Enhancing short‐term probabilistic residential load forecasting with quantile long–short‐term memory
2017; Institution of Engineering and Technology; Volume: 2017; Issue: 14 Linguagem: Inglês
10.1049/joe.2017.0833
ISSN2051-3305
AutoresDahua Gan, Yi Wang, Ning Zhang, Wenjun Zhu,
Tópico(s)Traffic Prediction and Management Techniques
ResumoThe Journal of EngineeringVolume 2017, Issue 14 p. 2622-2627 ArticleOpen Access Enhancing short-term probabilistic residential load forecasting with quantile long–short-term memory Dahua Gan, Dahua Gan Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorYi Wang, Yi Wang Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorNing Zhang, Corresponding Author Ning Zhang ningzhang@tsinghua.edu.cn Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorWenjun Zhu, Wenjun Zhu Electric Power Research Institute, Guangdong Power Grid Corporation, China Southern Power Grid, Guangzhou, People's Republic of ChinaSearch for more papers by this author Dahua Gan, Dahua Gan Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorYi Wang, Yi Wang Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorNing Zhang, Corresponding Author Ning Zhang ningzhang@tsinghua.edu.cn Department of Electrical Engineering, Tsinghua University, Beijing, People's Republic of ChinaSearch for more papers by this authorWenjun Zhu, Wenjun Zhu Electric Power Research Institute, Guangdong Power Grid Corporation, China Southern Power Grid, Guangzhou, People's Republic of ChinaSearch for more papers by this author First published: 19 January 2018 https://doi.org/10.1049/joe.2017.0833Citations: 21AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In the study of load forecasting, short-term (ST) load forecasting in the horizon of individuals is prone to manifest non-stationary and stochastic features compared with predicting the aggregated loads. Hence, better methodologies should be proposed to forecast ST residential loads more accurately, and refined representation of forecasting results should be reconsidered to make the prediction more reliable. A format of ST probabilistic forecasting results in terms of quantiles is offered, which can better describe the uncertainty of residential loads, and a deep-learning-based method, quantile long–ST memory, to implement probabilistic residential load forecasting. Experiments are conducted on an open dataset. Results show that the proposed method overrides traditional methods significantly in terms of average quantile score. 1 Introduction Electric load forecasting is playing an increasingly crucial role in the overall power system. It is of great significance to provide a forecasted load in advance, striking the balance between the demand and the cost; therefore, dispatching the energy more efficiently and minimising the energy wastage. Short-term (ST) load forecasting refers to the load forecasting with a cut-off horizon of two weeks [1], benefiting temporary demand response and scheduling in advance. However, most of the forecasting problems focused on aggregated loads of specific region, because this kind of load is more accessible than loads in residential level. With the prevalence of smart grid data, individual loads are more obtainable than ever, making it possible to make more accurate forecasts. The literature in ST residential load forecasting shows similarity in methodology with aggregated load forecasting problems. Concretely, approaches can be divided into two divisions. The first division is statistical modelling based. In [2], multivariable linear regression is utilised to find the linear relationship between ST load and corresponding variables such as the weather and previous loads. Hong et al. [3] proposed an interaction regression-based model, considering cross-effects of different variables. Other techniques used in the time-series analysis such as autoregressive moving average (ARMA) autoregressive integrated moving average model (ARIMA), and generalized autoregressive conditional heteroskedasticity (GARCH) have been utilised to capture trend and dependencies of elements in ST load series [4-6]. The second division is machine-learning related, especially with the help of artificial neural network (ANN) and more advanced deep-learning methods. Compared to gradient boosting [7], support vector machine [8], ANN [9], and other traditional machine-learning methods, deep learning has proved to be effective in ST load forecasting problems, especially for non-stationary and stochastic residential loads [10-12]. Recurrent NN (RNN) has been the most widely used deep-learning architecture due to its great adaptability to data in the form of time series. However, as is mentioned in the previous work doing residential load forecasting [10], due to the variety of residential behaviour, this kind of load has more uncertainty, hence decreasing the accuracy of prediction. To solve this issue, a more robust model should be proposed to enhance forecasting accuracy, and more reliable forecasting representation should be built to adapt to more intense uncertainty. Long–ST memory (LSTM), a special type of RNN, has achieved a significant enhancement in specific time-series related task such as machine translation, speech recognition [13, 14] due to its efficacy in memorising LT and ST temporal informations simultaneously. It is also utilised as the deep-learning-based model for most of the residential load forecasting tasks [10, 12, 15, 16]. Nevertheless, all of these works generate forecasting outputs in terms of single point loads, none of them consider a better representation of forecasted load to manifest the intense uncertainty. Compared to point forecasting, probabilistic forecasting describes the variation of the load by providing outputs in terms of probabilistic density function (PDF), confidential intervals, or quantiles of the distribution. It can better describe the uncertainty of loads. As is mentioned in [17], existing schemes follow the combination of following components: generating input scenario simulation, designing probabilistic models, and post-processing that can change single-load forecasts into probabilistic form. In [18, 19], input scenarios are generated according to historical data. Quantile regression is utilised in [20] to generate multiple forecasting results. In [19-21], simulation of historical-error distribution (HED) was implemented to convert point loads into intervals. It can be concluded that all these approaches are all implemented in aggregated level, while they neglect the potentials and necessities of conducting probabilistic residential load forecasting, since the latter kind of load is usually more volatile and unpredictable. Hence, this paper fills the scarcity by considering a probabilistic load forecasting scenario with LSTM in terms of quantiles, aiming to utilise the robustness of LSTM in forecasting ST non-stationary time series. The proposed probabilistic residential forecasting strategy is named as quantile LSTM (Q-LSTM) for short. Concretely, our method is established with deep LSTM with loss function resembling that of quantile regression, a commonly used method for generating probabilistic forecasting results [22]. Then, post-processing with calculating quantiles according to empirical DF is conducted to offset anomalous forecasting quantiles. We compared the results generated by Q-LSTM with traditional ANN with quantile loss and simulation of historical residuals, indicating the significant superiority of the proposed method. The main contributions of this paper can be concluded as follows: Probabilistic forecasting in terms of quantiles is first introduced to the field of residential forecasting, manifesting a more reliable result compared with single point forecast. Deep LSTM was extended with quantile loss, showing great potential in modelling stochastic residential loads in a short term. Quantile-loss-based method is able to present more robust results compared with the commonly used methods when keeping the same forecasting model. The following paper is organised as follows: Section 2 introduces the overall procedure of probabilistic residential load forecasting. Section 3 is responsible for illustrating the proposed methodology and benchmarks used in the case study. In Section 4, criteria for probabilistic forecasting are introduced. In Section 5, a case study forecasting loads in a probabilistic manner with real-world residential loads is implemented, and corresponding results and comparison are established. In the last section, the conclusion is drawn and future implications and extensions are illustrated. 2 Framework Fig. 1 is the process chart that indicates the overall process of with Q-LSTM. First, raw data loaded from the original dataset should be pre-processed including filling the missing value, normalisation, and converting the load sequences into shapes that meet the requirements for LSTM. Linear interpolation is utilised to fill the missing value since those missing values are not continuous. Aside from that, due to the sensitivity of LSTM to the values of input data, normalisation of data should be implemented. Specifically, raw data is converted into the range of [−1, 1] due to the limitation of tanh function inside LSTM. Moreover, historical sequences are processed as inputs for the model, whereas the current load is regarded as the target for training. Fig. 1Open in figure viewerPowerPoint Overall procedure of probabilistic residential load forecasting with Q-LSTM The following is the training and forecasting process, which is separated into models with different training loss functions, and generating the final prediction with the empirical distribution of outputs generated from different models. Finally, evaluation will be implemented by comparing predicted values and true load points. 3 Methodology In this section, overall ultra-ST probabilistic load forecasting problem is formulated. Then core methodologies proposed in this paper, together with benchmarks will be illustrated. 3.1 Problem formulation The residential load sequence in length of N can be formulated as time series with N dependent random variables . For each , the observation can be denoted as . In ultra-ST load forecasting problem, forecasting horizon is usually one time step, knowing d load points ahead. In other words, the aim is to forecast , given with specific forecasting model where t is the time to be forecasted and d is the length of historical load sequence. In the case of point forecasting, will be estimated in terms of single value . Yet when it comes to probabilistic forecasting, will be estimated by showing the PDF of the random variable – . However, in most cases, PDF cannot be easily described, and there are no direct criteria to evaluate the performance of probabilistic load forecasting. Instead, probabilistic forecasting can be given in terms of several quantiles of , that is to say, estimates in terms of vectors consisting of quantiles. It is much easier to generate than PDF since it does not require the specific formulation of the distribution of . Then the evaluation can be easily done by utilising and the observed value of – . 3.2 Probabilistic ultra-ST load forecasting with Q-fully connected (FC) NN As is mentioned in Section 1, there are many ways to do probabilistic load forecasting. However, in order to prove the superiority of the proposed model over commonly used ANN-based methods, we only discuss the ANN-related models. The most widely obtained architecture of ANN is an FCNN. The architecture of FCNN is demonstrated in Fig. 2. Fig. 2Open in figure viewerPowerPoint FCNN architecture for load forecasting We denote the input vector as where t is the time of load to be forecasted, n is the dimension of input vector, and d is the length of the historical load. If the historical load is considered to be the only components of the input, then . Then the FCNN model can be formulated as where n denotes the layer number and is the activation function in layer n. and are weight matrix and bias vector in layer n, which can be updated through back propagation (BP) mechanism. is the output of the NN, indicating the estimation of load in terms of a single value. The next step is updating the parameters in FCNN in order to minimise the loss function. This step can also be regarded as model training. For regression problem, the most commonly used loss function is mean squared error (MSE), which can be written as (1) where i denotes the i th target load in the training set. By minimising (1), the network can be learnt to estimate the load at time t. However, the aim is to generate a probabilistic forecasting result, thus special techniques should be used to meet this requirement. By borrowing the idea from linear quantile regression, a quantile loss for training the NN is proposed, which can be described by the following equation: (2) where is the quantile of the distribution of random variable to be calculated. From the expression above, we can get forecasting results , , …, . In this case, the uncertainty of load can be more explicitly described compared with single-load forecast. 3.3 Probabilistic ultra-ST load forecasting with FCNN with HED Since the residential load has a strong dependency on periodical temporal variables, the load points featured with same temporal variables are supposed to be similar. For instance, the residential loads of a specific user tend to have same 'low pattern' in the time interval of night. Nevertheless, the variation still exists after aggregating loads into different time intervals, and it is correlated with time as well. Concretely, the uncertainty of load tends to be quite small from 1:00–5:00 pm, for the user is in bed, and most appliances are likely to be turned off. Yet the load can vary a lot from 10:00–12:00 am because of different load consuming patterns caused by different kinds of behaviours. Hence, an intuition is to describe the load variation by adding a probabilistic error on single point load forecasting Now, the forecasting error can be described with a function with h as the independent variable. For each h, can be estimated by the empirical distribution calculated by all load points aggregated to time interval of h. As is mentioned in the first method, to make it easier to evaluate, the final forecasting results will be generated in terms of quantiles by calculating quantiles of error distribution . If there are observed error values at time interval h are denoted as , The quantile of the error is represented by , let , . Then the value of can be calculated by (3) Therefore, the final quantile predicting result made by FCNN-HED is calculated by (4) where is the estimated load with MSE as loss function and N is the number of samples feeding into the model. 3.4 Probabilistic ultra-ST load forecasting with Q-LSTM Although FCNN is the most prevailing architecture in load forecasting with an NN, the efficacy of FCNN is maintained with the assumption that the input data should be highly independent with each other. However, it is obvious that continuous values in a load sequence are temporal dependent. Therefore, another basic architecture in deep learning, RNN, is utilised to handle temporal-dependent forecasting problem, which has been proved to be effective in many other fields. LSTM is the most dealing with scenarios with long time sequences as inputs. It overcomes the most significant problems in the training process of naïve RNN – gradient vanishing. In forecasting field, this problem leads to the extreme decrease of the contribution of earlier load points when predicting the current point. In this case, crucial information might be lost, since residential load has LT periodic feature; therefore, load values at a specific time can be greatly determined by loads long time ago. Further explanation will be given to illustrate how LSTM can effectively utilise information from a long time ago, and capture features from temporally adjacent information. Concretely, probabilistic load forecasting based on Q-LSTM consists of two crucial elements. The first elements are structure and mechanism of LSTM, which is also the basic idea of combining ST and LT informations. Compared to single hidden state h in simple RNN, which is sensitive to ST input, an LSTM unit contains two hidden states: h and c. h is set for keeping ST information, yet c is designed for maintaining LT information, yet it contains an extra mechanism of forgetting unrelated information corresponding to current time. This dual-state connection is shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Dual hidden state of LSTM In Fig. 3, stands for the time steps moving forward, is the input, state c, and state h at specific time. Since the attribute of keeping adjacent temporal information is spontaneous due to the structure of RNN, the key of LSTM is to control the LT information. Therefore, LSTM innovatively creates the notion of 'controlling gate'. Gates are actually an FC layer, a vector with changeable elements. They are set in different parts of a single LSTM unit, controlling the information flowing through the gate by multiplying the information (in terms of vector or matrix) element wisely. The first gate in LSTM unit is the forget gate , determining how much information can keep from the last state c. The forget state at time t can be formulated as (5) where , , and stand for forget gate vector at time t, the output vector (also the state h vector) at time , and bias of the forget gate at time t, and is the weight matrix of the forget gate. is the concatenating operator for vectors. The second gate is the input gate , determining how much current information should be treated as the input to generate current state c. is defined by (6) It is obvious that has the same formulation with , yet the weights and bias are different, since these two gates are determined differently by . Combining these two controlling gates together, the current hidden state could be determined by adding the parts of information they controlled – the LT information controlled by and ST information controlled . The current state can be represented as where the operator ° stands for the element-wise product. The last thing of an LSTM unit is determining how much information can eventually be treated as the output. Another controlling gate is named as the output gate Since gates control the information flow by doing element-wise product, the final output of LSTM is defined by Fig. 4 shows the inner structure of an LSTM unit. By stacking LSTM units in temporal and spatial scopes, deep LSTM model can be generated, taking inputs from time to as inputs, and giving state h as the final output. To generate probabilistic forecasting result, pinball loss described in (2) is defined, all parameters in LSTM can be updated with BP through time with gradient-descent-based optimisation. The overall Q-LSTM for probabilistic forecasting in shown in Fig. 5. It contains two parts. The first part is RNN part, taking sequential loads at different time steps into RNN with LSTM units, outputting the hidden state at the last timestamp. The second part is FC part, taking hidden state generated from RNN part as the input, resolving it with non-linear function in order to decode high-dimensional hidden state into visible load values. Fig. 4Open in figure viewerPowerPoint Inner structure of LSTM unit Fig. 5Open in figure viewerPowerPoint Q-LSTM for probabilistic residential forecasting 4 Evaluation criteria In probabilistic forecasting problem, evaluation criteria are designed mainly in three aspects: reliability, resolution, and sharpness. Among all criteria, prediction interval coverage probability, interval score are in charge of reliability and sharpness, yet they only consider the upper and lower bounds of forecasting intervals, ignoring detailed information inside forecasting intervals. Therefore, a more advanced criterion is proposed in order to describe the overall performance of probabilistic forecasting considering contributions of all quantiles of the predicted distribution, called average quantile score (AQS). AQS calculated over N samples in testing set and quantiles are defined by (7) where is the actual residential load at time t and is the estimated load with respect to quantile. This criterion is proved to be effective in general load forecasting scenarios, and has been adopted as the ranking score for 2017 Global Energy Forecasting Competition (GEFCOM 2017). It is obvious that a lower AQS indicates a better forecasting result. This is the criterion being used to evaluate the proposed method and benchmarks in this paper. 5 Case study In this section, the implementation of the proposed methodology will be concluded including hardware and software settings for the experiment, data description as well as detailed implementation, and discussion on the dataset. 5.1 Hardware and software settings The experiment is implemented on a widely used configuration for deep-learning work. The whole forecasting programme is developed on the framework of Keras 2, with TensorFlow as the backend. The process of training the model and evaluating the performance is supported by CUDA8.0 and Nvidia GPU, TITAN X (Pascal). The parallel computing is conducted by assigning memories of GPU to several parallel processes, each of which is in charge of certain batches of quantiles (99 quantiles in total for specific residential load series). We choose the number of batches to be ten since it not only confirms enough computing memories for each process, but also maintains enough processes number to run at the same time. 5.2 Data description The residential load data used in this paper are from the Smart Metering Electricity Customer Behaviour Trails proposed by Commission for Energy Regulation in Ireland. The data recorded by the smart meter covers about 6000 residents enduring for more than 500 days, from 1st July 2009 to 31st December 2010. The temporal load is recorded per half an hour, in other words, there will be 48 load points in a single day. For this paper, only considering probabilistic forecasting on individuals, we randomly sample 20 residential consumers for performance evaluation. Besides, since this paper aims to solve ultra-ST load forecasting problem, original load sequences should be converted to historical sequences (features) and following target sequences (labels), and then be split to training sets and testing sets. To formulate the data generation process, the full load sequence for user can be defined as . Each element in can be indexed according to temporal sequence. We define the load point in as . The length of historical sequences is . Intuitively, the i th historical sequences can be generated in the following pattern: (8) Besides, the label load can be defined as Bringing these two loads together, samples for establishing forecasting models for user u can be denoted as . In total, there will be samples prepared for training and evaluating the forecasting model. Then, the samples are split according to specific ratio into training sets and testing sets. In the case study, we manually confirm the ratio of training and testing set as 4:1. 5.3 Implementation of ultra-ST forecasting with proposed model and benchmarks After generating the datasets, models are trained on training sets separately for 20 users, and then the probabilistic forecasting in terms of 10–90 percentiles will be given. Finally, evaluation of forecasting with pinball loss will be implemented to three methods. 5.3.1 Quantile-FCNN (Q-FCNN, benchmark I) For each normalised training sample , Q-FCNN takes in inputs with dimensions, referring to historical sequence with lookbacks. With loss function described in (2), ranges from 0.01 to 0.99, 99 values in all. For each , the training process minimises the loss with mini-batch-based Adam optimisation algorithm, by updating parameters of the network with BP. During the training process, the original training set is split into actual training set with data fitting into the model, and the rest as the validation set. To prevent overfitting, an early stopping mechanism is utilised by monitoring the average loss on validation set after each training epoch. If the validation loss fails to decrease for more than five epochs, the training process is terminated, preventing coming overfitting problem. The specific hyperparameters for the training of Q-FCNN are summarised in Table 1. Table 1. Parameter tuning details for Q-FCNN Hyperparameters and other tuneable settings Details layer number 2 dimension of hidden unit [16, 8] [32, 16] [64, 32] sequence length 48 activation function tanh training batch size 20, 40, 80, 160, 1000 After that, the 99 models with different quantile losses will generate 99 predictions based on feature sequences in the testing set. Those forecasting results will generate an empirical distribution for each user, and the pinball loss will be calculated on 10–90 percentiles of this distribution. The final result is reported with the combination of different parameters with the best forecasting result. 5.3.2 FCNN with HED (FCNN-HED, benchmark 2) FCNN-HED takes in the same inputs as Q-FCNN does, and maintains the same network structure, optimising the MSE loss described in (1). If the real load in training set and estimated load at time can be denoted as and , respectively, the training error is . Each contains a bunch of load points due to the long time interval of training set, thus the quantiles of the can be estimated with empirical distribution calculated by elements in . The final forecasting results in terms of quantiles will be generated according to (4), where equals 9 in the case study. 5.3.3 Quantile LSTM LSTM takes in the same sequential load data as FCNN does. As is mentioned in Section 3, the Q-LSTM contains two parts in network structure – RNN part and FC part; therefore, hyperparameters are tuned mainly in two parts. The detailed information of tuned parameters for Q-LSTM is recorded in Table 2. Table 2. Parameter tuning details for Q-LSTM Hyperparameters and other tuneable settings Details layer number of LSTM 1, 2, 3 dimension of hidden state in LSTM units 4, 8, 16, 32, 64 sequence length 48 layer number of FC layer 2 dimension of FC layer [16, 8] [32, 16] [64, 32] training batch size 20, 40, 80, 160, 1000 sequence length 48 5.3.4 Results and discussion We evaluate the performance of 20 random chosen residents on the proposed Q-LSTM and two benchmarks introduced in Section 3. The overall results in terms of AQS are represented in Table 3, where MaxRI is short for maximum relative improvement between the best forecasting model and the worst one. It is reported that Q-LSTM has the refinement to all users compared with the worst benchmarks, and has a significant improvement on average point compared with both benchmarks, which override Q-FCNN and FCNN-HED by 4.21 and 8.96%, respectively. This result validates two important proposals mentioned in this paper. The first one is that the framework quantile NN has a more solid performance in modelling highly stochastic time series such as residential loads compared with traditional historical-error-based method. Table 3. Overall results of three methods on residential loads in terms of AQS User ID Q-FCNN FCNN-HED Q-LSTM MaxRI, % 1 0.05053 0.05378 0.04795 10.84 2 0.01305 0.01380 0.01238 10.28 3 0.03349 0.03192 0.03163 0.91 4 0.07907 0.08165 0.07733 5.29 5 0.12962 0.13452 0.12644 6.01 6 0.12130 0.12699 0.11641 8.33 7 0.10433 0.10853 0.10461 3.61 8 0.12223 0.12900 0.11271 12.63 9 0.06631 0.06843 0.06318 7.67 10 0.10509 0.10941 0.10494 4.08 11 0.05442 0.05434 0.05227 3.81 12 0.02874 0.02662 0.02229 16.27 13 0.08161 0.08613 0.08021 6.87 14 0.08806 0.09175 0.08351 8.98 15 0.14227 0.14699 0.13927 5.25 16 0.05679 0.06492 0.05776 11.03 17 0.06946 0.06872 0.06035 12.18 18 0.06540 0.06787 0.06106 10.03 19 0.09963 0.14202 0.09253 34.85 20 0.06436 0.06268 0.06254 0.22 average 0.07879 0.08350 0.07547 8.96 Besides, by treating sequential historical loads with LSTM, the NN captures the dependency of different inputs better compared with treating them as parallel input; therefore, contributing to a smaller AQS in probabilistic forecasting. The forecasting results are shown in Fig. 6. It can be concluded that the interval widths vary temporally, indicating the ability to learn temporal error automatically, but in a more flexible manner compared with the historical-error-based method. Fig. 6Open in figure viewerPowerPoint Residential load forecasting with Q-LSTM 6 Conclusions and future work In this paper, the importance of probabilistic forecasting on residential load is emphasised, and a novel deep-learning-based method measured by AQS is first implemented, proven to be of great efficacy. Since it is just a relatively elementary experiment, lots of refinement, and exploration can be established. 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