 Original research
 Open Access
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Planning of distributed renewable energy systems under uncertainty based on statistical machine learning
Protection and Control of Modern Power Systems volume 7, Article number: 41 (2022)
Abstract
The development of distributed renewable energy, such as photovoltaic power and wind power generation, makes the energy system cleaner, and is of great significance in reducing carbon emissions. However, weather can affect distributed renewable energy power generation, and the uncertainty of output brings challenges to uncertainty planning for distributed renewable energy. Energy systems with high penetration of distributed renewable energy involve the highdimensional, nonlinear dynamics of largescale complex systems, and the optimal solution of the uncertainty model is a difficult problem. From the perspective of statistical machine learning, the theory of planning of distributed renewable energy systems under uncertainty is reviewed and some key technologies are put forward for applying advanced artificial intelligence to distributed renewable power uncertainty planning.
Introduction
In the context of the rapid development of renewable energy power generation, photovoltaic (PV) and wind power (WP) outputs fluctuate greatly and have strong randomness. This bring a series of problems in controlling, scheduling and planning. To address the randomness of distributed renewable energy (DRE), artificial intelligence (AI) has moved to the mainstream of renewable energy forecasting and prediction [1], while energy storage can also address the uncertainty of renewable energy systems [2]. Frequency control is one of the keys to WP integration [3], while multiobjective optimization is always a problem in a power grid with WP generation [4].
This paper is concerned with the influence of uncertain DRE output on traditional renewable energy network (REN) planning. Developing planning methods suitable for largescale gridconnected renewable energy has important engineering significance. However, the main theoretical problem of uncertainty planning is how to deal with uncertainty as the uncertainty of the source side leads to complex and changeable operating scenarios of REN. This brings challenges to the planning of DRE. In the literature, there are three major planning methods: (a) deterministic, (b) robust, and (c) probabilistic planning.
In terms of deterministic planning, daily typical load curves in spring, summer, autumn and winter play important roles in power grid planning. The operating scenarios are divided into typicalday scenarios and extreme scenarios [5], and multistage planning is carried out with the scenarios. In [6], multistage planning is divided into different time stages, and each stage includes different scenarios. A multiobjective planning model is presented in [7], which analyzes uncertainty using scenarios, and deterministic power flow (DPF) is carried out for each scenario. In essence, the above studies are scenario planning methods based on knowledgedriven schemes, and the effectiveness depends on whether the selected typical load curves can represent the load profile over the whole planning period. The fewer the number of operating scenarios, the more accurate the deterministic method can be. As new energy sources and demand response loads are widely integrated into REN, the uncertain disturbances on both power supply and load are becoming stronger, making the operation of energy systems increasingly complicated. The traditional typicalday method of spring, summer, autumn and winter is no longer suitable as it cannot cope with the complexity of the operating scenarios in REN. Since the effective refined operation scenarios cannot be passed to the planning decisionmakers, it is difficult to ensure the accuracy and safety of REN uncertainty planning. Reference [8] proposes a distributionfree scenario generation method based on generative adversarial networks (GAN), which can be deliberately modified according to statistical characteristics for power system planning and operation. The conditional variational automatic encoder method is used to simulate the refined random scenarios of renewable energy, while the simulation of complex operation scenarios based on datadriven theory has become the core of energy network uncertainty planning [9].
In terms of robust planning, interval power flow and interval optimization are the keys to its success. The idea of interval flow is to get the solution scheme of the interval range of the state variables based on mathematical planning with uncertain injection power as a feasible region [10]. To avoid nonlinear planning, a linear approximation is a feasible scheme to realize interval optimization [11]. It has benn proved that the interior point method can solve the nonlinear planning models [12]. In essence, the above studies use a planning method based on extreme condition scenarios, and the effectiveness depends on whether the selected extreme condition scenarios can represent the boundaries of the state variables over the whole planning period. Robust planning can be used for dispatching, and uncertainty planning is suitable for power grid planning. In addition, robust planning can ensure operational safety, while uncertainty planning can balance the needs between economy and safety.
In terms of probabilistic planning, it is well known that the classical way of solving the uncertainty problem is to use probability, and probabilistic power flow (PPF) may be one of the key issues. Both the Gaussian process emulator and Bayesian inference are the basic theories of PPF calculation [13]. When planning the best placement of electric vehicle (EV) charging stations and wind turbines, uncertainty planning should be involved in the planning framework [14]. An uncertainty planning model considering different operational constraints in multiperiods is presented in [15], and the planning model solves uncertain dual dynamic integer planning. Few studies exist on how to apply PPF to uncertainty planning, while researchers focus on how to enhance the calculation accuracy and efficiency of PPF. As PPF is largely absent in probabilistic planning, the framework combining PPF and probabilistic planning is a problem worthy of discussion.
In terms of probabilistic planning, it is easy to estimate the probability distribution function (PDF) of renewable energy generation when there is randomness, so it is suitable to use the probability theory random method [16]. However, it is difficult to determine the probability distribution and membership relationship of variables in many cases. Therefore, the interval method is more suitable for dealing with uncertainty. Compared with probability theory, the interval method used in robust programming finds it easier to describe various uncertain factors. Because it requires less data that are the upper and lower boundaries. The disadvantage is that interval expansion will occur [17]. The wider the uncertainty set is, the more conservative the optimal solution is in the economy.
University of California Berkeley considers statistical learning, also known as statistical machine learning (SML), to involve probability theory, statistics, approximation theory, convex analysis, algorithm complexity theory, machine learning and other disciplines [18]. The development of DRE planning theory has been accompanied by the development of SML algorithms in recent decades, showing a strong trend of development [19]. However, there is no systematic method for SML to deal with uncertainty planning.
REN with a high proportion of DRE generation and EV loads is a largescale dynamic system with high dimension, is nonlinear, and has uncertain and complex characteristics. Most traditional scenario analysis methods are based on probabilistic models [9], and the model capacity is small. Digital features can only capture local data features, which cannot fully characterize the complex highdimensional and large data features of renewable energy output. Thus, it is insufficient for solving the new energy uncertainty planning problem. Probability theory is modeloriented and relies on rigorous mathematical derivation, emphasizing model interpretation. However, it has limitations in the face of complex highdimensional data. Machine learning (ML) is algorithmoriented and pays attention to the prediction results, and has good controllability and scalability. Deep learning (DL) is a branch of ML and is based on a neural network, and can effectively solve highdimensional problems with a strong ability for autonomous learning, pattern recognition and optimization. However, ML has poor interpretability. A general knowledge of probabilistic theory combined with ML, SML emphasizes the intelligibility, precision and uncertainty of models, and this is consistent with the engineering requirements of uncertain DRE planning. New energy planning needs to provide decisionmakers with explanations, and SML calculation results can be understood and trusted [20, 21] (achieving ML understandingtohuman understanding).
The contributions of this paper can be summarized as follows.

(1)
Uncertainty analysis of renewable energy is conducted via statistical machine learning.

(2)
Novel techniques in uncertainty modeling are illustrated by principle demonstration.

(3)
Renewable energy allocation algorithms are analyzed in an uncertain environment.

(4)
Future development trend of uncertainty theory is analyzed for renewable energy integration.
Literature review
China actively supports regional integrated energy development at the policy level and has promulgated a series of policies to promote the development of power energy storage, the energy Internet and other technologies to push the development of smart energy management. With the current research background, renewable energy generation, integrated energy and EV have become important factors in regional planning. At present, many actual REN projects have been built in China. For example, the integrated energy system (IES) of Qiantang New District Power Supply Company, shown in Fig. 1, has built distributed clean energy services and EV services. Its power generation mainly covers PV and WP, and the load side mainly includes an intelligent energy storage system. Thus, the described research object directly targets practical systems.
Uncertainty in sourcenetworkload leads to complex and changeable operating scenarios of REN, which makes the planning problem uncertain. In this paper, the probability model is used to describe uncertainty. Based on the SML tool, the objective function and constraint conditions of the probabilistic planning model are established. Section 2.1 introduces and evaluates the common methods of distributed renewable energy and electric vehicle uncertainty modeling, including probabilistic and ML modeling. In Sect. 2.2, the characteristics of uncertainty planning methods of distributed renewable energy in REN and IES are analyzed, including models and solving algorithms. The planning model includes chanceconstrained, twostage, multistage, distributed robust planning, multiobjective and bilevel models. The solving algorithm is composed of the traditional, intelligent and hybrid algorithms. Probability theory and ML theory belong in the SML category.
Uncertainty modeling
Figure 2 depicts a high proportion of intermittent renewable energy, load side EV grid connection and air conditioning regulation in modern REN. On the source side, PV and WP generation are the typical uncertain power sources, which are affected by changing weather conditions. On the load side, the uncertainty of EV charging time is an important source of uncertainty. In addition to renewable energy and EV loads, there are a large number of other uncertainties in modern REN, such as air conditioning loads, and the uncertainties of source and load spread through transmission lines, leading to power flow uncertainty. Therefore, sourcenetworkload uncertainty makes power grid operation conditions complex and changeable, and the power flow presents strong uncertainty, which needs to be described by probability.
Distributed new energy modeling
Figure 3 shows the actual research project named "panoramic viewable", a renewable energy control substation of Qilinshan wind farm in Shangyi, Hebei Province. As seen, Fig. 3a shows the operational status of all PV inverters, and Fig. 3b, c show the voltage distribution map. In Fig. 3c, the red indicates a high voltage area and the blue indicates a low voltage area. Figure 3d and e show the active power distribution, while the threedimensional pipeline is used to represent the active power and grid loss of all feeders. The thicker pipelines with brighter color indicate that the active power flow and active power loss are greater. It can be seen that the system voltage and power flow are greatly affected by DRE in the projects. Therefore, the problem studied in this paper has great practical significance.
With DRE generation applied widely, the output characteristics of renewable energy shown in Fig. 4 fluctuate greatly and have strong randomness, which together lead to the uncertainty and complexity of energy system operation. IES planning characteristics have undergone fundamental changes, such as diversification and differentiation. Therefore, it is necessary to carry out renewable energy modeling accurately.
In scenario generation, random distribution is introduced to simulate the probability distribution of renewable energy. By sampling the continuous probability distribution of random variables, discrete sample sets and probability values are obtained, thereby turning the uncertainty problem into a certain problem. Reference [22] generates a set of dynamicwind speed scenarios that consider wind speed fluctuations to describe the WP uncertainty by using a multivariate normal distribution and inverse transform sampling, while a scenario generation method sampled from nonGaussian and interdependent conditional distributions is presented in [23]. The quantile prediction method of solar and WP generators is applied in [24]. In addition, a large amount of empirical research has shown that the probability distribution of scenario prediction errors doesn't follow any parameter density. The maximum entropy theory is introduced to analyze the uncertainty of WP output and EV loads, and a refined seasonal analysis of the complex operating scenarios of REN is conducted in [25]. In [26], a linear planning optimization model directly generates quantiles with different proportions, effectively generating nonparametric probability forecasts for WP generation, whereas a limited Boltzmann machine is used to capture the time characteristics of wind speed in [27]. This uses divergence and Gibbs sampling to fit the probability distribution of WP scenarios. Monte Carlo sampling (MCS) is combined with quantile estimation technology to estimate the dimensions of the hybrid renewable energy systems [28], while in [29], MCS is presented to represent the uncertainty of PV generation. In many cases, Latin hypercube sampling (LHS) is considered to be an improvement on rough MCS. In [30], LHS is used to improve sampling efficiency and deal with the relevance problem of DRE generation access nodes, while the methods of MCS and LHS are analyzed to generate WP scenarios in [31].
At present, many scholars have verified that the copula function in statistical science is effective in the correlation modeling between random variables. In [32], scenario generation is generated based on the multivariate estimated distribution of WP in multiple regions. Pair copula and vine copula were first introduced to the WP scenario generation to discuss their temporal coupling [33]. The presented model can easily obtain accurate and sufficient WP scenarios with time and space correlation. In [34], a combination of nuclear density estimation and regular vine copula is also used to describe the spatial correlation between WP plants, whereas in [35], Brownian motion is combined with copula to generate correlated dynamic PV scenarios. In [36], a dynamic factor model is presented to generate scenarios similar to the statistical characteristics of the actual scenarios. For some DRE power plants that do not have enough historical data, multiple scenarios and a knowledgebased scenario generation algorithm are presented, which capture the potential distribution of target wind speed with high precision [37].
The time series method is another emerging scenario generation method. The common methods used have been the autoregressive moving average model and Markov random process. In [38], random load and generated renewable energy scenarios are considered on multiple operating time scales. A doublelayer WP time series model is presented that considers daily weather changes and intraday WP fluctuations [39]. The Markov chain and time series are used to simulate the daily typical weather and intraday WP scenarios of each typical weather state, and then are integrated into a complete WP time series. An inhomogeneous Markov chain is also presented to depict the characteristics of diurnal and seasonal changes in wind speed [40]. Both models presented in the above two references have good ergodicity and consistency. In [41], an autoregressive moving average (ARMA) model is presented to grasp renewable energy spatiotemporal features. Based on this, [42] improves ARMA that can grasp the time correlation of WP scenarios. Generalized autoregressive scoring models are presented to generate renewable energy multivariate time series scenarios in [43], while [44] presents state space models to generate renewable energy WP scenarios.
Recently, the concept of machine learning has gradually become the direction pursued by the random scenario generation of the IES. In [45], the radial basis function (RBF) and tabu searchbased metaheuristic algorithm are combined to effectively capture the important features of the WP scenarios, while artificial neural networks are used to generate representative multidimensional DRE scenarios based on historical time series values [46]. Reference [47] adopts stacked autoencoders to improve the precision of WP scenario generation, whereas for reducing the prediction error in scenario generation, the concept of a generative confrontation network is presented in [48]. This was first used to generate continuous sequences in the field of artificial intelligence. In [49], an unsupervised scenario generation method of GANs is presented to capture the dynamic characteristics of DRE generation. However, the method has some problems such as vanishing gradients, difficulty in training, and lack of diversity in generated scenario samples. Therefore, Wasserstein distance is used instead of JS distance on the basis of GANs, and WassersteinGAN is used to train DRE scenarios. This effectively solves the above problems [50]. The scenario generation ability of GANs is further improved by adding Lipschitz constraints on the discriminator network [51], whereas in [52], a Bayesian formula is introduced into a genetic neural network to realize semisupervised learning. Reference [53] applies this method to DRE scenario generation. Even intentionally mixing PV and WP scenarios, this method can identify and generate different scenarios at the same time, so as to better represent the DRE generation process. Sequence generation is used for the network to capture temporal correlations. It adopts a long and shortterm memory structure and uses GAN combined with reinforcement learning [54]. In the work of [55], GAN based on two interconnected deep neural networks is used to generate real WP and PV power distributions with completely diverse behaviors. It also explains how to use labeled data in the training process to generate the required scenarios based on different interest conditions.
Research on accurate renewable energy uncertainty modeling has been carried out to a considerable extent, while scenario analysis methods provide an effective way to deal with DRE uncertainty modeling. Table 1 shows the characteristics of scenario analysis methods based on probability models, time series methods and AI algorithms.
Electric vehicle load modeling
More active devices and adjustable loads will be introduced into future intelligent energy systems, and the EV charging loads are the most typical uncertain loads. The loadside uncertainty caused by the characteristics of EV and the grouping characteristics of drivers' social networks, increases the operating pressure on the power grid. As shown in Fig. 5, grid connection of EV brings uncertainty to REN, with not only the uncertain grid connection time but also the random grid node. Among them, electric taxis are more uncertain and more difficult to predict than private cars and buses. Moreover, during the peak charging period of EV, the charging loads increase sharply, and thus the local peak loads increase significantly. During the lowpeak charging period, many EVs discharge power in reverse to the grid, which is filled with excess power load. The power in both periods may be higher than the equipment capacity of REN under traditional planning, resulting in local blocking and paralysis of the power grid in severe cases.
The EV charging load uncertainty models are based on the premise of realizing the charging stations' scientific planning in REN, and probability statistics are one of the key methods. Probability is the mathematics of uncertainty. It uses the actual historical data to simulate the PDF of the initial EV charging time and state of charge (SOC), and then models the charging loads by random sampling methods. In [64], MCS is used to fit the charging load distribution of each EV. The model assumes that the EV relevant parameters obey the above distribution, e.g., leaving home in the morning to stop charging and returning home at night to start charging. Reference [65] takes load samples by MCS and estimates the total average weight by a discrete probability formula, while [66] first selects whether the EV is V2G through roulette, then generates random numbers, and finally uses MCS to produce massive random scenarios. Reference [67] discretizes EV charging duration and chargingstart time, and obtains joint uncertainty by Cartesian product. In addition to the commonly used probability distribution function, the spherical simplex unscented transformation (UT) is used to approximate the probability distribution in [68]. The point set \(\sigma\) can accurately represent the statistical information required by random variables. In [69], the charging pullout time of the electric taxi is affected by the initial charging time and duration. Considering the disadvantages of MCS with more samples and long calculation time, LHS is presented. LHS is used to simulate the EV charging loads, and the EV relevant parameters obey the general probability distribution [70]. In [71], a combination of LHS and Kmeans is used to model the uncertain scenarios of EV charging loads, and the EV relevant parameters obey a truncated Gaussian distribution.
The above references, except [67], assume that the EV parameters are independent of each other, and the relationships between parameters are not considered. The copula is a function to describe the dependency between variables, and can create a multivariate distribution of data. Multivariate tcopula [72,73,74], Gaussian copula [75] and Archimedes copula [76] are used to describe the dependence between the EV starting time, arrival time and vehicle distance, to model the driving mode more accurately. In addition, a 3dim kernel density estimation (KDE) is considered to model the relationship between different variables [74]. In [77], the 3dim KDE is also used to describe the uncertainty of the EV charging mode. Traveling is a kind of activity arrangement in time and space, which connects two or more places, for home or nonhome travel purposes. First, the time and space probabilities are obtained respectively, and then the total probability is calculated to obtain the travel probability model of the EV. In [78,79,80,81], some prediction models of EV charging behavior are presented based on the travel chain. These methods can simulate the travel behavior of users after constructing the travel chain according to the different purposes of driving, so as to analyze the driving law of users more accurately. In the above probability models, the EV relevant parameters are fitted to synthesize the relevant probability distribution from the actual data. However, the accuracy of the fitting results remains to be discussed.
Considering the temporal and spatial uncertainty of EV charging loads, advanced machine learning has been used for their uncertainty modeling. Reference [82] uses a nonhomogeneous Markov chain to simulate and generate the EV usage pattern, and a fastforward selection method to reduce scenarios. The EV state is divided into normal charging, fast charging, driving and parking in [83], while in public charging stations, the continuoustime Markov chain method is used to depict the uncertainty of EV charging behaviors in [84]. The uncertainty of passengers, charging stations, and public transport usually leads to the uncertainty of electric taxi loads. Therefore, the operational behavior model of electric plugin taxis (PET) is presented based on environmental uncertainty [85]. In [86], regarding the movements of electric taxis as random walks, the Markov process is used to simulate the distribution of charging demand in static space, while in [87], the random forest (RF) method based on a regression tree is used to predict the driving characteristics of each EV, so as to obtain the travel mode data set of the EV. From the measured charging information and big data mining technology, the EV charging behavior model is presented based on RF and principal component analysis (PCA), which captures the EV with different charging characteristics based on a datadriven model [88]. The gradient boosting model (GBM) is used to simulate the SOC state at the end of a plugin EV (PEV) daily schedule [89], and in [90], a generalized regression neural network (GRNN) is used to realize power prediction. This can effectively describe the uncertainty on EV charging loads. DL has also received extensive attention in solving uncertainty problems. A back propagation (BP) neural network is presented to forecast EV charging loads in [91], while a convolutional neural network (CNN) is used to reduce the impact of uncertainty on EV charging demand accuracy in [92]. In [93], a method combining a CNN and a deep belief network (DBN) is presented to describe the uncertainty of EV charging demand. With the wide application of computer technology and social networks, multiagent technology has been developed. Based on multiagent technology, a spatiotemporal EV charging demand simulation model considering passenger travel demand is constructed [94], where the travel demand of passengers, taxi decisionmaking process and taxi queuing process are managed by a multiagent. Reference [95] uses multiagent and multistep Qlearning methods to describe the uncertainty of electric taxi charging loads on time and space scales. As the above datadriven methods only need to collect real data without the need to obey various distributions, the correlation between the starting time, arrival time, vehicle distance and other parameters of EV can be retained to ensure accuracy.
As shown in Table 2, uncertainty modeling methods for EV charging loads can be divided into two types: probability statistics theory and machine learning. There are six methods in probability statistics: UT, MCS, LHS, copula, 3dim KDE and the travel chain model. These methods approximate PDF through the actual historical data of EV parameters, and thus the planning results are greatly influenced by the accuracy of the PDF. Moreover, the relationship between the EV parameters would be ignored if only MCS or LHS were used. Therefore, MCS or LHS needs to be used together with methods that describe correlation, such as copula and 3dim KDE. Machine learning methods mainly include the Markov chain model, RBF, GBM, RF, DL and multiagent technology. These methods consider the temporal and EV spatial characteristics, and do not require the parameters of EVs to obey various distributions. However, the accuracy of the results depends on the adjustment of parameters and the extraction of features. In addition, from the perspective of the types of EVs, electric taxis have characteristics such as weak regularity of driving routes and diverse selection of charging stations, which are more uncertain than traditional private electric cars. Thus, the uncertainty modeling of electric taxis is more difficult.
Summary
The uncertainty of renewable energy power generation is very significant, and is the main source of uncertainty in REN planning. EV policies are being promoted all over the world, and it has become a trend for EVs to replace fuel vehicles. Consequently, EV loads have become the leading factor of load uncertainty. With the background of lowcarbon goals, renewable energy and EV loads have the most prominent impact on power systems. Uncertainty modeling theory has been analyzed in detail for renewable and EV loads in the previous sections. On the load side, air conditioning load is another important uncertainty factor. Because of the variations in weather and electricity price, the air conditioning load is always in a disordered operation state. The air conditioning load is sensitive to outdoor temperature and is related to residents' needs and habits. When the outdoor temperature changes, the air conditioning load will also change, and thus, the variation in outdoor temperature leads to an uncertainty of air conditioning load. Moreover, the behavior of residents is random and disordered, as the indoor temperature set point can randomly vary from 16 to 30 °C, so the air conditioning load is uncertain even in the same weather conditions [96]. The demand side response under different electricity price policies will increase the difficulty of analyzing the randomness of residents' behavior. In addition, network parameter uncertainty is also a source of uncertainty in REN, such as the parameter uncertainty of resistance, reactance or capacitance [97].
Uncertainty planning modeling
As shown in Fig. 6, the uncertainty of DRE makes IES operating scenarios complex and changeable, and the calculation burden of planning solutions caused by uncertainty increases greatly. The number of power flow calculations is the number of solutions traversed in the solution process and the number of operating scenarios. This multiplication makes the number of calculations increase explosively, while more decision variables and stronger uncertainty make the complexity of the solution process greater, in company with the increased grid connection capacity of renewable energy.
Distributed renewable energy planning in REN
Compared with traditional power supply, DRE power supplies have lower pollution emission, higher power supply efficiency, and are more flexible and convenient. On the basis of the known load prediction results and REN operating conditions, determining the installation location and configuration capacity of DRE supply can optimize the economy and reliability of REN in the whole planning period, thus improving power quality, economy and reliability.
To get the optimal configuration scheme of DRE, plant propagation and water cycle algorithms are presented in [98], while the power factors of DRE generators are also optimized to reduce power losses. A methodology is presented to find the optimal size, location and technology of distributed new energy generator units considering economic, technical, and environmental issues simultaneously, using multiobjective particle swarm optimization and fuzzy decisionmaking techniques to find the optimal solution [99]. In [100], a fuzzy multi criteria decision making approach is used to sort the load points and locations/feeder sections, and particle swarm optimization (PSO) is used to evaluate the optimal size and position of distributed new energy generators. To get the best configuration of DRE, reference [101] overcomes the shortcomings of the previous algorithms and presents the multileader PSO solution algorithm, while a distributional robust optimization model is presented, also known as a datadriven model in [102]. In addition, whale optimization algorithms are used to calculate the optimal layout and size of a multiobjective DRE generator set, which improves voltage distribution and minimizes power losses and operating costs [103].
To solve the largescale and complex DRE supply optimization configuration problem with multiple constraints, many optimization models have been presented from different perspectives. In [104, 105], bilayer planning models are presented to allocate more DRE, which generate power system operating scenarios in different periods by using the multistate models of renewable energy and loads. In [106], a bilevel planning method is also presented to achieve optimal distribution of hybrid distribution transformers. This integrates distribution and control issues. The method achieves an optimal distribution in a dynamic environment, reducing investment costs and improving voltage curves. Given the DRE power output intermittence and the load uncertainty, a bilevel planning approach is presented to achieve an integrated optimal decision at the upper level and comprehensively considers the benefits of DRE [107]. A multiobjective distributed generation planning model is established with the objective functions of minimizing both the annual total cost and the risk [108], while a riskbased multiobjective optimal allocation model is presented to optimize the placement and configuration of distributed new energy generators, to provide a reliable and costeffective system [109]. The collaborative planning of PV and energy storage is realized in [110].
For the problem of optimal longterm distribution of DRE, uncertainty twostage planning models are established [111, 112]. The formerly determined investment variables of DRE, before the scenarios, are known in the first stage, while the maintenance problem is solved, one that depended on scenario variables after the scenarios become known in the second stage. The latter also puts forward the solution to the problem of the scenario generation process, develops and launchs a general framework, as opensource DRE optimization of distribution. A twostage stochastic programming model is presented that considers the uncertainty of renewable distributed generator sets, demand and energy prices in [113], while in [114], a twostage stochastic programming model is developed to optimally determine the layout and size of PV modules in a distribution system, where the uncertainties of solar irradiance and load are modeled in finite scenarios to minimize installation and network operating costs while meeting the necessary operating constraints. This model can effectively reduce the cost of system operation. A novel multistage planning method is presented in [115], one which takes into account operational changeability and randomness introduced by emission price and demand growth of intermittent power generation sources through probabilistic and uncertainty methods.
For these models, various optimization algorithms are proposed. A comprehensive solution algorithm for optimal position setting and output power of DRE supply is presented, one which is suitable for all possible load changes of distributed generator sets, and uses a lightning search algorithm to find the optimal allocation of DRE generator units [116]. In [117], a multiobjective symbiont search method is presented to get the best placement of DRE, whereas a linearized load flow method is used to find the optimal allocation of DRE generator units in [118]. Reference [119] combines PSO with voltage and reactive power control of the DRE supply, and adjusts the voltage amplitude within the allowable range to get the optimal capacity and location of DRE. In [120], a chanceconstrained uncertainty planning model is introduced in the optimal placement of PV, and uses the combination of PSO and SVM algorithms to quickly obtain the optimal scheme. A sustainable unit planning method is presented for a distribution system, which tries to optimize a multiobjective index and uses Tabu Search [121]. An improved AI optimization algorithmbased ecosystem and a hybrid grey wolf metaheuristic algorithm are presented in [122] to get the best configuration of DRE to minimize the power loss of REN, while an evolutionary algorithmbased approach is used to solve optimization problems that can reduce the impact of the environmental system and the total cost [123].
Planning of DRE in the IES
With the correlation and coupling of ever increasing renewable energy and different types of loads, the uncertainty of energy and loads in the IES becomes increasingly prominent. This poses a great challenge to the planning and regulation of the IES. An adequately coordinated operation method of IES can ensure the supply and demand balance, and further promote the safe, economic, flexible and efficient operation of the energy Internet. However, it will affect the operation of the IES.
In practical application, uncertainty planning methods of DRE and loads have been applied. A column generation and sharing algorithm is presented to solve the computational burden of large capacity and multistage uncertainty mixedinteger planning model in [124], while a new congestion control method is developed to enable flexible DRE to participate in solving transmission system operator congestion problems, and to use the sequential least squares method for solving the optimization problem [125]. In [126], kmeans clustering is used to extract typical scenarios, and considers changes in sunshine, wind speed, and power demand to adjust the scale of the DRE. An iterative bilayer planning model is presented to deal with the nonlinear and nonconvex state space of the systems [127], while a combination of the hybrid big bangbig crunch solution algorithm and PSO is presented to obtain higher precision in the optimization performance of a highdimensional function [128]. In [129], distributed algorithms for multiple/single storages are used to realize the coordinated allocation of DRE without the need of a central coordinator. The modified Quasioppositional chaotic Selfishherd optimization algorithm in [130], with the advantages of both chaotic linear search and quasiopposition based learning supported, is a faster search process than normal uncertain search. In [131], PSO is improved based on the mapreduce, which reduces the particle search range of a single evolutionary algorithm with a faster calculation speed. Multi AgentHierarchical Task Network (HTN) adopts a heuristic search strategy based on depthfirst in [132], and compared with the classic HTN algorithm, it has higher quality and efficiency, better convergence and adaptability in solving complex problems. Based on the PSO algorithm, an iterative threestage planning model is used to get the optimal price of nonutility DRE contributions in different operation conditions [133]. To optimize the solution, the multiobjective PSO algorithm is used in [134], and proves that the comprehensive benefit of multiobjective optimization is better than that of single economic optimization system. In [135], the augmented epsilon constraint method is used to find the Pareto optimality region to analyse a complex mixed integer linear programming model. The developed multiobjective optimization method in [136] adopts the strength Pareto evolutionary algorithm 2 technique to promote and distribute the benefits of distributed new energy, and is expected to promote the update of current power industry regulatory proposals. An iterative constraintbased search method is presented to optimize the microgrid DRE configuration [137], while an improved teaching–learning optimization algorithm is used to enhance the performance of the algorithm in global search [138]. Limited by the nonanalytical mapping between uncertainty planning strategies, an iterative bilayer optimization algorithm is presented to deal with the problem in [139], while [140] uses a genetic algorithm on a mixedinteger linear planning model and gives the approximate optimal solution with accurate power flow. The harmony search algorithm and the firefly algorithm are combined in [141], and the solution results are of higher quality, have better convergence characteristics, and higher computational efficiency. An improved PSO is presented to overcome an inherent tendency [142].
Reference [143] presents a multitemporal scale operation planning model covering the upper, middle and lower levels, where the upper model optimizes the whole distributed IES of the previous day, the middle model optimizes each community in the daytime, and the lower model sets the power consumption of every community realtime adjustment. A multienergy microgrid optimal design mixedinteger linear planning model is established to achieve optimal scheduling in [144], while an uncertain twostage planning model is proposed in [145], where the first stage gets the virtual power plant configuration decision and the second stage deals with renewable energy power generator and storage. The integrated system tool is presented in [146], where a customer adoption model (CAM) optimizes dynamic modelintheloop, and realizes distributed energy scheduling optimization in power grid transactions. Based on the mixedinteger linear planning model, the role of DRE in the dynamic microgrid is used to improve the recovery ability of REN under extreme climate events in [147], while a mixedinteger nonlinear model of an island microgrid is established, which uses PDF to describe the uncertainty of DRE [148].
Summary
From different perspectives, many have used a variety of algorithms to model and solve the DRE planning problem in IES. Various classical mathematical, machine learning and intelligent algorithms have been adopted or improved to simulate and constrain a variety of uncertain factors of distributed resources, so as to solve the modeling, planning and configuration of distributed resources. Table 3 summarizes the common uncertainty planning models of renewable energy, while Table 4 compares the characteristics of the specific solution algorithms from the literature. The advantages and disadvantages of the traditional algorithms and intelligent algorithms are analyzed as follows:

(1)
Common traditional solution algorithms include the interior point method [149], and algorithms such as the sequential least squares [125], iterative bilayer optimization [127, 139], augmented epsilon constraint [135] and constraintbased iterative search algorithms [137]. Such algorithms generally require the first or second order gradient of the objective function to the control variable. The structure and parameters are fixed in the process of the solution, and the computational complexity and convergence can be proved by means of the analysis. In theory, a global optimal solution can be obtained quickly. However, these algorithms require the objective function to be derivable and continuous. This limits their application in largescale nonlinear and uncertain planning problems.

(2)
The intelligent solution algorithms are generally heuristic. The algorithms are independent of derivatives, without assumptions and approximations. They have a certain randomness and can jump out of the local optimum. In addition, their inherent parallelism helps to deal with complex planning problems in a discontinuous, nonsmooth and highly nonlinear solution space, showing good robustness, convergence characteristics and optimization ability. The main disadvantage is that the optimal solution relies on continuous parameter adjustment experiments and experience, resulting in high computational complexity and slow convergence rate. In addition, finding the global optimal solution cannot be guaranteed.
Advanced technology
The authors have conducted a series of studies in the uncertainty planning and inquiry field. In 2015, the support vector machine (SVM), a popular machine learning theory, was used to calculate power flow approximately and quickly [120], while in 2016, an analytic statistics method was presented in [151] to estimate and reduce the power loss in REN uncertainty planning. In 2017, the combination of information theory and probability theory was studied to analyze and solve the uncertainty problem of a power grid [152], whereas in 2018, a new datadriven approach was used to analyze how uncertain power flows affect power grid reliability [153]. In the 2020s, the theories of unsupervised dimensionality reduction, probability inequality and statistical estimation have been combined to deal with the uncertainty planning problem in distribution systems [96], and it was found that the above theories belong to the theoretical system of SML.
This section presents the primary advanced SML technologies for uncertainty planning. In Sect. 3.1, a Bayesian generative adversarial network (Bayesian GAN) and semisupervised algorithm are presented for renewable energy output scenario generation. This is the core of REN refined planning decision calculation. Section 3.2 presents information entropy theory for the uncertainty measurement of REN probabilistic power flow calculation results. This is an indispensable part of the constraint condition set in the uncertainty planning model. In Sect. 3.3, the economy of typical scenarios and the small probability of extreme scenarios for renewable energy uncertainty planning modeling are presented. The Bayesian GAN and semisupervised algorithm in Sect. 3.1, information entropy in Sect. 3.2 and small probability calculation in Sect. 3.3 all belong to the SML technologies.
Uncertainty modeling of new energy output
Scenario generation
The key to random scenario simulation is to learn the probability distribution. To analyze the characteristics of various DRE sources, SML is applied to study the uncertainty research of DRE. GAN combined with the Bayesian formula can use historical data to train and generate renewable energy operating scenarios in unsupervised and semisupervised learning methods.
The structure of the Bayesian GAN Is shown in Fig. 7. Bayesian formula probabilistic reasoning is introduced to set the weight parameters of the discriminator network (DN) and generator network (GN) iteratively sampled from the conditional posterior distribution. The update is as follows:
where \(x_{i}\) is the training sample, \(X\) is the new test sample, \(\theta_{d}\) and \(\theta_{g}\) are the weight parameters of DN and GN, respectively. \(\sigma_{d}\) and \(\sigma_{g}\) are the respective hyper parameters of the weight parameters of DN and GN, \(p\left\{ {\theta_{d} \left {\sigma_{d} } \right.} \right\}\) and \(p\left\{ {\theta_{g} \left {\sigma_{g} } \right.} \right\}\) are the respective prior distributions of the weight parameters of DN and GN, \(N_{d}\) and \(N_{g}\) are the numbers of input samples of DN and GN, respectively.
Combining the Bayesian formula for the edge processing of noise \(z_{j}^{noise}\), a simple Monte Carlo method can be used to marginalize:
The same can be obtained:
It is worth noting that when (3) and (4) are regarded as functions of noise \(z_{j}^{noise}\), the distribution of \(p\left\{ {\theta_{g} \left {z_{j}^{noise} ,\theta_{d} } \right.} \right\}\) and \(p\left\{ {\theta_{d} \left {z_{j}^{noise} ,X,\theta_{g} } \right.} \right\}\) should be broad, because \(z_{j}^{noise}\) is used to generate the candidate sample data. Therefore, each of the simple Monte Carlo sums usually makes a reasonable contribution to the total marginal posterior estimate. Through iterative sampling, the candidate sample data can be obtained from the approximate posterior of \(\theta_{g}\) and \(\theta_{d}\) in the extreme case.
Figure 8 depicts the comparison between the data generated by Bayesian GAN and GAN, and the original data from 3 \(\times\) 3 identical WP stations, where the abscissa represents the time sampling point, and the ordinate represents the power generation of each station. The difference between the real scenarios and the generated scenarios of multiple sites can be found in the figure. In general, the quality accuracy of renewable energy output scenarios generated by Bayesian GAN is higher than that of GAN. At the same time, this technology improves traditional GAN with the expressiveness of Bayesian formula probabilistic reasoning. The mathematical proof is clear and the interpretability is good, which prove the effectiveness of SML in realizing the uncertainty modeling of renewable energy output.
Typical scenario generation
For the IES uncertainty planning problem, PPF is calculated by using all the data of the large complex highdimensional scenarios. The more decision variables and calculations there are, the more complex renewable energy planning is, and the more difficult to solve. In addition, the amount of information passed to the IES planning in most scenarios is not completely used by decisionmakers, and the value density is low. Therefore, it is meaningful to effectively extract and use the key information of the large complex highdimensional scenarios.
Typical scenario generation is to generate scenarios with similar statistical, correlation and shape characteristics to the original scenarios, and the Wasserstein distance metric is adopted here. Suppose \(p_{a} \left( x \right)\) is a continuous PDF of onedimensional random variable x, then \(Z_{t}\) is used to calculate the corresponding probability \(p_{d}^{t}\) as:
In the optimal quantile theory presented in [154], the total of \(p_{d}^{t}\) calculated is not equal to 1. For IES, the energy variable always changes in an interval with upper and lower limits, so the following two different variable boundary approximation formulas are more consistent with the discretization of IES variables [155]:
As shown in Figs. 9 and 10, discretization of continuous variables is achieved through the Wasserstein distance. The PDF distributions represent the actual scenarios of CHP and GAS. Comparing PDF distribution shapes and discrete scenarios, one can see that the Wasserstein distance metric method is effective and accurate. Figure 11 achieves clustering by unsupervised learning kmeans (Euclidean distance). They all refine complex operating scenarios into typical scenarios in engineering. The number of typical scenarios is small, and are used for planning modeling. The calculation efficiency is high, while the accuracy of the planning results is close to the planning accuracy of complex operating scenarios, which meets the engineering needs.
Semisupervised learning algorithm
As shown in Fig. 12 and listed in Table 5, the conclusion is that semisupervised learning is more suitable than unsupervised learning for the typical operating scenario for generation of REN with renewable energy. Because power grid regulatory personnel require knowledge of the power flow of the grid, it needs to calculate the network power flow concerned with renewable energy power generation. The typical scenario extraction of renewable energy power generation based on unsupervised learning is not as good as the semisupervised learning method considering renewable energy output x and grid state variable y.
Uncertainty modeling of power flow
As shown in Fig. 13, PPF can reflect the influence of system operation caused by random changes in various factors in the power system. It can comprehensively consider the uncertain situations of variable changes such as power system network topology, component parameters, node load values, generator output, etc. At the same time, it can also analyze the randomness of PV, WP, EV charging loads and thermal loads caused by temperature change, wind speed fluctuation, and solar radiation change and travel behavior. The above can contribute to discovering the frail segments in REN and provide valuable information to the planning and dispatching departments to help them make decisions.
The uncertainty of power load and thermal load in networks will affect each other's energy networks through thermoelectric coupling. Figure 14 shows the bivariate copula function distribution of electrical load and thermal load. The larger the PDF value, the stronger the correlation.
Information entropy is perceived as the uncertainty of a random event or a measure of the amount of information. Greater information entropy indicates greater uncertainty, and greater uncertainty indicates greater complexity. The entropy \(H\left( x \right)\) can be expressed as:
where \(x{}_{k}\) is the \(k_{th}\) random variable.
As shown in Figs. 15, 16 and 17, information entropy is used to depict the strength of uncertainty. This paper sets three different scenarios with information entropy of [3.676, 7.015, 6.31]. In Fig. 15, CHP output characteristics are affected by uncertain variation of thermal load, and the entropies are in the area of 3.6. In Figs. 16 and 17, the uncertainties of the IES heating network and electricity network are jointly affected by PV, WP, EV charging loads and thermal loads. However, it can be seen that when the information entropy increases, that is, when the influencing factors increase, the entropies of scenarios II and III do not become larger. Therefore, it can be concluded that when more types and higher proportions of uncertain energies access the grid, the uncertainty of the IES overall operation will not increase.
Uncertainty modeling of renewable energy planning
A large amount of DRE integration will bring highdimensional characteristics and uncertainty to REN. As shown in Table 6, the following conclusions are verified by practical engineering experience: (a) the calculation efficacy of probability theory is good when there are few random variables; (b) in the highdimensional random variable modeling, the calculation efficacy is not good; and (c) with respect to the modeling of correlation, the method of the correlation coefficient is invalid. It can be concluded that dimensionality disaster is the difficulty of scientific data modeling. Modeling highdimensional correlation variables is the key to break through the accuracy issue of distributed power cluster uncertainty planning modeling, which needs to be modeled by SML technologies. The key SML technologies considered in this paper are shown in Fig. 18 and listed in Table 7.
The decision variables of uncertainty planning are the location and capacity of DRE. High dimensional correlation modeling can connect deterministic decisions with uncertain objectives and constraints, and then the uncertain planning model can be established. Distribution network operating scenarios can be divided into typical scenarios and extreme scenarios. The planning results need to make typical scenarios economical and maintain high reliability in extreme scenarios. The key scientific problem of typical scenario extraction of an REN is to realize the classification of operation data. Similar scenarios are treated as one category, with an obvious effect in improving the efficiency of planning and arriving at solutions. Small probability events in extreme scenarios are the key factors affecting the reliability of REN planning. Therefore, how to use SML to estimate small probability events and realize classification is key and constitutes the main research area of DRE uncertainty planning.
Discussion
The topic of this paper is the optimal placement method of DRE. A deterministic decisionmaking scheme should be given considering the uncertainty of DRE. The difficulty in this field is how to model the uncertainty so that the deterministic planning results can meet the safety and economic requirements in uncertain environments.
For uncertainty modeling, Monte Carlo and machine learning are the two most popular and effective methods. When the search string ''Monte Carlo uncertainty renewable energy'' was entered on December 9, 2021, there were 856 publications listed by Web of Science, while there were 302 publications corresponding to the search string ''machine learning uncertainty renewable energy''. It can be concluded that both Monte Carlo and machine learning have received significant attention in the area of renewable energy.
However, there are areas that need to be further improved, as follows:
(a) The Monte Carlo method is based on the probability distribution function in DRE uncertainty simulation. The capacity of the probability model is small, and all information depends on several digital characteristics, such as expectation, variance and correlation coefficient. The complex relationship of the time–space correlation of multiple renewable energies cannot be fully described via some digital characteristics. Engineering practice shows that the probability modeling of a single DRE is accurate, while multiple DRE models based on probability correlation are not. Thus, probabilistic models of uncertainty need to be further improved to ensure the simulation performance of multiple DREs.
(b) There are two general uses for machine learning in the uncertainty modeling of DREs. The first is DRE generation prediction, and the second is stochastic scenario simulation of DREs. However, it is documented that a prediction method based on machine learning performs poorly when renewable energy generation fluctuates violently. Concerning stochastic scenario simulation of DREs, machine learning has weak explanatory power for the results. Therefore, machine learning models need to be further improved to ensure DRE model reliability and interpretability.
Below is a discussion on the development prospects of SML in future planning technology. It can solve largescale planning problems subject to high renewable energy penetration. High renewable energy penetration is one of the most prominent features of the new generation REN, in which the number of operation scenarios is very large, and extreme scenarios are difficult to accurately predict. It is thus difficult to find a deterministic solution, one which can enhance the overall optimization under an ocean of operational scenarios and meet with a high probability strict constraint in each scenario.
Reference [164] considers that the probabilistic framework of machine intelligence has an attractive advantage in dealing with uncertainty modeling. It can be concluded that SML is an effective tool to deal with uncertainty. SML combines the advantages of machine learning and probability theory and is one of the effective tools to deal with uncertainty planning. On the one hand, the powerful data learning ability of machine learning can be used to deal with highdimensional, complex, and uncertainty modeling, which greatly improves calculation efficiency and maximizes the use of existing electricity datasets. On the other hand, rigorous probability theory is used to ensure the interpretability and precision of the solution results of the REN planning model. The planning results can be quantified as the probability level so that planners can make decisions to meet the engineering requirements. Based on the above analysis, the future development trend of DRE and grid connection technology can be explored. This is an information solution. SML and other artificial intelligence technologies need to be further developed to reduce the impact of DRE generation uncertainty.
Conclusions
Given the uncertainty planning problem of renewable energy, a robust planning method has been proposed in Chinese academia. In theory, robust planning is based on interval mathematics, and the idea of interval optimization is to use closed sets to express uncertainty. Solving the maximum and minimum state variables based on mathematical programming is essentially a planning method based on extreme condition scenarios, and its effectiveness depends on whether the selected extreme condition scenarios can represent the boundary of state variables throughout the planning period. Although the planning results are economically conservative, they can ensure the operational safety of distribution networks. The uncertainty optimization method, which is based on probability and statistics in theory, has been proposed by international academia. Probabilistic planning represents operational and security constraints at a specific expected confidence level so that decisionmakers can see the relationship between risk and possible planning results. However, probability theory cannot fully describe the complex relationship of renewable energy spatial–temporal correlations, so it may lead to potential safety hazards hidden in the planning results. Because of the high proportion of renewable energy, it is increasingly difficult to grasp the features of power grid operation, and economy and security cannot be guaranteed by the traditional modeldriven methods. Based on robust programming and probabilistic programming, distributional robust optimization has gradually attracted extensive attention by finding the decision results under the worst probability distribution of new energy uncertain parameter. Combining the advantages of both methods, the planning results show good performance in terms of economy and robustness. However, most of the distributional robust optimization processes need approximate transformation, and this reduces the accuracy of the results to some degree. SML is an effective way to solve the problem of uncertainty planning for DRE. In essence, it is a way to reduce the scope of DRE uncertainty, so that uncertainty planning can become closer to deterministic planning. This paper puts forward complex operating scenarios based on adversarial networks, extreme scenarios based on deep learning and representative scenarios based on semisupervised learning. These can guarantee the efficiency and accuracy of uncertainty planning of distributed generation. Extreme scenarios guarantee the security of planning results, and representative scenarios guarantee the economy of planning results.
Availability of data and materials
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
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Acknowledgements
The author would like to thank the referees and the editor of this journal for valuable comments.
Authors' information
X.Q. Fu (1985) received his B.S. and M.S. degrees from North China Electric Power University in 2008 and 2011, respectively. He received his Ph.D. degree from South China University of Technology in 2015. From 2011 to 2015, he was an electrical engineer with Guangzhou Power Supply Co. Ltd. From 2015 to 2017, he was a PostDoctoral Researcher with Tsinghua University. He is currently an Associate Professor at China Agricultural University. His current research interests include statistical machine learning, Agricultural Energy Internet, and PV system integration. He is an associate EditorinChief of “Information Processing in Agriculture", an editorial member of “Journal of Solar Energy Research Updates", and an associate editor of “Protection and Control of Modern Power Systems”.
X.P. Wu (1997), Major in the planning of integrated energy system based on the statistical machine learning.
C.Y. Zhang (2000), Major in the statistical machine learning.
S.Q. Fan (1997), Major in the planning of distribution network based on the statistical machine learning.
N. Liu (1981), PHD and Professor, Major in the Smart Power Distribution and Consumption and MultiEnergy System Integration.
Funding
This study is supported by the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources under Grant No.LAPS21016, the National Natural Science Foundation of China under Grant 52007193, and the 2115 Talent Development Program of China Agricultural University.
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XF: conceptualization, methodology, writing—review & editing, supervision. XW, CZ: methodology, formal analysis, investigation, writing—original draft. SF, NL: investigation, writing—original draft. All authors read and approved the final manuscript.
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Fu, X., Wu, X., Zhang, C. et al. Planning of distributed renewable energy systems under uncertainty based on statistical machine learning. Prot Control Mod Power Syst 7, 41 (2022). https://doi.org/10.1186/s4160102200262x
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DOI: https://doi.org/10.1186/s4160102200262x
Keywords
 Distributed renewable energy systems
 Statistical machine learning
 Uncertainty planning
 Renewable energy network