Abstract
Scenario generations for renewable energy sources and loads play an important role in the stable operation and risk assessment of integrated energy systems. This paper proposes a deep generative network based method to model time-series curves, e.g., power generation curves and load curves, of renewable energy sources and loads based on implicit maximum likelihood estimations (IMLEs), which can generate realistic scenarios with similar patterns as real ones. After training the model, any number of new scenarios can be obtained by simply inputting Gaussian noises into the data generator of IMLEs. The proposed approach does not require any model assumptions or prior knowledge of the form in the likelihood function being made during the training process, which leads to stronger applicability than explicit density model based methods. The extensive experiments show that the IMLEs accurately capture the complex shapes, frequency-domain characteristics, probability distributions, and correlations of renewable energy sources and loads. Moreover, the proposed approach can be easily generalized to scenario generation tasks of various renewable energy sources and loads by fine-tuning parameters and structures.
THE integrated energy system is a standard large-scale, non-linear, and dynamic control system with various flexible resources, such as electric vehicles, wind turbines, photovoltaic (PV) systems, and heat pumps [
The core thinking of the scenario generation aims to produce a range of new time-series curves, which look like historical time-series curves. With regard to whether statistical hypotheses about probability distribution are needed, existing methods of scenario generations can be subsumed under just two categories: explicit density model based methods and implicit density model based methods. A brief comparison between the above two methods is shown in
The implicit maximum likelihood estimation (IMLE) is a well-known deep generative network widely used in the computer vision field [
1) The IMLEs are generalized from scenario generation of a single wind farm into scenario generation of multiple correlated renewable energy sources and loads. The specific structure of IMLEs is designed to generate scenario for multiple units.
2) The proposed approach is completely based on data-driven generative networks, which project Gaussian noises into high-dimensional time-series curves. After unsupervised learning, IMLEs can directly generate realistic scenarios for PV plants, wind farms, heating, cooling, and power loads, which hold the same characteristics, e.g., correlations, frequency-domain characteristics, fluctuations, and probability distributions, as the real ones.
3) Different from explicit density model based methods, the proposed approach sidesteps manual assumption of the PDFs of renewable energy sources and loads, resulting in stronger generalization. By fine-tuning parameters and structures of the data generator, IMLEs can be easily used to produce new scenarios for multiple adjacent renewable energy sources and loads by simply inputting Gaussian noises to the pre-trained IMLEs. Besides, there is no limit to the number of new scenarios.
4) Compared with other implicit density model based methods (e.g., GAN), the proposed approach can sidestep unstable training problems, e.g., non-convergence and mode collapse, since the loss function ensures that each real example has a generated sample nearby at optimality, and the training process is a simple optimization problem that minimizes the loss function.
The rest of this paper is organized as follows. Section II formulates the scenario generation using IMLEs. Section III presents the case study. Section IV presents the conclusions and future works.
Generally, the scenario generation tasks can be divided into two categories: ① scenario generation for a single renewable energy resource or load; and ② scenario generation for multiple adjacent renewable energy resources or loads.
For the single renewable energy resource or load generation, training samples include a series of historical power generation curves or load curves , and the goal is to train a data generator based on IMLEs by using real samples from the training set. As shown in

Fig. 1 A simple structure of proposed approach.
In integrated energy systems, multiple adjacent renewable energy resources or correlated loads may need to be considered at the same time. Therefore, the scenario generation of a single time series can be generalized to the scenario generation of multiple time series. Specifically, historical samples of N renewable energy resources or loads are used to train a data generator whose goal is to simultaneously generate multiple scenarios. The generated samples should represent probability distributions of renewable energy resources or loads as well as the correlations. For scenario generation of renewable energy sources, the correlations include the spatial and temporal correlations of power generation curves. For the scenario generation of multiple loads, e.g., heating, cooling, and power loads, the correlations cover the temporal correlations of load curves and the correlations (also called coupling) between multi-class loads.
Normally, deep generative networks can be naturally viewed as a sampling procedure from a prior probability distribution, e.g., Gaussian distribution, and then noises are fed to the data generator to obtain the new samples [
(1) |
where is a Gaussian noise; and is a new scenario generated from the data generator of IMLEs.
The probability distribution of the real scenario x is expressed as:
(2) |
where is the Gaussian distribution; is the probability distribution of the real scenario x; and is a conditional Dirac distribution or conditional Gaussian distribution.
Theoretically, (2) can fit any probability distribution, covering the probability distribution of real samples. Furthermore, suppose is a Dirac distribution.
(3) |
where is the Dirac function; and is the expected value.
Actually, the Dirac function can be regarded as a Gaussian distribution with the variance close to zero, and its mathematical formula is expressed as:
(4) |
where is the standard deviation; and is the dimension of noises z.
Moreover, (4) is substituted into (3) to obtain the following results:
(5) |
Suppose p(x) is the probability distribution of real scenarios. The training process of generative networks is to minimize the following loss function by the gradient descent method:
(6) |
By substituting (5) into (6), a more specific loss function L can be obtained as:
(7) |
Obviously, there are few constants in (7) that will not affect the training results of IMLEs. In order to make the loss function more concise, these dispensable constants are ignored, and then a new form of the loss function is obtained as:
(8) |
In the training stage of IMLEs, n Gaussian noises and m real scenarios are fed to IMLEs to the last form of the loss function [
(9) |
In this case, this loss function can be used to update the weights of IMLEs by the back propagation algorithm.
The recent success of CNNs has boosted researches on the pattern recognition. Many data mining tasks such as target detection and speech recognition, which are once heavily dependent on artificial feature engineering to extract informative features, have been revolutionized by CNNs with powerful feature representing ability [
For the scenario generation task, the CNNs mainly consist of transposed convolutional (TransConv) layers and dense layers. Specifically, the input features are performed with TransConv operations, and then a bias vector is added to obtain the output data of the TransConv layers. Its mathematical formula is expressed as:
(10) |
where Ytran is the output data of TransConv layers; Xtran is the input data of TransConv layers; is the activation function of TransConv layers; Wtran is the weight of TransConv layers; Btran is the bias vector of TransConv layers; and * is the TransConv operation.
Similarly, the output data of dense layers can be obtained by multiplying input data with weights and adding a bias vector. Its mathematical formula is expressed as:
(11) |
where is the output data of dense layers; is the input data of dense layers; is the activation function of dense layers; is the weight of dense layers; and is the bias vector of dense layers.
To summarize the above description, the steps of scenario generation for multiple renewable energy sources and loads based on IMLEs are introduced as follows.
Firstly, the datasets of renewable energy sources and loads are imported. 80% of the samples are randomly selected as the training set, and the remaining samples are used as the test set. Before inputting real time-series curves of renewable energy sources and loads into IMLEs, they should be normalized; otherwise, IMLEs are hard to converge in the training process. Hence, the minimum-maximum normalization method is employed to project input data to values from zero to one [
It is widely known that IMLEs are originally used to produce images with the same size of columns and rows. Although the successful applications in image generation have proven that IMLEs can accurately capture complex characteristics of high-dimensional data through unsupervised learning, they have difficulty in directly processing the time-series curves where the size of rows and columns are different. Therefore, the time-series curves of renewable energy sources or loads should be reshaped into a square matrix before being input to IMLEs.
The rules of date transformation, i.e., converting time-series curves into matrices, are shown in

Fig. 2 Rules of data transformation. (a) A wind power curve to a 2-D matrix. (b) A PV power curve to a 2-D matrix. (c) A load curve to a 2-D matrix. (d) Multiple wind power curves to a 2-D matrix. (e) Multiple PV power curves to a 2-D matrix. (f) Multiple load curves to a 2-D matrix.
Before training IMLEs, some key parameters and structures such as choice of the optimizer, number of iterations, and number of hidden layers need to be initialized. Normally, the control variable method can be used to select these parameters [
To obtain a batch of noises , the Monte Carlo method is employed to sample the Gaussian distribution. Then, these Gaussian noises are input into the data generator of IMLEs to generate the corresponding new scenarios . Moreover, a batch of real scenarios is selected from the training samples randomly. The Euclidean distances between the real scenario and all new scenarios are calculated to find the closest new scenario .
To update the weights of data generator by the gradient descent method and back propagation method, the loss function of IMLEs is calculated. When the number of iterations is exceeded, the pre-trained data generator will be output.
After training IMLEs, batches of Gaussian noises are fed to the pre-trained data generator of IMLEs to obtain the corresponding new time-series curves. Furthermore, the new scenarios are also the matrices with the same size of columns and rows, and they need to be transformed into vectors through the inverse Reshape function. Last but not least, samples from the test set are used to evaluate the performance of the proposed approach.
To fully test the performance of the proposed approach for scenario generations of renewable energy sources and loads, numerical simulations are performed on three real datasets. Specifically, the solar integration data and wind integration data were published by the National Renewable Energy Laboratory of the United States [
In all datasets, 80% of the samples are randomly selected as the training set, and the remaining samples are used to evaluate the performance of the proposed approach. The programs are implemented in Spyder 3.2.8 with deep frameworks, including Tensorflow 1.0 and Keras 2.0.
The scenario generation of a single wind farm is utilized as an example to illustrate how to select the key parameters of IMLEs. Specifically, the basic units which are commonly used to construct the hidden layers in the field of deep learning include dense layers, TransConv layers, long short-term memory (LSTM) layers, and gated recurrent unit (GRU) layers. To find the suitable basic units for IMLEs, the loss functions of IMLEs consisting of different layers are calculated 20 times, and the average loss functions are given in
The following conclusions can be drawn from
1) The LSTM layer based IMLEs and the GRU layer based IMLEs have similar performance, which are inferior to the Dense-based IMLEs. In contrast, the TransConv layer based IMLEs significantly outperform other IMLEs in most cases, since CNNs are better at extracting the latent features of the data.
2) With the increase of the number of hidden layers, the loss function of TransConv layer based IMLEs first decreases and then increases again, because multiple TransConv layers can improve the representation ability of IMLEs. However, too many hidden layers lead to an over-fitting problem. Generally, three TransConv layers can be considered as a good starting point for IMLEs.
As shown in

Fig. 3 Structures and parameters of IMLEs for scenario generations. (a) Scenario generations for a single generation unit. (b) Scenario generations for nine adjacent generation units.
Taking
Note that the parameters in TransConv layers (e.g., filter, kernel, and stride) generally depend on experience and application (the shape of the output data) [
After initializing the structure of IMLEs, the gradient descent methods are applied to optimize the loss function of IMLEs. Furthermore, popular gradient descent methods include Adagrad, Adamax, SGD, Adam, Nadam, RMSProp, and AdaDelta [
IMLEs have good performances when Adamax, SGD, Nadam, RMSprop, and Adam are used as optimizers of IMLEs. In particular, the loss functions of Adam algorithm is the smallest, indicating that the Adam algorithm is the best choice for IMLEs. In addition, the loss functions of AdaDelta and Adagrad are significantly larger than those of other methods, which shows that they are not suitable for scenario generation via IMLEs.
In

Fig. 4 Training evolution of IMLEs on wind dataset.
Along with the increasing iteration times, the loss function of IMLEs decreases rapidly. When iteration times are more than 100, the loss function of IMLEs is getting close to a constant, which indicates that the data generator has converged. Unlike other generative networks (e.g., GAN) whose loss function involves instability or non-convergence [
Normally, relevant works visualize a part of real images and their closest new images, which are the final result of the actual optimization, to verify the effectiveness of different methods in computer visions [
In order to check if real scenarios and new scenarios generated from IMLEs have similar properties, 2500 Gaussian noises are used as inputs of the data generator respectively, so as to produce 2500 PV power generation curves and 2500 wind power generation curves. Next, a portion of real scenarios with specific properties (e.g., sharp fluctuation, fast ramps, and large peaks) from the test set are picked to calculate Euclidean distances with the new scenarios. Finally,

Fig. 5 Visualization of real scenarios and their closest new scenarios generated from IMLEs. (a) Three samples of wind farms. (b) Three samples of PV plants.
The first row of
Aside from the shapes of power generation curves, it is necessary to verify some common statistical properties between real and new power generation curves generated from IMLEs.
For example, the temporal correlation is a typical property at the wind and PV power generation curves, which are often represented by the auto-correlation function [
Duration curves of wind farms and PV plants reflect the variation of output powers in a downward form [
The frequency-domain properties and fluctuations of power generation curves play an important role in the stable operation of integrated energy systems. To represent the energy value of frequency components in wind and PV power generation curves, the power spectral densities (PSDs) are used to evaluate the frequency-domain characteristics of power generation curves [
From the fourth row of
The Pearson correlation coefficient is one of the popular indices to evaluate the relationship between continuous variables, and it is often used to measure the linear relationship of wind and PV power generation curves at different look-ahead times. To further verify if new scenarios generated from IMLEs have similar temporal characteristics as the real scenarios,

Fig. 6 Pearson correlation matrices of different scenarios. (a) Real wind power scenarios. (b) New wind power scenarios generated from IMLEs. (c) Real PV power scenarios. (d) New PV power scenarios generated from IMLEs.
The following conclusions can be drawn from
1) For PV power generation curves, their time horizons of Pearson correlation matrices are different from those of wind power generation curves. This is because PV power is zero in the morning and at night, and Pearson correlation coefficient does not exist during these periods. Moreover, with the increase of time, the Pearson correlation coefficients between current wind power and previous wind power gradually decrease, while the temporal characteristics of PV power decrease first and then increase.
2) The Pearson correlation matrices of real scenarios and generated scenarios are very similar, which shows that without any statistical hypotheses of probability distributions being made during the training process, IMLEs can capture the temporal dependency of wind and PV power generation curves accurately.
Previous publications have shown that there are strong spatial characteristics between adjacent PV plants or wind farms [
In order to qualitatively analyze the performance of IMLEs in capturing the spatial correlations of renewable energy sources,

Fig. 7 Real power generation curves of nine adjacent renewable energies. (a) Nine real wind power scenarios. (b) Nine new wind power scenarios generated from IMLEs. (c) Nine real PV power scenarios. (d) Nine new PV power scenarios generated from IMLEs.
It is obvious that the real power generation curves of nine wind farms and nine PV plants have the same trend, which indicates that there are strong spatial correlations among these adjacent wind farms and PV plants. IMLEs are able to account for the spatial correlations among multiple generation units while generating the stochastic scenarios for renewable energy sources, which is in line with the actual situation. Moreover, the nine generated PV power generation curves are lightly closer to each other than the real ones, but this small difference is acceptable, since the spatial correlation among multiple generation units is related to the time-varying environmental factors, e.g., cloud and light intensity.
Furthermore, the mean Pearson correlation coefficients of all samples are employed to quantitatively calculate the spatial correlations among multiple generation units, as shown in

Fig. 8 Pearson correlation matrix among nine renewable energies. (a) Nine real wind power scenarios. (b) Nine new wind power scenarios generated from IMLEs. (c) Nine real PV power scenarios. (d) Nine new PV power scenarios generated from IMLEs.
As can be observed from
In addition to the spatiotemporal correlations of power generation curves, their amplitudes are also important attributes. An explicit density model based method (e.g., Copula method [

Fig. 9 PDFs of PV and wind power generation curves. (a) Wind power generation curves. (b) PV power generation curves.
Besides,
To ensure that IMLEs have outstanding performance for scenario generation of heating, cooling, and power load curves, as shown in

Fig. 10 Structure and parameters of IMLEs for scenario generation of heating, cooling, and power load curves.
Obviously, the structures and parameters of IMLEs for scenario generation of heating, cooling, and power load curves can be obtained by fine-tuning the structures of IMLEs for renewable energy sources in
To check whether real load curves and new load curves generated from IMLEs have similar properties, 2500 Gaussian noises are used as inputs of the data generator to produce 2500 samples. Each generated sample includes a heating load curve, a cooling curve, and a power load curve. Then, a part of real scenarios with complex dynamics characteristics (e.g., large peaks and valleys) from the test set are selected to calculate Euclidean distances with the new load curves. Finally,

Fig. 11 Visualization of real load curves and their closest new load curves generated from IMLEs. (a) Sample 1. (b) Sample 2.
Obviously, the generated heating, cooling, and power load curves closely resemble the real ones from the test set, which are not used in the training process of IMLEs. Next, the second row of
In order to further verify whether the new heating, cooling, and power load curves generated from IMLEs have the similar temporal correlation as the real ones,

Fig. 12 Pearson correlation matrices of different scenarios. (a) Real heating load scenarios. (b) Real cooling load scenarios. (c) Real power load scenarios. (d) New heating load scenarios generated from IMLEs. (e) New cooling load scenarios generated from IMLEs. (f) New power load scenarios generated from IMLEs.
Although the Pearson correlation coefficients between previous and current loads decrease with time, they are always greater than 0.8, indicating that the heating, cooling, and power load curves are strongly time dependent. Similar element values in these Pearson correlation matrices show that the proposed scenario generation approach can capture the temporal correlations of heating, cooling, and power load curves accurately.
Previous publications have shown that there are strong correlations (also called coupling) among heating, cooling, and power loads [
Therefore, the correlations among heating, cooling, and power load curves need to be considered when generating stochastic scenarios for them.
In particular, the two samples in the first row of
Furthermore, the mean Pearson correlation coefficients of all samples are employed to quantitatively evaluate the correlations among heating, cooling, and power loads, as shown in

Fig. 13 Mean Pearson correlation coefficients among heating, cooling, and power loads. (a) Real samples. (b) New samples generated from IMLEs.
Apparently, the mean Pearson coefficient matrix of the new load curves does not differ much from the real ones.
The maximum error of Pearson coefficient matrix between the real samples and new one is 0.02, indicating that IMLEs can capture the correlations among heating, cooling, and power loads well.
In addition to the above properties,

Fig. 14 PDFs of load curves. (a) Heating load. (b) Cooling load. (c) Power load.
The probability distributions of the load curves generated from the different deep generative networks are generally consistent with the real situation, indicating that these models can simulate the distribution characteristics of the real load curves well. In addition, since the PDFs of IMLEs are the closest to those of heating, cooling, and power loads, it shows that IMLE outperforms Copula method, GANs, and VAEs in terms of probability distributions.
This paper proposes a novel data-driven approach to improve the quality of stochastic scenario generation for renewable energy sources and loads in integrated energy systems based on IMLEs. After performing simulation analysis on the real dataset, the following conclusions are obtained.
1) Compared with the dense layer, LSTM layer, and GRU layer, the TransConv layer is more suitable for the data generator of IMLEs. Normally, the number of TransConv layers should not be too small or too large, and three TransConv layers can be considered as a good starting point for different datasets. In addition, the Adam algorithm is more suitable as the optimization algorithm for IMLEs in scenario generation tasks than other algorithms.
2) Unlike previous implicit density model based methods (e.g., GANs) whose loss functions are difficult to converge, IMLEs converge very quickly and are relatively stable throughout the training process. In addition, IMLEs capture the probability distributions of renewable energy sources and loads more accurately than popular generative models, such as the Copula method, VAEs, and GANs.
3) Simulation results show that IMLEs are able to accurately capture the signature properties (e.g., fluctuations, large peaks, and fast ramps), frequency-domain characteristics, and temporal correlations of renewable energy sources and load curves in integrated energy systems. Besides, the energy consumption or energy generation of the generated samples is very similar to that of the real samples.
4) When simultaneously generating new stochastic scenarios for multiple adjacent renewable energy sources, IMLEs can consider the spatial correlations among them. Similarly, IMLEs also take good account of the correlations (also called coupling) among heating, cooling, and power loads.
For future works, the IMLEs may be extended into conditional scenario generations with specified properties (e.g., heavy loads).
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