Assessment of the Renewable Energy Consumption Capacity of Power Systems Considering the Uncertainty of Renewables and Symmetry of Active Power

Abstract

The rapid growth of renewable energy presents significant challenges for power grid operation, making the efficient integration of renewable energy crucial. This paper proposes a method to evaluate the power system’s capacity to accommodate renewable energy based on the Gaussian mixture model (GMM) from a symmetry perspective, underscoring the symmetrical interplay between load and renewable energy sources and highlighting the balance necessary for enhancing grid stability. First, a 10th-order GMM is identified as the optimal model for analyzing power system load and wind power data, balancing accuracy with computational efficiency. The Metropolis–Hastings (M-H) algorithm is used to generate sample spaces, which are integrated into power flow calculations to determine the maximum renewable energy integration capacity while ensuring system stability. Short-circuit ratio calculations and N-1 fault simulations validate system robustness under high renewable energy integration. The consistency between the results from the M-H algorithm, Gibbs sampling, and Monte Carlo simulation (MCS) confirms the approach’s accuracy.

1. Introduction

Symmetry in renewable energy integration refers to the balanced incorporation of renewable sources like wind and solar into the power grid, emphasizing equilibrium between variable supply and consistent demand to ensure stability. Lately, sources of renewable energy, including wind and solar, known for their lack of carbon emissions, have swiftly become part of the electrical grid [1,2]. They have occupied an important position in the energy systems of many countries. As of 2021’s close, the worldwide installed capacity for solar and wind energy reached 1674 GW, with European nations like Germany, Denmark, Norway, and Sweden extensively adopting these green energy solutions [3]. However, the increasing complexity and dynamic nature of modern power systems, driven by the integration of renewable energy, have been widely recognized [4,5]. The generation of renewable energy typically exhibits strong randomness, intermittency, and volatility [6]. The integration of a high proportion of renewable energy complicates the operational characteristics of power systems. Characteristics such as low disturbance rejection, low inertia, and low short-circuit capacity inherent in power systems will pose significant challenges to their balance capacity, support capacity, regulation ability, and safety and stability [7,8,9]. Stability, in this context, is vital because it ensures that the grid can reliably accommodate fluctuations in energy supply and demand without compromising the continuous delivery of electricity or the security of the power network. Hence, to lower the renewable energy reduction rate and guarantee the system’s safe functioning, it is critically important to precisely evaluate the system’s capacity to absorb renewable energy.

Symmetry in renewable energy integration refers to the balanced incorporation of renewable sources like wind and solar into the power grid, emphasizing equilibrium between variable supply and consistent demand to ensure stability. Lately, sources of renewable energy, including wind and solar, known for their lack of carbon emissions, have swiftly become part of the electrical grid [1,2]. They have occupied an important position in the energy systems of many countries. As of 2021’s close, the worldwide installed capacity for solar and wind energy reached 1674 GW, with European nations like Germany, Denmark, Norway, and Sweden extensively adopting these green energy solutions [3,4,5]. However, the generation of renewable energy typically exhibits strong randomness, intermittency, and volatility [6]. The integration of a high proportion of renewable energy complicates the operational characteristics of power systems. Characteristics such as low disturbance rejection, low inertia, and low short-circuit capacity inherent in power systems will pose significant challenges to their balance capacity, support capacity, regulation ability, and safety and stability [7,8,9]. Stability, in this context, is vital because it ensures that the grid can reliably accommodate fluctuations in energy supply and demand without compromising the continuous delivery of electricity or the security of the power network. Hence, to lower the renewable energy reduction rate and guarantee the system’s safe functioning, it is critically important to precisely evaluate the system’s capacity to absorb renewable energy.

In the field of power systems, assessing the absorption capacity of renewable energy is a significant research topic. Currently, the most commonly used analytical methods include typical day analysis and time series production simulation [10]. The typical day analysis method, as described in references [11,12], mainly evaluates the power system’s capacity to absorb renewable energy based on the peak load’s regulating margin on a typical day. However, this method does not fully consider the randomness and volatility of renewable energy output and load, resulting in conservative calculation outcomes [13]. In contrast, time series production simulation, as shown in reference [14], can more comprehensively simulate the operation of power systems, but its large data requirements, heavy computational load, and lengthy optimization solution time limit its practical application efficiency. Furthermore, the output of renewable energy is random. To comprehensively analyze the impact of system uncertainty factors on renewable energy absorption, it is necessary to use probabilistic analysis methods to precisely calculate the flow distribution of the system in a safe operating state.

Currently, the most common probabilistic analysis methods in the field of power system research are divided into two categories: analytical methods and simulation methods. Among them, the analytical method calculates the probability distribution of the output by processing the convolution between random variables. However, the assumption that input random variables are independent results in lower computational accuracy of the analytical method. The MCS method, as the most representative simulation method, achieves high computational accuracy with a sufficiently large sampling size [15]. However, due to the high computational cost of the MCS method, it has gradually become a standard for verifying the accuracy of other methods [16,17]. The M-H algorithm, an iterative sampling algorithm derived from the Metropolis algorithm, exhibits good sampling performance and high computational efficiency. It significantly increases the computation speed [18,19].

This paper addresses the effectiveness and precision required in assessing the renewable energy absorption capacity within power system operations. In simulating input random variables, it first constructs probability models of the load and wind power using the GMM, then employs the M-H algorithm to obtain a sample space from the probability density function of the input random variables. An assessment model for renewable energy absorption capacity is built under the constraints of safe system operation, taking into account the dynamic characteristics and complexity of the power system and enhancing the efficiency and accuracy of random variable simulation with the M-H algorithm. The model identifies renewable energy access points and initial outputs based on the network’s topology and uses power flow calculations to determine the maximum capacity of renewable energy that can be integrated for each sample set under the constraints of system safety, statistically analyzing the capacity for renewable energy integration. Finally, the model’s reliability and practicality are validated by analyzing the short-circuit ratio of renewable energy sites and performing N-1 fault simulations. The effectiveness of the proposed model in improving the precision of renewable energy absorption assessments and reducing computational complexity is verified through simulation experiments under various scenarios and conditions.

2. Probabilistic Modeling of Load and Wind Power

In the study of probabilistic assessment of renewable energy integration capacity, probabilistic modeling of uncertain factors is the primary issue to be addressed. The accuracy of this modeling directly affects the precision of probabilistic load flow calculations, which in turn determines the accuracy of the renewable energy integration capacity assessment results. The main uncertain factors affecting the integration capacity of renewable energy are the randomness of the load and the fluctuating nature of renewable energy. Currently, in power system studies considering load randomness, loads are mostly treated as following a normal distribution. However, actual load power distributions in general systems do not strictly follow a normal distribution but exhibit a multimodal nature. Some scholars have discovered the advantages of using a weighted GMM in modeling and have applied it to probabilistic load modeling, achieving excellent modeling results.

For the probabilistic modeling of wind power in renewable energy, existing methods can be divided into two categories: the first category establishes a probabilistic model of wind speed, and the second establishes a probabilistic model of wind power. Numerous studies have shown that measured wind speed data approximately follow a Weibull distribution. Based on the inherent relationship between wind speed and wind power, the probabilistic distribution of wind power can thus be derived. However, further research observations indicate that the probability distribution curve of measured wind speed data does not completely conform to a Weibull distribution. Some researchers have proposed using a probabilistic modeling method based on a weighted Gaussian mixture distribution. Compared to single distribution models, mixture distribution models offer greater flexibility in fitting curves, significantly improving the accuracy of wind power modeling.

GMM describes the joint probability distribution of a random vector x, represented as the convex sum of several Gaussian distributions [20]. The set of adjustable parameters for GMM is denoted by Ω = {ω_m, μ_m, σ_m; m = 1, 2, …, M}. The formula for GMM is outlined below:

$$f_{\mathrm{X}}(x)=f_{\mathrm{X}}(x)={\displaystyle \sum _{m=1}^M{\omega }_m}{N}_m\left(x;{\mu }_m,{\sigma }_m\right)$$

(1)

$${\displaystyle \sum _{m=1}^M{\omega }_m}=1,{\omega }_m>0$$

(2)

$$N_m\left(x;{\mathit{\mu }}_m,{\mathit{\sigma }}_m\right)={\displaystyle \frac{e^{-\frac{1}{2}{\left(x-{\mathit{\mu }}_m\right)}^T{\mathit{\sigma }}_m^{-1}\left(x-{\mathit{\mu }}_m\right)}}{{(2\mathsf{\pi })}^{W/2}\det {\left({\mathit{\sigma }}_m\right)}^{1/2}}}$$

(3)

In this context, $f_X(x)$ represents the joint probability density function of x; ω_m stands for the weight coefficient; N_m(∙) symbolizes the multi-dimensional Gaussian distribution, termed as the m-th Gaussian component within the GMM; det(∙) symbolizes the determinant of a matrix; M indicates the total count of Gaussian components; and μ_m and σ_m are the mean vector and covariance matrix of the m-th Gaussian component, respectively.

Figure 1 illustrates a GMM with four Gaussian components. In this diagram, the GMM’s probability density function (depicted by the black line) is a convex sum of the probability density functions of four Gaussian distributions (shown by red, blue, green, and purple lines). By tuning the parameter set Ω, GMM can effectively model the probability density function of any random variable.

In Figure 1, the Gaussian components from 1 to 4 have weight coefficients set at 0.4, 0.1, 0.3, and 0.2, respectively. Identifying the parameter set Ω for the GMM falls under the category of typical parameter estimation issues. Utilizing x, which represents the historical data of the load and wind power, one can establish the load’s Gaussian mixture model.

For analyzing power load and wind power using a GMM, the historical records of actual power consumption and wind power can be viewed as a series of observations. These observations reflect the power consumption and wind power in the power system over different periods. However, these data are not complete because it is not possible to directly observe which specific power value belongs to which category of power consumption or generation. Some data remain unrecorded or unobservable. In a GMM, this unobservable membership information is referred to as hidden variables or missing data. Let the category to which the input random variable belongs be Z, and then the complete data are (X, Z). If there are past observations of a random variable, these past observations can be used to establish a Gaussian mixture probability model for the variable through parameter estimation. Currently, the expectation–maximization (EM) algorithm is one of the main algorithms used for parameter estimation in GMMs. Using the EM algorithm to determine the parameters of the GMM for load and wind power is a classic and effective method. This algorithm optimizes the parameter set Ω through an iterative process, which includes the mean (μ), covariance matrix (Σ), and weight coefficients (π) for each Gaussian component [21]. First, these parameters are initialized, which can be performed randomly or through heuristic methods. The core of the EM algorithm consists of two main steps: the expectation step (E-step) and the maximization step (M-step). In the E-step, the hidden variables are estimated by calculating the probability that each data point belongs to each Gaussian component, referred to as responsibility. This is calculated as follows:

$${\gamma }_{ik}={\displaystyle \frac{{\pi }_k\cdot N(x_i|{\mu }_k,{\mathsf{\Sigma }}_k)}{{\displaystyle \sum _{j=1}^K}{\pi }_j\cdot N(x_i|{\mu }_j,{\mathsf{\Sigma }}_j)}}$$

(4)

where ${\gamma }_{ik}$ represents the probability that data point $x_i$ belongs to the kth Gaussian component, ${\pi }_k$ is the weight coefficient of the kth Gaussian component, and $N(x_i|{\mu }_k,{\mathsf{\Sigma }}_k)$ is the probability density function of the kth Gaussian component.

$${\pi }_k^{\mathrm{new}}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\gamma }_{ik}$$

(5)

$${\mu }_k^{\mathrm{new}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N}{\gamma }_{ik}{x}_i}{{\displaystyle \sum _{i=1}^N}{\gamma }_{ik}}}$$

(6)

$${\mathsf{\Sigma }}_k^{\mathrm{new}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N}{\gamma }_{ik}(x_i-{\mu }_k^{\mathrm{new}}){(x_i-{\mu }_k^{\mathrm{new}})}^T}{{\displaystyle \sum _{i=1}^N}{\gamma }_{ik}}}$$

(7)

where ${\pi }_k^{\mathrm{new}}$ is the updated weight coefficient, ${\mu }_k^{\mathrm{new}}$ is the updated mean, ${\mathsf{\Sigma }}_k^{\mathrm{new}}$ is the updated covariance matrix, and N is the total number of data points.

The E-step and M-step are repeated until the parameters converge, i.e., the change in parameters is less than a predefined threshold. Through this iterative process, the EM algorithm can effectively determine the parameters of the GMMfor load and wind power, thereby establishing a more accurate probabilistic model. This model can better reflect the randomness and volatility of load and wind power, thus improving the accuracy of renewable energy consumption capacity assessment.

3. Acquisition of Input Variables Sample Space

The M-H algorithm is an iterative sampling method derived from the Metropolis algorithm, known for its excellent sampling performance and high computational efficiency. This algorithm simulates transitions between system states based on a state transition probability matrix, derived from the Markov chain Monte Carlo (MCMC) method. It overcomes the limitations of the traditional MCS method, which can only model static scenarios, by allowing dynamic simulation of random variables transitioning between different states.

In the context of power systems, assuming that the first-order Markov chain is $X=\{x_l,x_2,\dots ,x_t,\dots ,x_N\}$, where N is the data length of the historical power series, x_t is the state value of the t-th data corresponding to dividing the historical sequence into S states, consists of S states forming the state space $E=\{E_l,E_2,\dots ,E_S\}$, each moment of historical data corresponds to a state, and there are S possibilities for each state to transition to the next state, which can be known according to the Markov property (the present state is solely connected to the immediate preceding state and is unrelated to any states prior to that), calculated as follows:

$$p_{ij}=P_r\left\{x_{t+1}=E_j\mid x_t=E_i\right\}=n_{ij}/\sum _{k=1}^S{n}_{ik}$$

(8)

In the formula, p_ij and n_ij are the probability of transferring from state E_i to state E_j and the number of transfers, respectively, and the transitions between the states form the state transfer probability matrix P, according to which the cumulative state transfer probability matrix Q can be formed, and P and Q are shown in Equations (5) and (6).

$$\mathit{P}=\left[\begin{array}{c}{p}_{11}{p}_{12}\cdots p_{1S}\\ p_{21}{p}_{22}\cdots p_{2S}\\ \begin{array}{cccc}⋮& ⋮& \cdots & ⋮\end{array}\\ p_{S1}{p}_{S2}\cdots p_{SS}\end{array}\right]$$

(9)

$$\mathit{Q}=\left[\begin{array}{cccc}{p}_{11}& p_{11}+p_{12}& \cdots & p_{11}+p_{12}+p_{1S}\\ p_{21}& p_{21}+p_{22}& \cdots & p_{21}+p_{22}+p_{2S}\\ ⋮& ⋮& \ddots & ⋮\\ p_{S1}& p_{S1}+p_{S2}& \cdots & p_{S1}+p_{S2}+p_{SS}\end{array}\right]$$

(10)

Assuming that the original power state sequence satisfies the f(x) distribution, and the initial state x satisfies f(x) > 0, according to the proposed probability distribution $g\left(y|x\right)$ (denoting the probability of sampling a known x to obtain y), sampling an alternative state y, the acceptance probability of y is calculated as follows:

$$h(x,y)=\min \left\{1,{\displaystyle \frac{f(y)g(x|y)}{f(x)g(y|x)}}\right\}$$

(11)

where x is the current state, y is the alternative state, f is the target probability density function, and g is the proposed probability density function. When Equation (7) satisfies the symmetry condition, i.e., $g(y|x)=g(x|y)$, it can be simplified as follows:

$$h(x,y)=\min \left\{1,{\displaystyle \frac{f(y)}{f(x)}}\right\}$$

(12)

We accept the alternative state with probability $h\left(x,y\right)$ as the state value at the next moment, otherwise keep the original state unchanged. Applying the M-H algorithm to extract load and wind power sample spaces in power systems can accurately simulate the dynamics and random behavior of load and wind power changes over time. This method significantly enhances the modeling and analysis of power systems, especially in evaluating and planning for the integration of variable renewable energy sources and other load management strategies.

4. Evaluation of Renewable Energy Consumption Capacity Considering Safety and Stability Constraints

4.1. Assessment Model for Renewable Energy Consumption Capacity

This paper employs a probabilistic assessment method that fully considers the uncertainties of load and wind power. Since the fluctuations in load and wind power result in varying amounts of wind power accommodation and the ultimate integration capacity of wind power is constrained by system operating conditions, the evaluation model proposed in this paper is based on the following concept: taking into account the impact of uncertain factors, the model assesses the maximum renewable energy integration capacity under the static security constraints of the power system.

(1)

Objective Function

The assessment of renewable energy integration capacity typically selects either the system operating cost or the amount of renewable energy integration as the objective function. The former optimizes the system’s economic operating cost, considering various safety constraints, and determines the optimal integration of renewable energy by minimizing the system’s operating cost. The latter directly aims to maximize the amount of renewable energy integration, determining the system’s maximum integration capacity while ensuring all operational indicators remain within safe operating limits. According to the evaluation model’s concept, this paper uses the maximum renewable energy integration as the objective function.

$$\max \left(P_w+\sum _{i\in C}\mathsf{\Delta }{P}_{wi}\right)$$

(13)

where $P_w$ represents the initial power of renewable energy integration, $\mathsf{\Delta }{P}_{wi}$ denotes the additional renewable energy integration capacity in the ith increment, and C signifies the number of increments in renewable energy power.

(2)

Constraints

To ensure the safe and stable operation of the power system, the operational data of each node and branch within the system must meet safety operation constraints. These constraints are divided into equality constraints and inequality constraints. The equality constraints represent the power balance equations at the nodes. The formula is expressed as follows:

$$\begin{array}{l}{P}_i=U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\cos {\delta }_{ij}+B_{ij}\sin {\delta }_{ij}\right) \\ Q_i=U_i{\displaystyle \sum _{j=1}^n}{U}_j\left(G_{ij}\sin {\delta }_{ij}+B_{ij}\cos {\delta }_{ij}\right)\end{array}$$

(14)

The inequality constraints include the active and reactive power output constraints of conventional generators, node voltage magnitude constraints, node voltage phase angle constraints, and branch power transmission constraints. The inequality constraints are expressed as follows:

$$\begin{array}{c}{P_{Gi}}^{\min }<P_{Gi}<{P_{Gi}}^{\max }\\ {Q_{Gi}}^{\min }<Q_{Gi}<{Q_{Gi}}^{\max }\\ {U_i}^{\min }<U_i<{U_i}^{\max }\\ \left|{\delta }_i-{\delta }_j\right|<{\left|{\delta }_i-{\delta }_j\right|}_{\max }\\ \left|F_k\right|\le F_{k_{\max }}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} k\in \mathsf{\Omega }\end{array}$$

(15)

where ${P_{Gi}}^{\max }$ and ${Q_{Gi}}^{\max }$ are the maximum active and reactive power outputs of the conventional generator at node i; ${P_{Gi}}^{\min }$ and ${Q_{Gi}}^{\min }$ are the minimum active and reactive power outputs of the conventional generator at node i; $U_i$ is the voltage magnitude at node i; ${U_i}^{\max }$ is the maximum voltage magnitude at node i; ${U_i}^{\min }$ is the minimum voltage magnitude at node i; ${\delta }_i$ is the voltage phase angle at node i; ${\delta }_j$ is the voltage phase angle at node j; $F_k$ is the active power flow through branch k; $F_{k_{\max }}$ is the maximum allowable power flow through branch k; and Ω represents the set of all branches in the system.

The mathematical model of renewable energy consumption capacity assessment in this paper can be simplified as the following nonlinear optimization model:

$$\left\{\begin{array}{c}\max f(s)=P_{\mathrm{w}}+{\displaystyle \sum _{i\in C}}\mathsf{\Delta }{P}_{\mathrm{w}i}\\ \mathrm{s}.\mathrm{t}.g(s)=0\\ h_{\min }⩽h(s)⩽h_{\max }\end{array}\right.$$

(16)

where s is all state and control variables; P_w is initial power of renewable energy access; ΔP_wi is third increase of renewable energy access capacity; C is number of renewable energy power increases; g(s) is trend equation of the system; and h(s) is inequality constraints determined by the static safety constraints of the system, with a lower limit of h_min and an upper limit of h_max. Equation (16) integrates key components from the preceding equations—specifically, the objective function in Equation (13) that maximizes renewable energy integration while keeping state variables within safety limits, and the constraints in Equations (14) and (15) that establish power balance and operational limits—into a comprehensive nonlinear optimization model; with the trend equation g(s) capturing the system’s dynamic behavior as renewable energy capacity increases and linking directly to the constraints to ensure that as ΔP_wi increases, the system remains within predefined operational boundaries, while the inequality constraints h(s) derived from Equations (14) and (15) further ensure the system operates within safe limits, thus providing a rigorous framework for assessing the maximum renewable energy integration capacity while maintaining system stability and security.

4.2. Short-Circuit Ratio Verification

For a long time, short-circuit ratio, as a static analysis method, has provided an important reference for the planning and operation of power grids due to its simplicity and intuition. The multi-site short-circuit ratio for renewable energy, as suggested in the referenced literature [21], presents a more comprehensive M_RSCR calculation formula that removes all assumptions inherent in the traditional CIGRE short-circuit ratio derivation. This approach accounts for both amplitude and phase differences among electrical quantities across various nodes. It also considers the impact of reactive power from renewable energy generation equipment, making it applicable for evaluating and calculating voltage strength in diverse scenarios of multi-renewable energy site access systems. Figure 2 shows the simplified equivalent model of i (i = 1, 2, …, n) renewable energy stations simultaneously connected to the AC system.

In Figure 2, ${\dot{S}}_{\mathrm{RE}i}$, P_REi, Q_REi, and ${\dot{U}}_{\mathrm{RE}i}$ represent the apparent power, active power, reactive power, and voltage at the bus of renewable energy generation equipment/station i, respectively. ${\dot{Z}}_{ij}$ denotes the equivalent impedance between grid-connected points i and j, while ${\dot{Z}}_{ij}$ indicates the system-side equivalent impedance between the main grid-equivalent power source i and its associated grid-connected point j. The renewable energy grid-connected buses in Figure 2 can symbolize either the grid-side access points of renewable energy power generation equipment or the grid-connection points of renewable energy stations. Assuming the currents injected into the AC system by each renewable energy grid-connected bus are ${\dot{I}}_1$, ${\dot{I}}_2$, …${\dot{I}}_n$, then the nodal voltages at each grid-connected bus ${\dot{U}}_{\mathrm{REl}}$, ${\dot{U}}_{\mathrm{RE}2}$,…, ${\dot{U}}_{\mathrm{REn}}$, ${\dot{S}}_{\mathrm{RE}n}$ can be expressed as follows:

$$\left[\begin{array}{c}{\dot{U}}_{\mathrm{REl}}\\ {\dot{U}}_{\mathrm{RE}2}\\ ⋮\\ {\dot{U}}_{\mathrm{RE}n}\end{array}\right]=\left[\begin{array}{cccc}{\dot{Z}}_{\mathrm{eq}11}& {\dot{Z}}_{\mathrm{eq}12}& \cdots & {\dot{Z}}_{\mathrm{eq}1n}\\ {\dot{Z}}_{\mathrm{eq}21}& {\dot{Z}}_{\mathrm{eq}22}& \cdots & {\dot{Z}}_{\mathrm{eq}2n}\\ ⋮& ⋮& & ⋮\\ {\dot{Z}}_{\mathrm{eq}n1}& {\dot{Z}}_{\mathrm{eq}n2}& \cdots & {\dot{Z}}_{\mathrm{eq}nn}\end{array}\right]\left[\begin{array}{c}{\dot{I}}_1\\ {\dot{I}}_2\\ ⋮\\ {\dot{I}}_n\end{array}\right]$$

(17)

In the equation, ${\dot{Z}}_{\mathrm{eq}ij}$ denotes the element located in the i-th row and j-th column of the equivalent impedance matrix ${\dot{Z}}_{\mathrm{eq}ij}$ at the bus where renewable energy sources are interconnected with the AC grid. The short-circuit ratio is used to measure the relative magnitude between the system’s nominal voltage and the voltage generated by the equipment upon integration into the system. Reflecting on the physical significance described, the maximum renewable source connection ratio (M_RSCR) at the i-th renewable energy interconnection bus within the system is defined as follows:

$${\mathrm{M}}_{\mathrm{RSCRi}}={\displaystyle \frac{\left|{\dot{U}}_{Ni}\right|}{\left|{\dot{U}}_{REi}\right|}}={\displaystyle \frac{\left|{\dot{U}}_{Ni}\right|}{\left|{\dot{Z}}_{eqii}{\dot{I}}_i+{\displaystyle \sum _{j=1,j\ne i}^n}{\dot{Z}}_{eqj}{\dot{I}}_j\right|}}$$

(18)

where ${\dot{U}}_{Ni}$ is the nominal voltage at the i-th grid-connected bus node; ${\dot{U}}_{\mathrm{RE}i}$ is the voltage produced by the renewable energy generation equipment at node i, with the subscript RE denoting the renewable energy power station; and ${\dot{I}}_i$ is the short-circuit current supplied by the i-th renewable energy power station. The actual operating voltage at the i-th grid-connected bus node, ${\dot{U}}_i$ is derived by multiplying both the numerator and denominator of Equation (18) by $\left|{\dot{U}}_i^{\ast }/{\dot{Z}}_{\mathrm{e}\mathrm{q}ii}\right|$, calculated as follows:

$$\begin{array}{cc}\hfill {\mathrm{M}}_{{\mathrm{RSCR}}_i}& ={\displaystyle \frac{\left|{\dot{U}}_i^{\ast }{\dot{U}}_{\mathrm{N}i}/{\dot{Z}}_{\mathrm{eq}ii}\right|}{\left|{\dot{U}}_i^{\ast }{\dot{I}}_i+{\displaystyle \sum _{j=1,j\ne i}^n}\frac{{\dot{Z}}_{\mathrm{eq}ij}}{{\dot{Z}}_{\mathrm{eq}i}}{\dot{U}}_i^{\ast }{\dot{I}}_j\right|}}\hfill \\ & =\left|{\displaystyle \frac{{\dot{U}}_i^{\ast }{\dot{U}}_{\mathrm{N}i}}{{\dot{Z}}_{\mathrm{eq}ii}}}\right|/\left|{\dot{S}}_{\mathrm{RE}i}+{\displaystyle \sum _{j=1,j\ne i}^n}{ \mathsf{\Pi } \dot{}}_{ij}{\dot{S}}_{\mathrm{RE}j}\right|\hfill \end{array}$$

(19)

where ${\dot{S}}_{\mathrm{RE}i}$ represents the actual apparent power of renewable energy injected at the i-th renewable energy grid-connected bus node; ${ \mathsf{\Pi } \dot{}}_{ij}$ = ${\dot{Z}}_{\mathrm{eq}ij}$${\dot{U}}_i^{\ast }$/${\dot{Z}}_{\mathrm{eq}ii}$${\dot{U}}_j^{\ast }$ is the complex power conversion factor between renewable energy grid-connected buses i and j, indicating the phase and amplitude differences among the electrical quantities at each grid-side access point of renewable energy generation equipment or at each renewable energy station grid-connected node. Depending on the specific requirements, the M_RSCRG (maximum renewable source connection grid ratio) for the grid-side access point of the renewable energy generation equipment or the M_RSCRS (maximum renewable source connection station ratio) for the grid-connected point of the renewable energy station can be determined using Equation (19). It is noted that, when considering the connection of just a single renewable energy power station to the AC system, the corresponding renewable energy station short-circuit ratio (R_SCR) is defined as follows:

$${\mathrm{R}}_{\mathrm{SCR}}=\left|\begin{array}{c}{\dot{U}}^{\ast }{\dot{U}}_{\mathrm{N}}/{\dot{Z}}_{\mathrm{eq}}\end{array}\right|/\left|\begin{array}{c}{\dot{S}}_{\mathrm{RE}}\end{array}\right|$$

(20)

where ${\dot{Z}}_{\mathrm{eq}}$ = R + jX represents the system-side equivalent impedance from the grid-side access point of the renewable energy generation equipment or the connection point of the renewable energy station.

In this paper, this quantitative index is used to carry out safety and stability verification of the maximum capacity of renewable energy field station access under the condition of satisfying the system safety operation constraints.

4.3. Assessment Process for Considering Safety and Stability Constraints in the Consumption Capacity Evaluation

Based on the above analysis, a set of probabilistic assessment methods for renewable energy consumption capacity and an assessment flow system for system security and stability analysis can be derived, as shown in Figure 3.

5. Case Analysis

5.1. Probabilistic Modeling of Load

While higher-order GMM models tend to offer greater accuracy, balancing modeling precision with computational efficiency is crucial. Thus, this paper opts for a 10th-order Gaussian mixture model for load modeling and calculation. The load’s probability density function is defined as follows:

$$f_{\mathrm{L}}(x)={\omega }_1{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_1}}{\mathrm{e}}^{-{\displaystyle \frac{{(x-{\mu }_1)}^2}{2{\sigma }_1^2}}}+{\omega }_2{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_2}}{\mathrm{e}}^{-{\displaystyle \frac{{(x-{\mu }_2)}^2}{2{\sigma }_2^2}}}+\dots +{\omega }_{10}{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{10}}}{\mathrm{e}}^{-{\displaystyle \frac{{(x-{\mu }_{10})}^2}{2{\sigma }_{10}^2}}}$$

(21)

In this paper, the 5th-order Gaussian mixture model, the 8th-order Gaussian mixture model, the 10th-order Gaussian mixture model, the 12th-order Gaussian mixture model, the 15th-order Gaussian mixture model, and the normal distribution are selected to fit the probability density curves of the load measured data, respectively. The selection of these specific model orders is based on a balance between model accuracy and computational efficiency. The 5th-order model was chosen as it provides a simple yet reasonably accurate fit, capturing the basic structure of the data. The 8th-order model was introduced to improve upon the 5th-order, addressing some of the complexities in the data that the lower-order model might miss. The 10th-order model, identified through preliminary analysis, was expected to provide the best overall fit, balancing accuracy and computational demands. The 12th-order model was added to examine the effects of a slight increase in model complexity beyond the 10th-order, though it introduces some risk of overfitting, particularly evident in its slight overestimation of the highest peak. Finally, the 15th-order model was included to explore the impact of higher complexity on the model’s accuracy and computational cost. These selections allow for a comprehensive comparison of the fitting effects, providing insights into the trade-offs between model complexity, accuracy, and computational efficiency.

Figure 4 shows the probability density distribution curves of the load measured data and the six selected fitting models. As can be seen from Figure 4, the probability density distribution curve of the load measured data contains two peaks. The 5th-order GMM and the normal distribution both underestimate these two peaks of the original data. The 8th-order GMM improves the fit over the 5th-order, providing a better approximation to the lower peak, but it still slightly underestimates this peak. The 10th-order GMM achieves the best overall fit, accurately capturing both peaks with minimal deviation from the original data. The 12th-order Gaussian mixture model, while offering a good fit, slightly overestimates the highest peak, which reduces its overall accuracy compared to the 10th-order model. The 15th-order GMM also overestimates the highest peak, though it is in good agreement with the rest of the distribution. To verify the accuracy of these models more convincingly, the fitting indexes of each model were analyzed.

The metrics for evaluating a model’s fit to reality are root mean square error (RMSE), summed squared error (SSE), and coefficient of determination (R²). The closer the RMSE and SSE are to 0, the better the model fits the original data, and the closer R² is to 1, the better the ability to explain the original data, and the formula is as follows:

$$\begin{array}{l}\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left({\widehat{x}}_i-x_i\right)}^2}\\ \mathrm{SSE}={\displaystyle \sum _{i=1}^n}{({\widehat{x}}_i-x_i)}^2\\ R^2={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{({\widehat{x}}_i-{\overline{x}}_i)}^2}{{\displaystyle \sum _{i=1}^n}{(x_i-{\overline{x}}_i)}^2}}\end{array}$$

(22)

where ${\widehat{x}}_i$ is sample data; $x_i$ is raw data; ${\overline{x}}_i$ is average of raw data.

Table 1. Accuracy evaluation index of load model.

Fit Model	RMSE	SSE	R2
normal distribution	0.000753	0.000318	0.963
5th-order Gaussian distribution	0.000560	0.000127	0.988
8th-order Gaussian distribution	0.000382	0.000106	0.991
10th-order Gaussian distribution	0.000159	0.000048	0.995
12th-order Gaussian distribution	0.000147	0.000027	0.993
15th-order Gaussian distribution	0.000132	0.000013	0.982

illustrates that, when weighing simulation accuracy against computational expense, the 10th-order GMM emerges as the optimal modeling method.

5.2. Probabilistic Modeling of Wind Power

Based on the analysis above, given the satisfactory fitting performance of the 10th-order Gaussian mixture model, the probability model for wind power output is also adopted as a 10th-order Gaussian mixture model. A probability model of the GMM is established using actual output power data from a wind farm.

The wind power model based on the Gaussian mixture distribution demonstrates better accuracy in fitting the probability density curves of the load-measured data compared to models that utilize the Weibull distribution. Both the Weibull distribution and GMM were used to fit the original data, with the fitting curves displayed in Figure 5. Clearly, although the probability distribution of wind power is similar to the Weibull distribution, the Weibull distribution fails to capture the smaller fluctuations in wind power. This limitation is evident in Figure 5, where the Weibull distribution underestimates the short-term variations that arise due to local atmospheric conditions or turbulence. In contrast, the Gaussian mixture distribution, due to its flexible and versatile structure, accurately models these fluctuations. A comprehensive analysis of Figure 5 and

Table 2. Evaluation of wind power model accuracy.

Fit Model	RMSE	SSE	${\mathit{R}}^2$
Weibull distribution	0.000533	0.000258	0.9677
10th-order Gaussian distribution	0.000163	0.000052	0.9989

confirms that the Gaussian mixture distribution is more suitable for fitting the probability distribution of wind power than the Weibull distribution.

5.3. Validity Test of Renewable Energy Consumption Capacity Assessment Method

To validate the effectiveness of the proposed methodology, we applied the Metropolis–Hastings (M-H) algorithm to a modified IEEE 39-bus system to evaluate its ability to integrate renewable energy sources. The method’s accuracy was subsequently corroborated using Monte Carlo simulation (MCS) as a benchmark. Additionally, Gibbs sampling was employed to provide a further comparison and validation of the results. In this simulation, the active and reactive power at nodes 21 and 23 followed Gaussian mixture distributions, derived from actual load data measurements. Moreover, the conventional generator at node 36 was replaced by a wind farm, with the farm’s output modeled as a random variable also adhering to a Gaussian mixture distribution. To further analyze the impact of stochastic variations in load and wind on the system’s capacity to accommodate renewable energy, node 35 was designated as a renewable energy access point with a power factor of 0.94, and the capacity for renewable energy was incrementally increased by 2 MW in each calculation iteration. The modified IEEE 39-bus system is shown in Figure 6.

A sampling space for active load and wind power was created by sampling from the load and wind power variables’ probability density functions. These samples were then used sequentially to adjust the power at the renewable energy access points, determining the maximum capacity of renewable energy that could be safely supported. Finally, kernel density estimation was employed to illustrate the probability density curves for renewable energy absorption.

As shown in Figure 7 and Figure 8, the probability density curves generated by the Metropolis–Hastings (M-H) algorithm, the Monte Carlo simulation (MCS) method, and the Gibbs sampling method are nearly identical, indicating that all three methods provide a consistent and accurate assessment of the fluctuations in renewable energy consumption in response to changes in load and wind power. This comparison highlights the reliability of the M-H algorithm, which was validated against the well-established MCS method and further corroborated by the Gibbs sampling approach.

Table 3. Statistical indicators of three assessment methods.

Algorithm	Number of Samples	Execution Time/ms	Mean/MW	Standard Deviation/MW	Maximum/MW	Min/MW
MCS	10,000	0.324	801.14	79.49	1088.94	518.04
M-H	2000	0.169	800.71	80.83	1054.87	535.96
Gibbs	2000	0.175	798.39	75.67	1035.28	558.37

presents the probabilistic statistical indicators of renewable energy consumption obtained from all three assessment methods, further confirming the accuracy and consistency of the M-H algorithm in evaluating renewable energy integration.

As can be seen from Table 3, when the sampling scale of the MCS method is 10,000 and the sampling scale of the M-H algorithm is 2000, the probability indices of the M-H algorithm and the results of the MCS method are very close to each other, which verifies the accuracy of the proposed method and determines the maximum renewable energy carrying capacity of this system to be 1054.87 MW. Additionally, the Gibbs sampling method, with a sampling scale of 2000, also produces consistent results. Notably, the M-H algorithm not only achieves high accuracy but also requires the least execution time among the methods compared, demonstrating its efficiency and effectiveness.

When the renewable energy field station output in the system is 1054.87 MW, its R_SCR can be calculated according to Equation (20) as follows:

$${\mathrm{R}}_{\mathrm{SCR}}=\left|\begin{array}{c}{\dot{U}}^{\ast }{\dot{U}}_{\mathrm{N}}/{\dot{Z}}_{\mathrm{eq}}\end{array}\right|/\left|\begin{array}{c}{\dot{S}}_{\mathrm{RE}}\end{array}\right| = 2.79$$

(23)

This result not only satisfies the condition of R_SCR > 2.5 proposed in the literature [16], indicating that the system still maintains high voltage stability and safety margin under a larger proportion of renewable energy access, but also further confirms the validity and reliability of the proposed evaluation method.

In order to assess the security and robustness of the power system under maximum renewable energy access conditions, an N-1 fault validation is necessary. This process simulates the power system’s ability to respond and adjust in the event of failure of any single component (e.g., substation, transmission line, etc.) to ensure that the stability of the power system is not affected by a single point of failure. Under the maximum renewable energy access capacity, this paper simulates an N-1 fault on lines 4–14 in the IEEE 39-bus system, whose fault type is a three-phase ground fault with a fault onset time of 2 s. After the fault, the changes in each generator parameter are shown in Figure 9, Figure 10 and Figure 11.

After the N-1 fault simulation of the modified IEEE 39-bus system, the analysis of the three parameter variation plots shows that the response of each generator in the system exhibits significant transient fluctuations after the occurrence of a three-phase ground fault. Firstly, Figure 9 demonstrates that the rotational speed undergoes a brief overshoot and then stabilizes after the fault, which reflects that the generators have sufficient inertia and regulation to resist the transient perturbations. Figure 10 shows that the generators’ power angle differences increase significantly during the initial phase of the fault, posing a brief challenge to system stability, but gradually return to a more stable state after the fault is cleared. Figure 11 shows that the voltages of all generators show a sharp drop, followed by a gradual recovery of the voltage to a level close to the pre-fault level. It shows that the system has the ability to cope with N-1 faults and maintain stable operation under the calculated maximum carrying capacity of the renewable energy source.

Figure 12 represents a power system that integrates renewable energy and conventional power sources, transmitting electricity externally through DC transmission. The system consists of 102 nodes, with the 500 kV main grid comprising 42 nodes. It includes three DC transmission lines: LCC1, LCC2, and LCC3, each with a transmission capacity of 800 MW. Through these three DC transmission channels, the system stably transmits a total of 2400 MW of electricity. To further validate the effectiveness of the proposed methodology, the M-H algorithm was applied to the system to assess its capacity for integrating renewable energy. The method’s accuracy was subsequently verified using both MCS and Gibbs sampling as benchmarks. In this simulation, the active and reactive power of the loads on bus 1B-1 were modeled using a GMM derived from actual load data measurements. Similarly, the output of the wind power plant WTGA-5 was also modeled using a GMM, as shown in Figure 5. To analyze the impact of stochastic variations in load and wind power on the system’s ability to accommodate renewable energy, Gen1B-1 was designated as a renewable energy access point with a power factor of 0.94. The renewable energy capacity was incrementally increased by 2 MW in each calculation iteration.

To more intuitively understand the system’s performance under different conditions, we performed kernel density estimation on the results of the three algorithms and plotted the probability density distribution curves and cumulative distribution curves. As shown in Figure 13 and Figure 14, the probability density curves of the M-H algorithm, Gibbs sampling, and the MCS method basically overlap, indicating that the M-H algorithm can achieve accuracy comparable to both the MCS method and Gibbs sampling.

As seen in

Table 4. Statistical indicators of three assessment methods.

Algorithm	Number of Samples	Execution Time/ms	Mean/MW	Standard Deviation/MW	Maximum/MW	Min/MW
MCS	10,000	0.379	756.58	289.31	1253.11	254.43
M-H	2000	0.182	757.88	286.69	1250.91	256.38
Gibbs	2000	0.237	758.39	291.67	1259.28	259.15

, all three methods yield very similar statistical indicators, further validating the effectiveness of the M-H algorithm. Notably, Table 4 also shows that the M-H algorithm has the shortest execution time, making it more efficient than both the MCS method and Gibbs sampling.

When the renewable energy field station output in the system is 1250.91 MW, its R_SCR can be calculated according to Equation (20) as follows:

$${\mathrm{R}}_{\mathrm{SCR}}=\left|\begin{array}{c}{\dot{U}}^{\ast }{\dot{U}}_{\mathrm{N}}/{\dot{Z}}_{\mathrm{eq}}\end{array}\right|/\left|\begin{array}{c}{\dot{S}}_{\mathrm{RE}}\end{array}\right| = 2.74$$

(24)

This result also satisfies the condition of RSCR > 2.5 proposed in the literature [16].

Similarly, under the maximum renewable energy access capacity, this paper simulates an N-1 fault on bus 1B-8 in the AC–DC hybrid system shown in Figure 12. The fault type is a three-phase ground fault with a fault onset time of 2 s. After the fault, the changes in bus voltages, generator active power, and generator speed are shown in Figure 15, Figure 16 and Figure 17. After the N-1 fault simulation of the AC–DC hybrid system, Figure 15 shows that the bus voltage fluctuates transiently when the fault occurs, which is manifested as an instantaneous drop, and then gradually recovers and tends to stabilize, and recovers to a level close to the pre-fault level, which indicates that the system has a good voltage recovery capability and steady-state performance; Figure 16 shows that the active power of each generator fluctuates to varying degrees at the time of the fault occurrence, especially that some of the generators (e.g., Gen1B-4 and Gen1B-6) showed a significant drop in active power and then a rapid recovery, after the fault is cleared, the active power of the generators gradually recovers and tends to a new steady state, which indicates that the system is able to re-distribute the power load after the fault and gradually return to a steady state; Figure 17 shows that the generator speed remains basically unchanged before and after the fault, showing a horizontal straight line, which indicates that the generator speed is not significantly affected in the simulation of the N-1 fault, indicating that the system has strong inertia and regulation ability, and is able to effectively resist the transient shocks caused by the fault, and maintain the stable operation of the generator. In summary, the simulation results show that the AC–DC hybrid system exhibits good voltage recovery capability, power load redistribution capability, and generator stability under N-1 fault, which verifies the safety and robustness of the system under the maximum renewable energy access capacity.

6. Conclusions

This paper proposes a probabilistic assessment method for evaluating the renewable energy integration capacity, considering the uncertainty of renewables and the symmetry of active power. The proposed method achieves high simulation accuracy for the uncertainties of node injection powers, such as load and wind power. Compared to existing renewable energy integration assessment methods, the proposed method has the following characteristics:

• Through the simulation of the random characteristics of load and wind power and the calculation of probabilistic load flow, the proposed method can comprehensively and accurately reflect the changes in renewable energy integration capacity influenced by the operating state of the power system and its uncertainties.
• A 10-order GMM is used to construct the probabilistic model for load. The research results show that the RMSE and SSE of the 10-order Gaussian distribution load fitting model are 0.000159 and 0.000048, respectively, with a R² reaching 0.995. Compared to other fitting models, this approach provides more accurate load modeling. Similarly, the 10-order Gaussian distribution wind power fitting model has an RMSE and SSE of 0.000163 and 0.000052, respectively, with an R² reaching 0.994. Compared to the Weibull distribution, this model offers more precise wind power modeling.
• During the simulation calculations on both the 102 nodes system and IEEE 39-bus system with renewable energy integration, the maximum renewable energy integration capacities were determined to be 1250.91 MW and 1054.87 MW, respectively. The results of different simulation case studies also show that the probability metrics obtained by the M-H algorithm, Gibbs sampling, and the MCS method are basically the same, verifying the accuracy of the proposed method.
• Under the maximum renewable energy integration capacity in different scenarios, it was found through short-circuit ratio and N-1 fault validation that the system can consistently meet stability requirements.

While the proposed method offers a comprehensive and accurate assessment of renewable energy integration capacity, it is not without limitations. One limitation is the computational intensity associated with high-order Gaussian mixture models, which may lead to increased processing times in larger systems or more complex scenarios. Additionally, the method’s reliance on accurate probabilistic models means that its effectiveness is contingent upon the quality of the input data, which can be challenging in cases of sparse or highly variable data. To address these limitations, future research could explore the development of more efficient computational techniques or hybrid models that balance accuracy with computational demand. Additionally, extending the methodology to incorporate other types of uncertainties, such as those related to grid infrastructure or market conditions, could further enhance its applicability. Researchers might also consider the integration of machine learning techniques to improve the accuracy of input data modeling and reduce dependency on large datasets.

In conclusion, the proposed probabilistic assessment method represents a significant step forward in renewable energy integration analysis, offering a robust framework for future studies while highlighting areas for continued improvement.