A two-step smoothing Levenberg-Marquardt algorithm for real-time pricing in smart grid

Linsen Song; Gaoli Sheng; Linsen Song; Gaoli Sheng

doi:10.3934/math.2024230

AIMS Mathematics

2024, Volume 9, Issue 2: 4762-4780. doi: 10.3934/math.2024230

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Research article

A two-step smoothing Levenberg-Marquardt algorithm for real-time pricing in smart grid

Linsen Song ^,,
Gaoli Sheng

School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, China

Received: 07 September 2023 Revised: 09 December 2023 Accepted: 04 January 2024 Published: 22 January 2024
MSC : 90C33

As is well known, the utility function is significant for solving the real-time pricing problem of smart grids. Based on a new utility function, the social welfare maximization model is considered in this paper. First, we transform the social welfare maximization model into a smooth system of equations using Krush-Kuhn-Tucker (KKT) conditions, then propose a two-step smoothing Levenberg-Marquardt method with global convergence, where an LM step and an approximate LM step are computed at every iteration. The local convergence of the algorithm is cubic under the local error bound condition, which is weaker than the nonsingularity. The simulation results show that, the algorithm can not only reduce the user's electricity consumption but also improve the total social welfare at the most time when compared with the fixed pricing method. Additionally, when different values of the approximating parameter are adopted in a smoothing quasi-Newton method, the price tends to that obtained by the present algorithm. Furthermore, the CPU time of the one-step smoothing Levenberg-Marquardt algorithm and the proposed algorithm are also listed.
- smart grid,
- real-time pricing,
- shadow price,
- KKT conditons,
- complementarity problems
Citation: Linsen Song, Gaoli Sheng. A two-step smoothing Levenberg-Marquardt algorithm for real-time pricing in smart grid[J]. AIMS Mathematics, 2024, 9(2): 4762-4780. doi: 10.3934/math.2024230

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Abstract

As is well known, the utility function is significant for solving the real-time pricing problem of smart grids. Based on a new utility function, the social welfare maximization model is considered in this paper. First, we transform the social welfare maximization model into a smooth system of equations using Krush-Kuhn-Tucker (KKT) conditions, then propose a two-step smoothing Levenberg-Marquardt method with global convergence, where an LM step and an approximate LM step are computed at every iteration. The local convergence of the algorithm is cubic under the local error bound condition, which is weaker than the nonsingularity. The simulation results show that, the algorithm can not only reduce the user's electricity consumption but also improve the total social welfare at the most time when compared with the fixed pricing method. Additionally, when different values of the approximating parameter are adopted in a smoothing quasi-Newton method, the price tends to that obtained by the present algorithm. Furthermore, the CPU time of the one-step smoothing Levenberg-Marquardt algorithm and the proposed algorithm are also listed.

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