Survey on Modeling of Temporally and Spatially Interdependent Uncertainties in Renewable Power Systems

Abstract

Constructing a renewable energy-based power system has become an important development path for the power industry’s low-carbon transformation. However, as the proportion of renewable energy generation (REG) increases, the power grid gradually changes to uncertainty. Technologies to address this issue have been introduced. However, the majority of existing reviews focus on specific uncertainty modeling approaches and applications, lacking the consideration of temporal and spatial interdependence. Therefore, this paper provides a comprehensive review of the uncertainty modeling of temporal and spatial interdependence. It includes the discrete and continuous stochastic process-based methods to address temporal interdependence, the correlation coefficient and copula functions in modeling spatial interdependence, and the Itô process and random fields theory to describe temporal and spatial interdependence. Finally, their applications in power system stability, control, and economic scheduling are summarized.

1. Introduction

Under the implementation of the Paris Climate Agreement, countries are moving toward a sustainable and low-carbon development. The construction of a renewable energy-based power system has become an important step toward that goal [1].

However, when the proportion of renewable energy generation (REG) increases, the power system is gradually changing to uncertainty [2]. On the power supply side, different from conventional hydro and thermal power, REG is influenced by the climate and environment, which shows strong intermittent and fluctuating features. On the load side, with the rapid development of distributed generations (DG) and energy storage, the load has integrated power supply and demand, making the load dynamic more complex.

The uncertainties brought by both the source and load sides force the operators to make decisions under uncertain conditions. Hakami et al. (2022) [3] and Alsharif et al. (2022) [4] point out that the uncertainties may lead to misjudging the voltage and frequency stability margins. The bus voltage may exceed limit [5], and the nadir of frequency may exceed the safety threshold after a small disturbance [6]. Tan et al. (2023) [7] and Su et al. (2023) [8] point out that a stable power angle in deterministic stability analysis may turn unstable under uncertainties. In economic scheduling and planning, the power imbalance caused by REG and load uncertainties are difficult to address by conventional dispatching methods [9,10]. The surplus and shortage uncertainties of REG affect the generation planning and further reduce the profit [11].

Technologies to address uncertainties have been introduced and their applications in a specific research field or object have been reviewed. Deng et al. (2020) [12] and Ehsan et al. (2019) [13] summarize the state-of-the-art techniques for modeling uncertainties in the planning of large power grids and active distribution networks, respectively. Liu et al. (2019) [14] and Huang et al. (2021) [15] review the forecasting approaches of wind power and photovoltaic outputs. Qiu et al. (2022) [16] summarizes the robust optimization (RO) to address uncertainties in the optimal dispatch. Yu et al. (2019) [17] summarizes the impact of electricity price and load demand uncertainties on the operation and control of virtual power plants. Wu et al. (2017) [18] proposes an analytical method based on generalized polynomial chaos (gPC) for probabilistic load flow. To analyze the influence of parametric uncertainties, an arbitrarily sparse polynomial chaos expansion (PCE) method for general high-dimensional parametric problems is proposed in Shen et al. (2022) [19]. The Galerkin method-based polynomial approximation, and the PCE are summarized by Wu et al. (2020) [20] and Shen et al. (2020) [21], respectively. Soroudi et al. (2013) [22] and Aien et al. (2016) [23] classify uncertainty modeling into five categories: probability and possibility methods, information gap decision theory-based approaches, robust optimization and interval analysis, and their mathematical properties and applications are reviewed.

Most of the above references lack discussions on the temporal and spatial interdependence. However, the interdependence between different uncertainties in the power system is common. For example, the outputs of REGs with different geographic distributions are affected by the temporal correlation of meteorological conditions [24]. Different wind farms and photovoltaic stations in the same area have a strong spatial correlation due to the similarity of wind energy and irradiation [25]. At present, many papers have studied the spatially and temporally interdependent uncertainties. However, a comprehensive review paper is lacking. For example, Song et al. (2019) [26] summarizes the modeling of temporally interdependent uncertainties based on stochastic processes, but lacks a discussion on spatial interdependence.

To fill this gap, this paper presents a comprehensive review as shown in Figure 1. For independent random variables modeling, the probability distribution-based methods are briefly reviewed. For the interdependence, the discrete-time and continuous-time stochastic processes are reviewed. Then, methods to deal with spatial interdependence such as the linear correlation, rank correlation, and copula functions are summarized in detail. The Itô process and random fields to simultaneously describe the temporal and spatial interdependence are discussed. Finally, their applications in power system stability, control, and economic scheduling are reviewed.

The remainder of the paper is organized as follows: Section 2 summarizes the key issues in power system uncertainty research. Section 3 reviews the uncertainty modeling methods from the perspectives of temporal and spatial interdependence. The applications of the uncertainty modeling approaches in power system stability, control, and economic scheduling are presented in Section 4. The conclusions and outlooks are given in Section 5.

2. Classification of Power System Uncertainty Problems

Power system uncertainty research faces two challenges: uncertainty modeling and solving methods. In this section, a brief discussion of four key mathematical problems is given.

2.1. Power System Model with Uncertainties

Considering the uncertainties brought by the stochastic fluctuation of REGs and loads, the power system model can be written as:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{x}}{dt}=\mathit{f}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })}\hfill \\ \mathbf{0}=\mathit{g}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })\hfill \end{array}\right.$$

(1)

where $\mathit{x}$ is the state variable; $\mathit{y}$ is the algebraic variable; $\mathit{u}$ is the decision variable; $\mathit{p}$ is the parameter; $\mathit{\xi }$ is the random variable, which represents the prediction errors of REGs, loads, and electricity price, etc.

According to the sources of uncertainties, $\mathit{\xi }$ can be modeled as either one-dimensional or multidimensional random variables. Since the uncertainty sources effect the power system analysis and control in different ways, Equation (1) can be modeled in different forms to imply different problems.

2.2. Key Mathematical Problems in Power System Uncertainty

2.2.1. Quantification of Uncertainties

The uncertainty quantification problem is to obtain the numerical relationship between random variables and the output of the system. It can be modeled as (2) [27],

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{x}}{dt}=\mathit{f}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })}\hfill \\ \mathbf{0}=\mathit{g}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })\hfill \\ \mathit{x}\left(0\right)={\mathit{x}}_0\hfill \\ w\left(t\right)=W({\mathit{x}}_0,{\left\{\mathit{\xi }\left(\tau \right)\right\}}_{\tau \in [0,t]})\hfill \end{array}\right.$$

(2)

where ${\mathit{x}}_0$ is the initial state $\mathit{x}\left(0\right)$; $W(·)$ is the random response function such as the rotor angle and system frequency; $w\left(t\right)$ is the value of the function $W(·)$.

It is necessary to consider the temporal and spatial interdependence in solving (2). One reason is that the disturbances in the power system are not single at the same time. The power system faces the superposition and coupling of multiple random disturbances [28,29]. For example, the outputs of wind farms and photovoltaic stations are uncertain, and the load fluctuation cannot be accurately predicted. Thus, the uncertainties come from the source and load sides simultaneously. The fluctuation of REG output affects the electricity price, which in turn affects the load demand. Therefore, these uncertainties cannot be described accurately by independent random variables [30].

The other reason is that the disturbances propagate with time. For example, the photovoltaic power at different locations is affected by the temporal correlation of meteorological conditions. A typical example is that the cloud flow influences photovoltaic outputs. In addition, due to the influence of the wake effect in large wind farms, the uncertainties of upstream wind turbines affect the downstream’s [31]. Therefore, how to accurately quantify the influence of uncertainties is a key problem.

2.2.2. Stochastic Inverse Problems

The stochastic inverse problem means to explore the internal laws of the system from observations under an uncertain environment, expressed as [32],

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{x}}{dt}=\mathit{f}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })}\hfill \\ \mathbf{0}=\mathit{g}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })\hfill \\ \mathit{p}\left(t\right)=P(V\left({\left\{\widehat{\mathit{x}}\left(\tau \right)\right\}}_{\tau \in [0,t]}\right),{\left\{\mathit{\xi }\left(\tau \right)\right\}}_{\tau \in [0,t]})\hfill \end{array}\right.$$

(3)

where $P(·)$ describes the internal laws of the parameter or state; $\widehat{\mathit{x}}\left(t\right)$ is the observation of the state; $V(·)$ is the performance functional of $\widehat{\mathit{x}}\left(t\right)$ such as the expectation or variance.

System operating parameters and state change with the environment. For example, the system parameters of jointly operated wind farms [33] and the demand response of residential loads [34] only can be estimated by measurement. Thus, accurately describing the statistical or probability information of random variables has become a major difficulty.

2.2.3. Stochastic Optimization

The stochastic optimization (SO) is aimed at obtaining the optimal operation plan or operation state under the influence of random variables, which can be uniformly expressed as [16,17]:

$$\begin{array}{cc}\hfill & \underset{\mathit{x},\mathit{u}}{min}J\left(F(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })\right)\hfill \\ \hfill & s.t.\left\{\begin{array}{c}G(g_i(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })\le 0),i=1...m\hfill \\ H(h_j(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi })=0),j=1...n\hfill \end{array}\right.\hfill \end{array}$$

(4)

where F is the objective function; h is the equality constraint; g is the inequality constraint. J, G, and H are the performance functionals, such as expectation, conditional value at risk (CVaR), chance constraint, distributionally robust constraint, etc.

There are different requirements for modeling and solving (4) under different objectives. For example, in power system planning, the time scale is usually in years. Thus, the modeling of random variables focuses on the long-term statistical characteristics. However, for scheduling or control, the time scale is often second or minute. Thus, Equation (4) is required to reflect the changing of random variables in a short time period with a high computing efficiency. Therefore, balancing the accuracy and solving efficiency is still a challenging task.

2.2.4. Decision-Dependent Uncertainty

In the above three problems, the uncertainties are exogenous, which means the uncertainty is mainly affected by external factors. Sometimes, they are also affected by decision-making, that is, endogenous uncertainty [35], written as:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathit{x}}{dt}=\mathit{f}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi }\left(\mathit{u}\right))}\hfill \\ \mathbf{0}=\mathit{g}(\mathit{x},\mathit{y},\mathit{u},\mathit{p};\mathit{\xi }(\mathit{u}\left)\right)\hfill \end{array}\right.$$

(5)

where $\mathit{u}$ is a control variable that affects the uncertainties.

A typical example of endogenous uncertainty is that the fluctuation of wind farm output is affected by the decision-making of expansion planning, scheduling, and control [36]. As the scale of the wind farm increases, the spatial correlation of wind speed tends to decrease. The expansion decreases the error of wind farm output. In addition, when the wind turbine participates in frequency regulation, the wind speed brings uncertainty to its available reserve capacity. The load shedding of wind turbines directly affects the correlation between wind speed and the available wind power, thereby influencing the frequency regulation effectiveness [37].

On the load side, the uncertainty of electric vehicle charging is affected by the owner’s decision under incomplete information [38]. The decision-making of industrial load participating in demand-side response affects the load uncertainty [39]. At the operation level, the uncertainties of transmission line faults under extreme weather are affected by the decision of whether to strengthen the equipment [40,41]. Thus, how to describe the coupling between decision-making and uncertainties is also a key problem.

3. Modeling of Temporally and Spatially Interdependent Uncertainties

This section briefly introduces the probability distribution-based methods first. Then, several methods to address the temporally and spatially interdependent uncertainties are reviewed. They include the discrete and continuous stochastic process-based methods, correlation coefficients, copula function-based methods, and the random fields.

3.1. Independent Random Variable

The probability distribution-based methods can be divided into parametric and nonparametric. The parametric methods need to select a specific probability distribution first, then use fitting approaches to fit the historical data, and finally obtain the key parameters of the selected probability distribution.

For the parametric methods, the normal distribution is most widely used [42,43,44]. The Weibull distribution is usually used to model the wind speed uncertainty [45]. The Gamma distribution and the Beta distribution have been introduced to model the photovoltaic outputs and loads uncertainties [46,47,48,49].

For the nonparametric methods, kernel density estimation describes the multimodal distribution, and is widely used in wind power and wind speed forecasting [50,51]. Wang et al. (2016) [52] and Almeida et al. (2017) [53] give a more detailed introduction to the applications of probability distribution-based methods in REG prediction.

3.2. Temporally Interdependent Uncertainties

3.2.1. Discrete Temporal Interdependence

Discrete-time stochastic process models the changing of random variables at discrete time points. Markov chain and time series are two typical discrete-time stochastic processes. They are suitable for modeling the electricity price [54,55,56,57,58], REG output [59,60], and automatic generation control (AGC) signal [61,62], etc.

The Markov chain model is as follows. Supposing $\mathit{S}=\{s_1,s_2,\dots ,s_m\}$ is the state of a discrete series $\left\{\mathit{\xi }\right(t\left)\right\}$. Modeling $\left\{\mathit{\xi }\right(t\left)\right\}$ changes from $\mathit{\xi }\left(t\right)=s_i$ to $\mathit{\xi }(t+1)=s_j$ by a Markov chain can be expressed as:

$$Pr\{\mathit{\xi }(t+1)=s_j|\mathit{\xi }\left(t\right)=s_i\}=\frac{n_{ij}\left(\mathit{\xi }\left(t\right)\right)}{{\sum }_{k=1}^m{n}_{ik}\left(\mathit{\xi }\left(t\right)\right)}$$

(6)

$${\mathit{P}}_{\mathit{im}}=\{P_{i1},P_{i2},\dots ,P_{im}\}$$

(7)

$$P_{ij}=ma{x}_{1\le k\le m}\left(P_{ik}\right)$$

(8)

where $Pr\{·\}$ means probability; $n_{ij}\left(\mathit{\xi }\left(t\right)\right)$ means the transition times from state $s_i$ to $s_j$ in the historical observation of $\left\{\mathit{\xi }\right(t\left)\right\}$; ${\mathit{P}}_{\mathit{im}}$ is the state transfer matrix; $P_{im}$ represents the probability of $\left\{\mathit{\xi }\right(t\left)\right\}$ changes from state $s_i$ to $s_m$. Mor et al. (2018) [63] gives a detailed introduction to higher-order and improved Markov chains.

The Markov chain describes random variables based on the transfer state matrix. In [60], two Markov chains with different orders are used to forecast the ultra-short-term wind speed and wind power, respectively. Yang et al. (2015) [64] uses the enhanced support vector machine (SVM) to modify the state transfer matrix coefficient of the Markov chain, for short-term wind power forecasting. Li et al. (2021) [65] and Chen et al. (2019) [66] predict the short-term wind power and photovoltaic output based on the Markov chain, respectively. Power quality influenced by weather conditions such as lightning strikes and ice storms based on the hidden Markov chain is analyzed by Xiao et al. (2018) [67].

Time series models are applied to analyze the historical data and extract the temporal interdependence. They include the autoregressive (AR) model [68], moving average (MA) model [69,70], autoregressive moving average (ARMA) model [71], and autoregressive integrated moving average (ARIMA) model [72,73].

• AR model

The AR model is developed from linear regression analysis, expressed as:

$$\mathit{\xi }\left(t\right)=\sum _{j=1}^p{\mathit{a}}_j\mathit{\xi }(t-j)+\mathit{\epsilon }\left(t\right)$$

(9)

where (9) is called a p-order AR model; $\left\{\mathit{\epsilon }\right(t\left)\right\}$ represents the white noise series; the AR model is abbreviated as $AR\left(p\right)$.

• MA model

The MA model fits certain time-series data through a white noise with different orders of time-delay, expressed as:

$$\mathbf{\xi }\left(t\right)=\mathbf{\epsilon }\left(t\right)+\sum _{j=1}^q{\mathbf{b}}_j\mathbf{\epsilon }(t-j)$$

(10)

where (10) is called a q-order MA model, abbreviated as $MA\left(q\right)$.

• ARMA model

The ARMA model combines with the AR and MA models, expressed as:

$$\mathit{\xi }\left(t\right)=\sum _{j=1}^p{\mathit{a}}_j\mathit{\xi }(t-j)+\sum _{j=1}^q{\mathit{b}}_j\mathit{\epsilon }(t-j)+\mathit{\epsilon }\left(t\right)$$

(11)

where p and q are the maximum order of AR and MA; ${\mathit{a}}_j$ and ${\mathit{b}}_j$ are the AR coefficient and MA coefficient, respectively; the ARMA model is abbreviated as $ARMA(p,q)$.

• ARIMA model

The ARIMA model is an extension of the ARMA model, expressed as:

$$\mathit{A}\left(\mathit{L}\right){(1-\mathit{L})}^d\mathit{\xi }\left(t\right)=\mathit{B}\left(\mathit{L}\right)$$

(12)

where $\mathit{L}$ is a lag operator; $\mathit{A}\left(\mathit{L}\right)$ and $\mathit{B}\left(\mathit{L}\right)$ are $AR\left(p\right)$ and $MA\left(q\right)$ models, respectively; d indicates the order of differential processing of the observations; the ARIMA model is abbreviated as $ARIMA(p,d,q)$.

Time series models are usually used to simulate REG outputs and loads. In Latimier et al. (2020) [62], a Markov switching AR method is proposed to predict wind power. It is applied to the stochastic dynamic programming of the energy storage in a wind farm, which reduces the operating cost by 15%. Dong et al. (2022) [70] classifies wind power output data by a logic function, and uses ARMA models to predict the wind power. In Arora et al. (2018) [74], a rule-based ARMA model is proposed to forecast the load on holidays and working days in France. Dash et al. (2019) [75] proposes an adaptive-cube-Kalman-filter-learning-based method to train an adaptive ARMA model, in order to predict electricity prices under the influence of short-term load demand.

The AR, MA, and ARMA models are widely used for modeling temporal interdependence. However, the selection of appropriate model orders (p and q) can be challenging, and the accuracy of the predictions depends on the quality of historical data and the stationarity assumption.

3.2.2. Continuous Temporal Interdependence-Stochastic Differential Equations

Continuous-time stochastic processes are generally modeled in the form of stochastic differential equations (SDEs), which unify the analysis and control of the uncertainty and power system dynamics model. They have a higher accuracy compared with the discrete-time stochastic processes, and are commonly used in REG real-time control [76,77,78], electricity price forecasting [79,80], and system frequency regulation [81,82].

• Itô process

The mathematical expression of an Itô process is given as an SDE:

$$d\mathit{\xi }\left(t\right)=\mathit{\mu }(\mathit{\xi },t)dt+\mathit{\sigma }(\mathit{\xi },t)d\mathit{W}\left(t\right)$$

(13)

where $\mathit{W}\left(t\right)$ is a standard Wiener process; $\mathit{\mu }(\mathit{\xi },t)$ is the drift term, which represents the mean reverting characteristic; $\mathit{\sigma }(\mathit{\xi },t)$ is the diffusion term. By setting $\mathit{\mu }(\mathit{\xi },t)$ and $\mathit{\sigma }(\mathit{\xi },t)$, the Itô process satisfies different probability distributions and temporal interdependence [26].

The Itô process has been widely used in finance, especially in options pricing [83]. In recent years, scholars introduce it into the power system and carry out a series of exploratory works on the modeling of wind power [84], photovoltaic output [85,86], and the stochastic analysis and control [27,81,82,87]. Chen et al. (2019) [81] and Qiu et al. (2021) [88] establish a wind power model based on the Itô process. In Qiu et al. (2020) [87], the uncertainty of photovoltaic output in a cascaded run-of-the-river hydropower system is modeled. The joint control considering river dynamics and photovoltaic uncertainty is realized. The Gaussian or non-Gaussian stochastic disturbance is modeled with an Itô process by Qiu et al. (2021) [27]. And a nonintrusive uncertainty quantification method that leverages an off-the-shelf commercial power system simulator is proposed.

A simple example of the Itô process, named Ornstein-Uhlenbeck (OU) process [89], written as:

$$d\mathit{\xi }\left(t\right)=-\mathit{\lambda }\mathit{\xi }\left(t\right)dt+\mathit{\gamma }d\mathit{W}\left(t\right)$$

(14)

where $\mathit{\lambda }$ and $\mathit{\gamma }$ are constants, which subject to $\mathit{\lambda }>\mathbf{0}$ and $\mathit{\gamma }>\mathbf{0}$. The OU process has a Gaussian distribution and can turn into a non-Gaussian distribution through the equimeasure transformation.

In Wang et al. (2017) [90], the wind speed uncertainty is modeled based on an OU process, and has an exponential autocorrelation. In [91], Wang et al. (2015) merges the OU process-based wind power model into the dynamic equations. They theoretically show that the stochastic model can be approximated by a corresponding deterministic model for stability analysis under the mild conditions.

The Itô process offers high simulation accuracy and flexibility in capturing different probability distributions and temporal interdependencies. However, setting the drift and diffusion terms ($\mathit{\mu }(\mathit{\xi },t)$ and $\mathit{\sigma }(\mathit{\xi },t)$) can be challenging and may require expert knowledge. Qiu et al. (2022) [86] gives a way to map public weather reports to SDE parameters. But this method is only suitable for photovoltaic power.

• Jump-diffusion process

Itô process cannot describe load stochastic jumps. For this reason, scholars have introduced jump-diffusion processes, which are continuous-time stochastic processes consisting of a pure jump process and a diffusion process, expressed by the following SDE:

$$d\mathit{\xi }\left(t\right)=\mathit{\mu }(\mathit{\xi },t)dt+\mathit{\sigma }(\mathit{\xi },t)d\mathit{W}\left(t\right)+\mathit{\pi }(\mathit{\xi },t)d\mathit{C}\left(t\right)$$

(15)

where $\mathit{C}\left(t\right)$ is a compound Poisson process; $\mathit{\pi }(\mathit{\xi },t)$ describes the jump amplitude.

The jump-diffusion process is used in electricity price [92,93], and load [94] modeling. It is also suitable for describing the start-up and shut-down of REGs and loads. However, there are still a few applications.

Finally, note that the discrete-time methods are flexible and easy to combine with machine learning (ML). It is widely used in REG and electricity price forecasting [74,75]. The continuous-time models are more complex, but it achieves a more general description form of the random change and are more suitable for a continuous-time application, such as stability analysis [87,88].

3.3. Spatially Interdependent Uncertainties

3.3.1. Two-Dimensional Random Variable

Classification and segmentation are two simple methods to model the two-dimensional random variables. Mohammadi et al. (2020) [95] uses fuzzy sets to describe the relationship between energy demand and electricity price. Martinez-Mares et al. (2013) [96] and Soroudi et al. (2014) [97] model the relationship between wind power uncertainty and electricity price based on scenario sets. He et al. (2019) [98] uses the least absolute shrinkage and selection operator (LASSO) algorithm to obtain the influence of wind power on load fluctuation by generalized cross-validation.

The linear correlation coefficient, rank correlation coefficient, and bivariate copula functions are common models for the two-dimensional random variables.

• Linear correlation

The Pearson correlation coefficient describes linear correlation. It is denoted as $r_p(\mathit{\xi },\mathit{\omega })$, and can be expressed as:

$$r_p(\mathit{\xi },\mathit{\omega })=\frac{{\sum }_{i=1}^n\left({\xi }_i-\overline{\xi }\right)\left({\omega }_i-\overline{\omega }\right)}{\sqrt{{\sum }_{i=1}^n{\left({\xi }_i-\overline{\xi }\right)}^2}\sqrt{{\sum }_{i=1}^n{\left({\omega }_i-\overline{\omega }\right)}^2}}$$

(16)

where $\overline{\mathit{\xi }}$ and $\overline{\mathit{\omega }}$ are the mean values of series $\left\{\mathit{\xi }\right(t\left)\right\}$ and $\left\{\mathit{\omega }\right(t\left)\right\}$, respectively.

Liu et al. (2015) [99] uses the Pearson correlation coefficient to describe the relationship between the aerosol index and the irradiation when analyzing photovoltaic output fluctuation. Zhang et al. (2021) [100] uses multiple Pearson linear correlation coefficients to model the relationships between wind power and wind speed, wind direction, and temperature, respectively. Then the Shapley value is employed to determine their weight.

However, the linear correlation coefficient cannot describe the nonlinear relationship. It is based on the assumption of a normal distribution, which has many limitations [101]. For this reason, the rank correlation coefficient is proposed.

• Rank correlation

Rank correlation describes the relationship of samples sorted by the data values. It can be modeled as:

$$r_s(\mathit{\xi },\mathit{\omega })=\frac{{\sum }_{i=1}^n\left(R_i-\overline{R}\right)\left(S_i-\overline{S}\right)}{\sqrt{{\sum }_{i=1}^n{\left(R_i-\overline{R}\right)}^2}\sqrt{{\sum }_{i=1}^n{\left(S_i-\overline{S}\right)}^2}}$$

(17)

where $r_s(\mathit{\xi },\mathit{\omega })$ is named as Spearman’s rank correlation coefficient; R and S are the datasets obtained by sorting series $\left\{\mathit{\xi }\right(t\left)\right\}$ and $\left\{\mathit{\omega }\right(t\left)\right\}$ in descending or ascending order, respectively; $R_i$ and $S_i$ are the i th data values; $\overline{R}$ and $\overline{S}$ are the mean values of the corresponding dataset.

Spearman’s rank correlation coefficient has the same expression as the Pearson correlation coefficient. However, it is only related to the ordering, which avoids the limitation that requires random variables to follow the normal distribution. In [102], the correlation of wind speed time series is described based on Spearman’s rank correlation coefficient. In [103], the relationship between Google search engine index and load is described by rank correlation. Pan et al. (2019) [104] studies the correlation between the investment of a provincial grid and its local GDP, resident population, and electricity consumption based on the Spearman’s rank correlation coefficient.

• Bivariate copula

Copula functions model the correlation between interdependent random variables. The bivariate copula functions are designed for two-dimensional random variables and are mainly divided into two categories: elliptic copula and Archimedean copula [105]. To use the copula method, the marginal distribution should be modeled first. Then, an appropriate copula function should be selected to establish the joint distribution. Sriboonchitta et al. (2013) [105] gives a detailed introduction to selecting copula functions.

The bivariate copula functions have been applied in the forecasting of electricity price [106], REG output [107,108], and load [109]. Han et al. (2019) [107] uses the binary normal and t-copula functions to establish a joint distribution. The maximum likelihood estimation is used to obtain the parameters of each copula function. Wei et al. (2022) [108] models the probability distribution of wind power and speed based on the Weibull distribution and asymmetric Gaussian distributions, respectively. Then a mixed copula composed of Gaussian copula, Gumbel copula, Clayton copula, and Frank copula is used to simulate the joint distributions. It gets a more accurate prediction compared with the SVM and backpropagation (BP) neural network-based approaches. Ouyang et al. (2019) [109] uses a Gumbel-Houggard mixed copula function to model the correlation between load, electricity price, and temperature. Yu et al. (2019) [110] establishes a joint distribution between wind speed and load based on the Gumbel copula.

• Time-varying copula

The time-varying copula function is based on the static bivariate copula function. Its parameters are considered to change with time such as following the ARMA process [105]. It is mainly used in decisions of energy market risk investment [111] and research of energy market risk portfolio [112].

Notably, correlation coefficient-based methods are simple, but they only capture the global correlation, and cannot describe the tail correlation. The copula methods can flexibly describe the nonlinear interdependence, but the modeling of them is complex and requires careful selection and accurate modeling of marginal distributions.

3.3.2. Multidimensional Random Variable

In higher dimensions, a graphical tool is employed to identify different multivariate copula constructions. Such a tool is named as the vine tree structure [113]. There are two subclasses of vine tree structures. One is the canonical (C-) vine tree structure, the other is the drawable (D-) vine tree structure. Copula functions aggregated on a tree structure are defined as the vine copula. The C-vine copula determines the root node of each tree according to the main variable. The D-vine copula is a linear structure that is convenient for modeling. In Jiang et al. (2018) [114], considering the interdependence between electric vehicle charging loads in adjacent periods, a pair-copula decomposition method is proposed to construct the charging scenarios of electric vehicles, and the joint probability distribution between charging loads in different periods is described by the D-vine copula structure. Cherubini et al. (2011) [115] studies the relationship between photovoltaic power and the meteorological conditions based on the D-vine copula. The spatial interdependence of wind power is analyzed based on the D-vine copula model by Arrieta-Prieto et al. (2022) [116].

The vine copula models consider the time-domain variation, but the computational complexity limits their application. The dynamic copula theory can overcome the computational performance deficiency. In Li et al. (2020) [117], the dynamic symmetrized Joe-Clayton copula (SJC-Copula) and dynamic Gaussian copula show superior performance in the modeling of wind power. The development of copula theory and its more detailed applications in energy can be seen in Bhatti et al. (2019) [118].

Besides, the joint probability density of multidimensional random variables can be implied by the inner product space corresponding to multi-parameter orthogonal polynomials. The arbitrary polynomial chaos (aPC) represents arbitrary second moment distributions and nonlinear interdependence. For example, Wang et al. (2019) [119] proposes a data-driven polynomial chaos expansion (DD-PCE) method for assessing the stability margin of a photovoltaic system considering the impact of uncertain solar irradiance. Shen et al. (2022) [19] proposes an arbitrarily sparse PCE method that effectively relieves the curse of dimensionality for high-dimensional parametric problems.

Some methods to deal with temporal interdependence can address spatial interdependence simultaneously, which are discussed in Section 3.4.

3.4. Temporally and Spatially Interdependent Uncertainties

The Itô process and random fields model the temporally and spatially interdependent uncertainties. For the Itô process, expanding the random variable $\mathit{\xi }$ in (2) from one-dimensional to multi-dimensional can address the spatial interdependence. Chen et al. (2022) [84] describes setting the diffusion term $\mathit{\sigma }(\mathit{\xi },t)$ and the drift term $\mathit{\mu }(\mathit{\xi },t)$ of the Itô process model based on the covariance matrix and the autocorrelation function, to describe the spatial and temporal interdependence uncertainties simultaneously.

A random field is a generalized random process with two or more arguments. It expands the 1-dimensional argument (usually time) in the random process to multiple dimensional, including time and space coordination. It is widely used in artificial intelligence and image processing [120,121]. In recent years, it has been introduced into power system research. Zhang et al. (2016) [122] uses a Gaussian conditional random field to forecast the photovoltaic output. Gao et al. (2017) [123] introduces a Markov random field to detect and estimate the status of electrical equipment. In addition, the Markov random field is also applied to modeling distribution systems topology correlation by Zhao et al. (2020) [124]. Detailed introductions of random fields can be found in Li et al. (2022) [125] and Liu et al. (2019) [126].

In summary, the temporal and spatial interdependence modeling methods introduced in Section 3 are listed in

Table 1. Comparison of the temporal and spatial interdependence modeling methods.

Interdependence	Type	Model	Mathematical Expression	Applications	Solving Methods
Temporally	Discrete	Markov chain	(6)	Wind power, photovoltaic output [60,66] and wind speed [65] forecast	Numerical analysis based on scenarios [74] or historical datasets [60,62,65,66]
		AR	(9)	Wind power forecast [62]
		ARMA	(11)	Wind power [70], load [74] and electricity price [75] forecast
		ARIMA	(12)	Wind power [72], generation capacity [73] forecast
	Continue	Itô process	(13)	Wind power and photovoltaic output modeling [84,85], stochastic analysis and control of power system [81,82,87]	Trajectory sensitivity decomposition [87] or time domain simulation [92,93]
		Jump-diffusion process	(15)	Electricity price forecast [92,93], load jump modeling [94]
Spatially	Correlation coefficient	Pearson linear correlation	(5)	Wind power and photovoltaic output forecast [99,100,101]	Numerical analysis based on historical datasets [99,100,101,102,103]
		Spearman’s rank correlation	(17)		Wind speed and load forecast [102,103]
	Copula	Bivariate/Multivariate copula	[105]	Electricity price [106], wind power, photovoltaic output [107,108], load [109] and wind speed [110] forecast	Fitting method based on historical datasets [106,107,108,114]
		Vine copula	[116]	Electric vehicles charging load [114], wind power and photovoltaic output modeling [115,116,117]
	Polynomial chaos (PC)	gPC/aPC	[18]/[119]	Probabilistic power flow [18], REG uncertainty [119]	Model or data driven-based methods [19,119]
Temporally and Spatially	/	Itô process	(13)	Wind power and photovoltaic output modeling [84,85], stochastic analysisa and control of power system [81,82,87]	Trajectory sensitivity decomposition [87] or scenario-based simulation [123,124]
		Random fields	[120]	Photovoltaic output forecast [122], Equipment status estimation [123], topology modeling [124]

.

4. Application in Power Systems

4.1. Power System Stability

According to the classification of the IEEE/CIGRE joint working group, power system stability can be divided into transient stability, small disturbance stability, frequency stability, and voltage stability [127,128].

For the transient stability, the electromagnetic transient simulation is the most widely used way. Han et al. (2014) [129] uses a slope signal, discrete cosine signal, and step signal to simulate wind speed, and analyzes the transient stability of wind turbines by simulation. The Monte Carlo method is used to generate the sample states to simulate the probabilistic transient stability of the large-scale system by Vaahedi et al. (2000) [130] and Chiodo et al. (1999) [131]. Yue et al. (2020) [132] derives the analytical expression of the probabilistic transient stability considering wind power uncertainty. In addition, Xia et al. (2018) [133] analysis the influence of parameters on the transient behavior by the Galerkin method-based polynomial approximation. Adeen et al. (2021) [134] employs the Itô process to model uncertainties in transient stability analysis.

For the small disturbance stability, the existing research focuses on the damping or eigenvalue. In Huang et al. (2013) [135], a probabilistic small disturbance stability of electric vehicle access is analyzed based on the quasi-Monte Carlo sampling. In Zheng et al. (2012) [136], the third-order probabilistic collocation method is used to simulate the system eigenvalues changing under the wind power uncertainty. For the small signal stability under parameter variation, Shen et al. (2022) [137] proposes accurate polynomial approximations of bifurcation hypersurfaces in the multi-dimensional parameter space. The OU process is used to simulate wind power, and a small signal stability analysis method is proposed by Verdejo et al. (2016) [138].

For the frequency and voltage, whether the fluctuation of them can be kept within the allowable interval is the key point. For this reason, Alsharif et al. [4], Deng et al. (2017) [139] and Deng et al. (2009) [140] use Monte Carlo method to analyze the influence of source and load sides uncertainties on frequency and voltage variation. Ali et al. (2021) [141] designs a robust load controller to address the uncertainty of frequency variation. Qiu et al. (2016) [142] and Qiu et al. (2018) [143] use the polynomial approximation to describe the static voltage stability region boundaries. The random parameters of virtual inertial control are modeled by the Wiener process to analyze the small signal stability of wind turbines by Ma et al. (2017) [144]. The Fokker-Planck equation and an SDE are utilized to analyze the frequency stability by Farmer et al. (2021) [145].

Hasan et al. (2019) [146] presents a state-of-the-art review of the probabilistic stability analysis framework, where interested readers are referred to.

4.2. Optimal Control

Different control methods are proposed to deal with uncertainty from the system or equipment level. They can be divided into optimization-based and rule-based methods.

Model predictive control (MPC) and stochastic MPC (SMPC) are two typical optimization-based methods. In [147], the primary frequency controller of wind farms is designed based on MPC. The AGC controller is designed based on MPC in [82,87]. Fuzzy logic and proportional integral (PI) control are two rule-based methods. In [148], a microgrid centralized controller is designed by fuzzy PI control to improve stability in an uncertain environment. Mesbah et al. (2016) [149] and Nguyen et al. (2019) [150] summarize the development and applications of SMPC and fuzzy logic control in detail. The Markov chain is used to describe the discrete time delay and a time delay-based load frequency controller is proposed by Lv et al. (2021) [151]. Besides, the ARMA model is used to predict power fluctuation in static voltage control by Naderipour et al. (2020) [152].

The parameters of a robust controller are obtained by solving an optimal problem under the worst influence of uncertain factors, but remain fixed in the process of control. This method copes with uncertainty but may be too conservative in some scenarios. Ho et al. (2021) [153] designs an online robust controller for large-scale nonlinear systems based on machine learning. In Moradi et al. (2015) [154], an H-∞ robust pitch angle controller is designed. The simulation results show it has significant robustness and stability advantages compared with the PI controller. A comprehensive review of robust control is presented by Mohammadi et al. (2021) [155].

4.3. Economic Scheduling

SO and RO are two classical methods to deal with uncertainties in economic scheduling.

For the SO, the probability distribution of the random variable is usually assumed in advance, and scenario-based methods such as the Monte Carlo are used to quantify uncertainty [156,157]. However, the assumed probability distribution sometimes cannot accurately describe the actual uncertain factors, and the scenario-based methods cause a large computational burden.

The RO uses a set to limit the uncertainties, which avoids assuming probability distribution. Qiu et al. (2022) [16] gives a review of the modeling and solving of RO in scheduling. However, the results of RO are sometimes too conservative [158,159]. In Ardabili et al. (2021) [160], the randomness of wind power is described by the Markov chain, and its volatility is measured by the variance. In the modeling of wind speed, the Latin hypercube sampling method is proposed based on the Kantorovich distance predicted by ARMA patterns in Ju et al. (2019) [161]. Different variants of SO and RO such as the recourse model, chance-constrained programming, and multistage model are discussed in Zheng et al. (2014) [162] and Reddy et al. (2017) [163].

In recent years, distributionally robust optimization (DRO) has been proposed to overcome the shortcomings of low computational efficiency and high conservative in SO and RO. It is widely used in unit commitment, optimal power flow, and integrated energy system (IES) scheduling. The key to this method is to reasonably model a set of probability distributions. Wang et al. (2022) [164] and Zhang et al. (2020) [165] measure the distance between the candidate and empirical distribution based on the Wasserstein distance. Chen et al. (2018) [166] constructs the uncertainty set based on the Kullback-Leibler (KL) divergence. The uncertainty sets are constrained by 1-norm and ∞-norm by He et al. (2019) [167,168]. Rahimian et al. (2019) [169] presents a comprehensive review of the DRO.

A review of the latest economic scheduling research can be found in Xue et al. (2023) [170] and Yang et al. (2020) [171].

5. Conclusions and Outlook

A comprehensive review of uncertainty modeling from the perspectives of temporal and spatial interdependence is given, and their applications in power system stability, control, and economic scheduling are summarized. The main conclusions and future research directions are given below:

Methods to balance the modeling simplicity and calculation accuracy for stochastic jumps are still lacking. Thus, it is hard to accurately simulate the dynamics that incorporate both continuous and event-driven, such as the start-up and shut-down in the continuous operation of generators and loads. Jump-diffusion process seems to be a potential approach, but there is still a lack of research.

Achieving an accurate and efficient modeling of temporally and spatially interdependent uncertainties still needs further study. The correlation coefficients are simple to calculate. However, stochastic process or copula function-based methods with higher accuracy usually cause a great computational burden or are hard to solve.

To fill the shortcomings, it is valuable to make use of emerging technologies such as machine learning and artificial intelligence to improve the computing efficiency of the stochastic jumps model. Exploring new modeling approaches and developing advanced computing technologies is important to promote the accuracy of the temporal and spatial interdependent uncertainties modeling, and is beneficial for the economic operation and control of the renewable power systems.