these quantum algorithms in comparison to classical methods. The simulation results presented here which substantially improve the results in [1] indicate that quantum approaches yield similar solutions and optimal costs compared to classical methods, suggesting the potential use case of quantum computing for power system optimization.
Practical quantum computing applications to power grids are nonexistent at the moment. This paper investigates how a fundamental grid problem, namely DC power flow, can be solved using quantum computing. Power flow is the most widely used power system analysis technique, either as a stand-alone application or embedded in other applications; therefore, its fast and accurate solution is of utmost significance for grid operators. We base our studies on the Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which has a proven theoretical speedup over classical algorithms in solving a system of linear equations. Practical studies on a quantum computer are conducted using the WSCC 9-bus system.