- Original Research
- Open Access
Time series modeling and filtering method of electric power load stochastic noise
- Li Huang^{1}Email author,
- Yongbiao Yang^{1},
- Honglei Zhao^{2},
- Xudong Wang^{2} and
- Hongjuan Zheng^{1}
https://doi.org/10.1186/s41601-017-0059-8
© The Author(s) 2017
- Received: 11 January 2017
- Accepted: 30 June 2017
- Published: 11 July 2017
Abstract
Stochastic noises have a great adverse effect on the prediction accuracy of electric power load. Modeling online and filtering real-time can effectively improve measurement accuracy. Firstly, pretreating and inspecting statistically the electric power load data is essential to characterize the stochastic noise of electric power load. Then, set order for the time series model by Akaike information criterion (AIC) rule and acquire model coefficients to establish ARMA (2,1) model. Next, test the applicability of the established model. Finally, Kalman filter is adopted to process the electric power load data. Simulation results of total variance demonstrate that stochastic noise is obviously decreased after Kalman filtering based on ARMA (2,1) model. Besides, variance is reduced by two orders, and every coefficient of stochastic noise is reduced by one order. The filter method based on time series model does reduce stochastic noise of electric power load, and increase measurement accuracy.
Keywords
- Electric power load
- Stochastic noise
- ARMA model
- Kalman filter
1 Introduction
Power load operation is complex. Accurate power load forecasting has great significance for designing power supply program and making a good power balance between supply and demand. The power load sequence contains relatively obvious white noise. With longer sampling time interval, the white noise becomes more intense [1, 2]. The prediction accuracy of power load is related to the length of historical observation data. With noise and chaos in the observed data, different time series have different upper limit of prediction accuracy [3, 4]. It is important to estimate the noise intensity directly from the observed data and to separate the noise from the observed data, which is very important to improve the accuracy of the power load forecasting result.
To improve the quality of power load data, stochastic noise present in the load data must be identified and filtered out [5, 6]. At present, there are mainly following methods in the power load forecasting field, such as regression analysis, combined forecasting, exponentially smoothing, neural network and wavelet methods, and so on. Moreover, in view of the uncertainties and randomness of short-term load, innovative data processing strategies are proposed, such as frequency domain decomposition method and property matrix hierarchical analysis method [7–9]. However, the existing time series modeling methods may not meet the requirements of time series stationary. These methods neglect the pretreatment of load data and statistical checking [10]. Independent, steady, normal, zero-mean and trend-item processing of the required data is required, and non-stationary, non-random and non-normal characteristics of power load data are needed to be tested.
Time series method and Kalman filter algorithm are proposed to filter the power load stochastic noise by pretreating and statistically testing of power load data, then, the total variance method is used to evaluate the stochastic errors of the load data before and after filtering effectively.
2 Methods
2.1 Stochastic noise time series method in power load data
2.1.1 Timing sequence processing
Traditional load forecasting method adopts the regression analysis and the least square method. However, this method is difficult to reflect the new information of the load change during the operation of the power system to the model, and the prediction accuracy is low. According to the characteristics of power load data, the statistical parameter model reflecting the running state of the system is established, and the time series of electric load is constructed. Then, the shortcomings of the existing methods can be effectively overcomed [11–13].
The prerequisite for establishing the ARMA model is that the load data satisfy the requirements of stationarity and normality. Power load output data usually do not meet these requirements, then, it is necessary to make pre-processing operations and test of the corresponding characteristics for sampled data.
If ∣u∣ ≤ 1.96, there is no significant difference between μ _{ i }, and {x _{ n }} can be determined to be a stationary sequence.
The second step is trend item extraction. The data sequence is processed by difference to get the new sequence. Data sequence subtracts the mean of the new sequence, then, obtains the mean value of the difference to complete the trend item extraction.
The third step is normality test. The power load data sequence was tested for normality [14], mainly including standard skewness coefficient ξ and standard kurtosis coefficient ν.
ξ ≈ 0 and ν ≈ 0 indicates stochastic sequence satisfies the normality requirement.
2.1.2 Online timing modeling
After pretreatment and statistical tests of power load data, model order and parameters also need to be calculated. In addition, the applicability of the new model still need to be tested [14, 15]. Based on the new model, the system state equation and output equation can be established, and Kalman filter method can be used to deal with the power load data.
The AIC criterion takes into account the interaction between model order and residuals, and the smallest AIC value is to be selected.
The applicability of the model is also a critical task for online modeling of power load data. The criterion is to check whether the model residuals are white noise. If the model residuals are white noise, the model is available; otherwise, it is not applicable.
2.2 Kalman filtering based on time series model
Kalman filtering method, a kind of effective recursive filtering method, estimates the system state according to a series of measurements including stochastic noise. Kalman filtering selects proper state space, builds state equation and measurement equation, based on the period and characteristic of load prediction. Parameter estimation and load forecasting are implemented in the filtering, to be an organic whole.
According to the ARMA model, Kalman filtering method is adopted to suppress the stochastic noise of power load. System state equation is built by white noise of the stochastic noise of power load [16, 17].
The mean of both v _{ k } and W _{ k } is zero, white noise with constant autocorrelation function is independent of each other. The statistical properties satisfy the mean equals to zero, E(W _{ k }) = E(v _{ k }) = 0. Autocorrelation function φ _{ vv } = Rδ _{ ki } , φ _{ vv } = Qδ _{ kj }, and cross-correlation function φ _{ vw }(k, j) = 0.
Kalman prediction process is the filtering process of state reconstruction. Known from Eq.(16), estimated information \( \widehat{X} \) of state phasor X is updated constantly. Considering feedback unit, this part can avoid the effect of dynamic noise v _{ k }. However, for estimation value \( \widehat{Y} \) of output phasor Y, it can only be approximated owing to the influence of dynamic noise v _{ k }.
3 Result
3.1 Application and analysis of time series model and Kalman filtering
AIC values of ARMA model of power load
p | q | AIC value | p | q | AIC value |
---|---|---|---|---|---|
- | - | - | 2 | 0 | 0.2145 |
0 | 1 | −0.0268 | 2 | 1 | −0.0295 |
0 | 2 | −0.0265 | 2 | 2 | −0.0212 |
0 | 3 | −0.0144 | 2 | 3 | −0.0273 |
1 | 0 | 0.3546 | 3 | 0 | −0.0157 |
1 | 1 | −0.0221 | 3 | 1 | −0.0219 |
1 | 2 | −0.0233 | 3 | 2 | −0.0249 |
1 | 3 | −0.0275 | 3 | 3 | −0.0263 |
Initial value of co-variance matrix P is \( \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill \end{array}\right] \), initial value of matrix X is \( {\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^{\mathrm{T}} \),value of matrix R is variance of estimation error, and value of progress noise Q equals to \( \left[\begin{array}{cc}\hfill {\sigma}_a^2\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\sigma}_a^2\hfill \end{array}\right] \).
Stochastic noise coefficients before and after Kalman filtering
Noise coefficient | Before filtering | After filtering |
---|---|---|
Q | 1.50e-3 | 1.91e-4 |
L | 1.01e-5 | 1.26e-6 |
B | 4.74e-4 | 5.66e-5 |
K | 7.50e-3 | 8.89e-4 |
R | 4.31e-2 | 5.10e-3 |
Known from Table 2 and Fig.5, selected power load data mainly contains quantization noise, rate random walk and bias instability. However, each stochastic coefficient in the power load data is effectively reduced through time series modeling and Kalman filtering, each coefficient value is reduced by an order of magnitude. The proposed method can eliminate the stochastic noise of power load data and promote power load accuracy.
4 Conclusion
Suppressing power load stochastic noise is one of the important links in power load modeling and forecasting. Based on the characteristics of power load data, time series analysis is used to model the data of power load on-line, realizing the pretreatment and inspection analysis of power load data. ARMA (2,1) model is established and Kalman filtering method is used to denoise load data. And total variance method is adopted to verify the effect of modeling and filtering, namely the stochastic error coefficients before and after filtering.
The results show that stochastic noise amplitude of power load data after time series modeling and Kalman filtering is significantly reduced, the variance value is decreased by two orders of magnitude, and each stochastic error coefficient of power load is reduced by an order of magnitude. The proposed time series modeling and filtering method can effectively suppress the stochastic noise of power load data and improve the prediction accuracy of power load.
Declarations
Acknowledgments
This work was financially supported by Science and Technology Project of SGCC (SGTJDK00DWJS1600014).
Authors’ contributions
LH, YY and HZ mainly wrote the paper together, LH is responsible for the most of paper, including abstract, Part 1 Stochastic noise time series method in power load data and Part 3 Application and analysis of time series model and Kalman filtering individually. In addition, LH and YY are responsible for the revised manuscript, finishing the reviewers’ comments. YY is responsible for the Part 2 Kalman filtering based on time series model and participates in the abstract with LH. HZ writes Part 0. Introduction and participates in the Part 4 Conclusion with YY. And helps to submit the revised manuscript in the web. XW gives advice on the paper structure and helps to check the whole paper’s grammar. HZ helps to check the whole paper’s grammar and words spelling. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests. And the authors certify that none of the material in the paper has been published or is under consideration for publication elsewhere.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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