Original Research Article

Article volume = 2022 and issue = 2

Pages: 181–190

Article publication Date: November 21, 2022

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# Discrete Sampling Analysis for Electricity Market Forecasting with Reproducing Kernel Hilbert Space

#### Mohammadreza Foroutan(a) and Farzad Farzanfar(b)

(a) Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.

(b) Department of Computer Engineering and Information Technology, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.

##### Abstract:

Analyse discrete sampling theories in the reproducing kernel Hilbert space are applied here to whole-sale electricity market forecasting problem. We consider the optimal approximation of any function be longing to the kernel across pricing nodes and hours via a sampling method. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding reproducing kernel Hilbert space of function is investigated. The key idea of our work is adopting the reproducing kernel Hilbert space corresponding to the Gramian matrix of the additive tensor kernel and considering the orthogonal projector by the kernel functions. We also give numerical examples, using the sampling theorem, to confirm the behavior of the proposed method.

##### Keywords:

Gramian matrix, Hilbert space, Orthogonal projector, Reproducing kernel, discrete Sampling theorem.

##### References:

- [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68(3) (1950), 337–404. 1
- [2] P. Atsawathawichok, P. Teekaput, T. Ploysuwan, Long term peak load forecasting in Thailand using multiple kernel Gaussian Process, In ECTI-CON, (2014), 1–4. 1
- [3] J. Bastian, J. Zhu, V. Banunarayanan, R. Mukerji, Forecasting energy prices in a competitive market, IEEE computer applications in power 12(3) (1999), 40–45. 1
- [4] C. Berg, J. P. R. Christensen, P. Ressel, Harmonic analysis on semigroups, Springer-Verlag, New-York, 1984. 3
- [5] J. Contreras, R. Espínola, F. J. Nogales, A. J. Conejo, ARIMA models to predict next-day electricity prices, IEEE Trans. Power Syst., 18 (2003), 1014–1020. 1
- [6] N. Cressie, Statistics for Spatial Data, Wiley, 1991. 1
- [7] J. B. Fiot, F. Dinuzzo, Electricity demand forecasting by multi-task learning, IEEE Trans. Smart Grid, 99 (2016). 1
- [8] M. R. Foroutan, R. Asadi, A. Ebadian, A reproducing kernel Hilbert space method for solving the nonlinear threepoint boundary value problems, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(3) (2019), 1–18. 1
- [9] M. R. Foroutan, A. Ebadian, R. Asadi, Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four-point boundary value problems, International Journal of Computer Mathematics, 95(10) (2018), 2128–2142. 1
- [10] M. R. Foroutan, A. Ebadian, H. Rahmani Fazli, Generalized Jacobi reproducing kernel method in Hilbert spaces for solving the black-scholes option pricing problem arising in financial modeling, Mathematical Modelling and Analysis, 23(4) (2018), 538–553. 1
- [11] P. S. Georgilakis, Artificial intelligence solution to electricity price forecasting problem, Appl. Artif. Intell., 21 (2007), 707–727. 1
- [12] P. S. Georgilakis, Market Clearing Price Forecasting in Deregulated Electricity Markets Using Adaptively Trained Neural Networks, In Proceedings of the fourth Helenic Conference on Artificial Intelligence, Heraklion, Greece, 18-20 (2006), 56–66. 1
- [13] Y. Hong, C. P. Wu, Day-ahead electricity price forecasting using a hybrid principal component analysis network, Energies, 5 (2012), 4711–4725. 1
- [14] J. Lago, F. De Ridder, P. Vrancx, B. De Schutter, Forecasting day-ahead electricity prices in Europe: the importance of considering market integration, Appl. Energy, 211 (2018), 890–903. 1
- [15] P. Mandal, A. K. Srivastava, T. Senjyu, M. Negnevitsky, A new recursive neural network algorithm to forecast electricity price for PJM day-ahead market, Int. J. Energy Res., 34 (2010), 507–522. 1
- [16] M. Z. Nashed, Q. Sun , Function spaces for sampling expansions, Multiscale Signal Analysis and Modelling, Lecture Notes in EE, Springer (2012), 81–104. 1
- [17] M. Z. Nashed, Q. Sun, Sampling and reconstruction of signals in a reproducing kernel subspace of LpRd, J. Funct. Anal., 258(7) (2010), 2422–2452. 1
- [18] M. Z. Nashed, G. G. Walter, General sampling theorem for functions in reproducing kernel Hilbert space, Math. Control, Signals Syst., 4(4) (1991), 363–390. 1
- [19] A. L. Ott, Experience with PJM market operation, system design, and implementation, IEEE Trans. Power Syst., 18(2) (2003), 528–534. 3
- [20] Y. Pan, M. Jiang, LRR-TTK DL for face recognition, IET Biometrics, 6(3) (2017), 165. 3
- [21] S. Schneider, Power spot price models with negative prices, J. Energy Mark, 4 (2012), 77–102. 1
- [22] M. Shahidehpour, H. Yamin, Z. Li, Market overview in electric power systems. Market operations in electric power systems, New York (USA): John Wiley and Sons, Inc., 2 (2002), 1–20. 1
- [23] A. Tanaka, H. Imai, M. Miyakoshi, Kernel-induced sampling theorem, IEEE Trans. Signal Process, 58(7) (2009), 3569–-3577. 4, 5, 5
- [24] G. Wahba, Spline Models for Observational Data, Society for Industrial and Applied Mathematics, PA 1990. 1
- [25] R. Weron, Electricity price forecasting: a review of the state-of-the-art with a look into the future, Int. J. Forecast, 30(4) (2014), 1030–1081. 1
- [26] H. Zhang, F. Gao, J. Wu, K. Liu, X. Liu, Optimal bidding strategies for wind power producers in the day-ahead electricity market, Energies, 5 (2012), 4804–4823. 1

##### Cite this article as:

- Mohammadreza Foroutan and Farzad Farzanfar, Discrete Sampling Analysis for Electricity Market Forecasting with Reproducing Kernel Hilbert Space, Communications in Combinatorics, Cryptography & Computer Science, 2022(2), PP.181–190, 2022
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