Neural Computing and Applications

, Volume 20, Issue 1, pp 79–89 | Cite as

Nonlinear maximum likelihood estimation of electricity spot prices using recurrent neural networks

  • Derrick MirikitaniEmail author
  • Nikolay Nikolaev
Original Article


Electricity spot prices are complex processes characterized by nonlinearity and extreme volatility. Previous work on nonlinear modeling of electricity spot prices has shown encouraging results, and we build on this area by proposing an Expectation Maximization algorithm for maximum likelihood estimation of recurrent neural networks utilizing the Kalman filter and smoother. This involves inference of both parameters and hyper-parameters of the model which takes into account the model uncertainty and noise in the data. The Expectation Maximization algorithm uses a forward filtering and backward smoothing (Expectation) step, followed by a hyper-parameter estimation (Maximization) step. The model is validated across two data sets of different power exchanges. It is found that after learning a posteriori hyper-parameters, the proposed algorithm outperforms the real-time recurrent learning and the extended Kalman Filtering algorithm for recurrent networks, as well as other contemporary models that have been previously applied to the modeling of electricity spot prices.


Electricity spot price Recurrent neural network Expectation maximization 


  1. 1.
    Amjady N (2006) Day-ahead price forecasting of electricity markets by a fuzzy neural network. IEEE Trans Power Syst 21(2)Google Scholar
  2. 2.
    Amjady N, Hemmati M (2008) Day-ahead price forecasting of electricity markets by a hybrid intelligent system. Eur Trans Elec PowerGoogle Scholar
  3. 3.
    Aggarwal SK, Sani LM, Kumar A (2008) Electricity price forecasting in Ontario electricity market using wavelet transform in artificial neural network based model. Int J Control Auto Syst 6(5):639–650Google Scholar
  4. 4.
    Bunn D (2004) Modelling prices in competitive electricity markets. Wiley, LondonGoogle Scholar
  5. 5.
    Catalao JPS, Mariano SJPS, Mendes VMF, Ferreira LAFM (2007) Short-term electricity prices forecasting in a competitive market: a neural network approach. Elec Power Syst Res 77(10):1297–1304CrossRefGoogle Scholar
  6. 6.
    Casdagli M (1989) Nonlinear prediction of chaotic time series. Physica D 35:335–356zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cernansky M, Benuskova L (2003) Simple recurrent network trained by RTRL and extended Kalman filter algorithms. Neural Network World 13(3):223–234Google Scholar
  8. 8.
    Contreras J, Conejo AJ, Espinola R (2002) Forecasting next-day prices by time-series models. IEEE Trans Power Syst 17(2):342–348CrossRefGoogle Scholar
  9. 9.
    Contreras J, Espínola R, Nogales FJ, Conejo AJ (2003) ARIMA mdels to predict next-day electricity prices. IEEE Trans Power Syst 18(3):1014–1020CrossRefGoogle Scholar
  10. 10.
    Conejo AJ, Plazas MA, Espinola R, Molina AB (2005) Day-ahead electricity price forecasting using the wavelet transform and ARIMA models. IEEE Trans Power Syst 20(2):1035–1042CrossRefGoogle Scholar
  11. 11.
    de Freitas JFG, Niranjan M, Gee AH (2000) Dynamic learning with the EM algorithm for neural networks. J VlSI Signal Proc 26:119–131zbMATHCrossRefGoogle Scholar
  12. 12.
    Dempster AP, Laird NM, Rubi DB (1977) Mazimum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B 39:1–38zbMATHGoogle Scholar
  13. 13.
    Francisco NJ, Javier C (2002) Forecasting next-day electricity prices by time series models. IEEE Trans Power Syst 17(2):342–348CrossRefGoogle Scholar
  14. 14.
    Gao F, Guan X, Cao XR, Papalexopoulos A (2000) Forecasting power market clearing price and quantity using a neural network. IEEE PES Summer Meeting SeattleGoogle Scholar
  15. 15.
    Georgilakis P (2006) Market clearing price forecasting in deregulated electricity markets using adaptively trained neural networks. SETN, pp 56–66Google Scholar
  16. 16.
    Ghahramani Z, Hinton GE (1996) Parameter estimation for linear dynamical systems University of Toronto Technical Report CRG-TR-96-2Google Scholar
  17. 17.
    Gonzalez AM, San Roque AM, Conzales JG (2005) Modeling and forecasting electricity prices with input/output hidden Markov models. IEEE Trans Power Syst 20(1):13–24CrossRefGoogle Scholar
  18. 18.
    Haykin S (2001) Kalman filtering and neural networks. Wiley, New YorkCrossRefGoogle Scholar
  19. 19.
    Hong YY, Lee CF (2005) A neuro-fuzzy price forecasting approach in deregulated electricity markets. Elec Power Syst Res 73:151–157CrossRefGoogle Scholar
  20. 20.
    Hong YY, Hisao CY (2001) Locational marginal price forecasting in deregulated electric markets using a recurrent neural network. PESWM 2:539–544Google Scholar
  21. 21.
    Karsaz A, Mashhadi HR, Eshraghnia R (2006) Cooperative co-evolutionary approach to electricity load and price forecasting in deregulated electricity markets. Power India Conference, pp 1–6Google Scholar
  22. 22.
    Kim B, Velas JP, Lee J, Park J, Shin J, Lee KY (2006) Short-term system marginal price forecasting using system-type neural network architecture. PSCE, pp 1753–1758Google Scholar
  23. 23.
    Li C, Wang S (2006) Next-day power market clearing price forecasting using artificial fish-swarm based neural network. ISSN, pp 1290–1295Google Scholar
  24. 24.
    Liu Z, Yang H, Lai M (2005) Electricity price forecasting model based on chaos theory. IPEC, pp 1–5Google Scholar
  25. 25.
    Mandal P, Senjyu T, Urasaki N, Funabashi T, Srivastava AK (2007) A novel approach to forecast electricity price for PJM using neural network and similar days Method. IEEE Trans Power Syst 22(4):2058–2065CrossRefGoogle Scholar
  26. 26.
    Mandal P, Senjyu T, Funabashi T (2006) Neural netowrks approach to forecast several hour ahead electricity prices and loads in deregulated market. Energ Conve Manag 47:15–16Google Scholar
  27. 27.
    Mori H, Awata A (2006) A hybrid method of clipping and artificial neural network for electricity pricew zone forecasting. PMAPS, pp 1–6Google Scholar
  28. 28.
    Niimura T, Ko HS, Ozawa K (2002) A day-ahead electricity price prediction based on a fuzzy-neuro autoregressive model in a deregulated electrictiy market. IJCNN, pp 1362–1366Google Scholar
  29. 29.
    Nikolaev N, de Menezes L (2008) Sequential Bayesian Kernel Modelling with non-gaussian noise. Neural Networks 21(1): 36–47Google Scholar
  30. 30.
    Nikolaev N, Iba H (2006) Adaptive learning of polynomial networks: genetic programming, backpropagation and Bayesian methods. Springer, New YorkzbMATHGoogle Scholar
  31. 31.
    Nogales FJ, Contreras J, Conejo AJ, Espinola R (2002) Forecasting next-day electricity prices by time-series models. IEEE Trans Power Syst 17(2):342–348CrossRefGoogle Scholar
  32. 32.
    Pindoriya NM, Singh SN, Singh SK (2008) An adaptive wavelet neural network-based energy price forecasting in electricity markets. IEEE Trans Power Syst 23(3):1423–1432CrossRefGoogle Scholar
  33. 33.
    Pino R, Parreno J, Gomez A, Priore P (2008) Forecasting next-day price of electricity in the Spanish energy market using artificial neural networks. Eng Appl Artif Intell 21(1):53–62CrossRefGoogle Scholar
  34. 34.
    Puskorius GV, Feldkamp LA (1994) Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks. IEEE T Neural Networ 5(2):279–297CrossRefGoogle Scholar
  35. 35.
    Rauch HE, Tung F, Striebel CT (1965) Maximum likelihood estimates of linear dynamic models. AIAA J 3(8):1445–1450CrossRefMathSciNetGoogle Scholar
  36. 36.
    Rodriguez CP, Anders GJ (2006) Energy price forecasting in the Ontario competitive power system market. IEEE Trans Power Syst 21(2):887–896CrossRefGoogle Scholar
  37. 37.
    Ruck DW, Rogers SK, Kabrisky M, Maybeck P, Oxle ME (1992) Comparative analysis of backpropgation and the extended Kalman filter for training multilayer perceptrons. IEEE Trans Patt Anal Mach Intell 14(6):686–691CrossRefGoogle Scholar
  38. 38.
    Schafer AM, Zimmerman G (2007) Recurrent neural networks are universal approximators. Int J Neural Syst 17(4):253–263CrossRefGoogle Scholar
  39. 39.
    Schottky B, Saad D (1999) Statistical mechanics of EKF learning in neural networks. J Phys A 32(9):1605–1621zbMATHCrossRefGoogle Scholar
  40. 40.
    Zhang L, Luh PB, Kasiviswanathan K (2003) Energy clearing price predication and confidence interval estimation with cascaded neural networks. IEEE Trans Power Syst 18(1):99–105CrossRefGoogle Scholar
  41. 41.
    Shumway RH, Stoffer DS (1982) An approach to time series smoothing and forecasting using the EM algorithm. J Time Ser Anal 3(4):253–264zbMATHCrossRefGoogle Scholar
  42. 42.
    Szkuta BR, Sanabria LA, Dillon TS (1999) Electricity price short-term forecasting using artificial neural networks. IEEE Trans Power Syst 14(3):2116–2120CrossRefGoogle Scholar
  43. 43.
    Wang A, Ramsay B (1997) Prediction of system marginal price in the UK power pool, vol 4, In: International conference on neural networks, pp 2116–2120Google Scholar
  44. 44.
    Williams RJ, Zipser D (1989) A learning algorithm for continuously running fully connected recurrent neural networks. Neural Comp 1:270–280CrossRefGoogle Scholar
  45. 45.
    Srinivasan D, Yong FC, Liew AC (2007) Electricity price forecasting using evolved neural networks. ISAP, pp 1–7Google Scholar
  46. 46.
    Xu YY, Hsieh R, Lyu YL, Shen YC, Chuang SC, Pao HT (2004) Forecasting electricity market prices: a neural network based approach. IJCNN 4:2789–2794Google Scholar
  47. 47.
    Yamin HY, Shahidehpour SM, Li L (2004) Adaptive short-term electricty price forecasting using artificial neural networks in teh restructured power markets. Elec Power Energ Syst 25:571–581CrossRefGoogle Scholar
  48. 48.
    Yang B, Chen Y, Zhao Z, Han Q (2007) Forecasting of market clearing price by using GA based neural network, in proc. ICIC 2:1278–1286Google Scholar
  49. 49.
    Yang H, Lai M (2005) Chaotic characteristics of electricity price and its forecasting model. Aus J Elec Elec Eng 2(2):117–125Google Scholar
  50. 50.
    Zareipour H, Canizares CA, Bhattacharya K (2006) Application of public-domain market information to forecast Ontario’s wholesale electricity prices. IEEE Trans Power Syst 21(4):1707–1717CrossRefGoogle Scholar
  51. 51.
    Zhang L, Luh PB, Kasiviswanthan K (2003) Energy clearing price prediction and confidence interval estimation with cascaded neural networks. IEEE Trans Power Syst 18(1):99–105CrossRefGoogle Scholar
  52. 52.
    Zhang L, Luh PB (2005) Neural network-based market clearing price prediction and confidence interval estimation with an improved extended Kalman filter method. IEEE Trans Power Syst 20(1):29–66CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Department of ComputingGoldsmiths CollegeNew Cross, LondonUK

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