An Efficient Method for the Reliability Evaluation of Power Systems Considering the Variable Photovoltaic Power Output

Abstract

The operational reliability of power systems is threatened by the random failure of components and uncertain power output of renewable energies, such as photovoltaics. Under such circumstances, reliability evaluation is necessary for maintaining a continuous and stable energy supply. However, traditional reliability evaluation methods are usually extremely time-consuming, considering the numerous system states that need to be analysed. Hence, the reliability evaluation process cannot follow up the dynamic changes in PV output, which makes the timeline of the evaluation disappointing. This paper proposes an efficient reliability evaluation method for power systems with PV integration. The method reveals the analytical relationship between the reliability levels of the power system and the uncertainty factors that influence the reliability, such as the PV output. In this way, the dynamic reliability evaluation is achieved, and the evaluation results can be updated timely when the output of PV changes. First, a Gaussian mixture-hidden Markov model (GMM-HMM) is used to model the distribution characteristics of PV output. Then, the state enumeration and the hyperbolic truncated polynomial chaos expansion method are used to determine the analytical relationship between the reliability indices and PV output. Lastly, based on the analytical function, the operational reliability of the power systems is dynamically evaluated considering the real-time PV output. The effectiveness of the proposed method is verified using the modified IEEE 30 system as an example.

1. Introduction

Photovoltaic (PV) and other types of renewable energy sources are increasingly integrated into power systems in order to address energy and environmental issues. In some provinces in China, such as Shandong province, the accumulative installed capacity of PV has reached 42.699 GW, which accounts for 22.5% of the total power generation capacity [1]. The fluctuation of PV output, along with the random failures of the components, increases the risk of power supply interruptions [2]. Hence, maintaining the operational reliability of power systems with the integration of PV becomes a challenging task. The key to ensuring the reliable operation of the power system is to propose an efficient operation reliability evaluation method that evaluates and forecasts the system’s reliability in time and provides operators with a quantitative basis for formulating operation and maintenance plans.

The methods for evaluating system reliability can be roughly divided into two categories: the Monte Carlo simulation (MCS)-based methods [3] and the state-enumeration (SE)-based method. For example, Ref. [4] proposes a state enumeration method based on a Lagrange multiplier to accelerate the reliability evaluation considering multiple load types and renewable energy generation. Ref. [5] introduces the application of a state-enumeration-based reliability evaluation method in power systems. Ref. [6] presents a bilevel optimisation model for power system risk evaluation. The proposed method reduces the computational burden by not enumerating all contingencies. Ref. [7] proposes a multi-scenario risk-oriented clustering algorithm considering renewable energy. The enumeration method is used to evaluate power system reliability for different scenarios. Ref. [8] proposed an improved method combining MCS and parallel computing, which can quickly evaluate the reliability of the power system. Ref. [9] integrates the Markov process and MCS method to evaluate the reliability of the power system.

Due to the optimal power flow (OPF) state analysis for large-scale system states, the SE and MCS methods are computationally intensive. One of the valuable suggestions for improving these methods is increasing the sampling efficiency and decreasing the number of system states that need to be calculated. Some researchers use enhanced MCS methods to enhance computational efficiency. For example, Ref. [10] proposes a method for evaluating reliability based on the non-parametric importance of stratified sampling and the MCS reliability evaluation techniques. Ref. [11] effectively reduces the number of system states by the impact-increment method and state enumeration method.

The aforementioned studies take a giant stride forward in achieving efficient reliability evaluation of power systems. Nevertheless, there are a couple of limitations related to previous studies that need to be addressed. Firstly, the aforementioned advanced variants of MCS methods still require extensive computations to estimate the small load-shedding probability. Lastly, and most importantly, those methods are not adapted to the variable renewable power, such as PV. To be specific, those methods evaluate the power system reliability based on the forecasted renewable power, which means the whole reliability calculation process needs to be performed again when the forecast of renewable power changes. Such a situation occurs frequently during the power system operation. For the above reasons, those reliability evaluation methods cannot follow up the dynamic changes in renewable power output and can be used only for the planning stage, which makes the timeline of the evaluation disappointing.

The contributions of this paper are as follows: First, this paper is focused on the dynamic reliability evaluation of power systems with PV integration and proposes an efficient reliability evaluation method for power systems with PV integration. Second, a polynomial chaos expansion (PCE)-based method is customised to reveal the analytical relationship between the reliability levels of the power system and the PV output, wherein the dynamic characteristics of the PV output are described based on the Gaussian mixture-hidden Markov model (GMM-HMM). Moreover, the quick sorting technology is applied to screen the system states with high priority to further improve the reliability evaluation efficiency. Based on the analytical-based reliability evaluation method, this paper evaluates the critical components that, upon failure, will lead to serious load shedding based on the proportional allocation method. In this way, the proposed reliability evaluation method can be applied to the operational phase to achieve the dynamic reliability evaluation and critical component identification. On this basis, the reliability-oriented power system dispatch and emergency reinforcement can be decided to improve reliability.

The remainder of this paper is organised as follows: In Section 2, the distribution characteristics considering correlation and variability of PV are modelled by GMM-HMM. Section 3 establishes an operational reliability efficiency evaluation model based on the improved PCE method. In Section 4, calculations and analyses for test cases are performed. Section 5 gives the conclusion.

2. PV Model of the Power System

2.1. Definition of the HMM

The hidden Markov model (HMM) is a statistical model used to describe a Markov process with hidden, unknown state variables [12]. The state sequence of the system is recorded as $I=\{I_1,I_2,\dots ,I_T\}$, where T is the length of the observation sequence. Each state sequence is assigned a unique emission probability, which generates the observation sequence denoted by the symbol $O=\{O_1,O_2,\dots ,O_T\}$.

The HMM can be uniquely determined by the initial state probability vector π, the state transition probability matrix A, and the emission probability matrix B.

HMM is represented as:

$$\lambda =(\pi ,A,B)$$

(1)

where initial state probability vector π:

$${\pi }_i=P\left(I_1=i\right),1\le i\le N$$

(2)

State transition probability matrix A:

$$a_{ij}=P(I_{t+1}=j|P(I_t=i),1<i,j<N$$

(3)

The two basic assumptions of the HMM are:

Assumption 1.

The state I_t of the hidden Markov chain at any given time t depends only on the state I_t−1 of the previous time and is independent of the state or observations at other times.

$$P\left(I_t\mid I_{t-1},O_{t-1},\cdots ,I_1,O_1\right)=P\left(I_t\mid I_{t-1}\right),t=1,2,\cdots ,T$$

(4)

Assumption 2.

The observation O_t at any given time is dependent only on the state I_t of the Markov chain at time t and is independent of the observations and states at other times.

$$P\left(O_t\mid I_T,O_T,\cdots ,I_{t-1},O_{t-1},\cdots ,I_1,O_1\right)=P\left(O_t\mid I_t\right)$$

(5)

2.2. The PV Output Modeling Based on GMM-HMM

It is presumed that N unobservable scenarios exist for factors affecting PV output, such as the weather. The discrete natural numbers from 1 to N are used as state variables. Moreover, the output $O=\{O_1,O_2,\dots ,O_T\}$ of multiple PVs is utilised as an observation variable, where $O_T={[O_T^{}(1),O_T^{}(2),\dots ,O_T^{}(m)]}^T$ is the column vector comprised of m PV output at time T, and the hidden Markov model of PV output is depicted in Figure 1.

Given that the observation variable in the research is continuous, the emission probability from the state variable to the observation variable is no longer represented by the form of the observation probability matrix. Instead, it is described by the continuous probability distribution, which is specifically the joint probability distribution of the observation variable. The GMM-HMM is a common type of statistical learning model, and Ref. [13] has shown that it performs significantly better than the Gaussian HMM. The Gaussian hidden Markov model (HMM) uses the Gaussian distribution as the emission distribution in each hidden state. However, the Gaussian distribution does not adhere to the actual PV output distribution, which results in a significant amount of inaccuracy. When modelling the PV output with HMM, the mixed Gaussian distribution is employed to better fit the features of any shape distribution. This helps enhance the modelling accuracy of the PV output.

Assuming the mixed Gaussian model is composed of K Gaussian models, the Gaussian mixture model probability density function is as follows:

$$P(X\mid \theta )={\displaystyle \sum _{m=1}^K{\omega }_m}\varphi \left(X\mid {\theta }_m\right)$$

(6)

where ${\omega }_m$ is the coefficient of weight, ${\omega }_m⩾0$, and ${\displaystyle \sum _{m=1}^K{\omega }_m}=1$, ${\theta }_m=\left({\mu }_m,{\sigma }_{\kappa }^2\right)$, $\varphi \left(X\mid {\theta }_m\right)$ is the Gaussian distribution probability density function:

$$\varphi \left(X\mid {\theta }_m\right)=1/\sqrt{2\pi }{\sigma }_m\ast \exp \left(-\frac{{\left(X-{\mu }_m\right)}^2}{2{\sigma }_m^2}\right)$$

(7)

In this paper, the joint probability distribution of the observation variables is characterised by a multidimensional mixed Gaussian distribution. Therefore, the probability distribution of the observation variable can be explained as follows when the state at time t is i:

$$b_i={\displaystyle \sum _{m=1}^K{\omega }_m{N}_i({\mu }_m,{\sum }_m),}1<i<N$$

(8)

where $N_i({\mu }_m,{\sum }_m)$ is the m-th Gaussian distribution in the mixed Gaussian model when the state is i at time t. ${\mu }_m$ = {${\mu }_m^1$,${\mu }_m^2$,…,${\mu }_m^M$} and ${\sum }_m$ are the mean vector and covariance matrix, and M is the number of PV. Equation (8) shows that when the state at time t is i, the observation variable O_t follows a mixed Gaussian distribution with a weight coefficient of ${\alpha }_k$, a mean value of Gaussian distributions of all dimensions of ${\mu }_m$, and a covariance matrix of ${\sum }_m$.

2.3. Calculation of GMM-HMM Parameters

When the observed historical data are O = {O₁, O₂,…, O_T} in the training phase of the HMM model, the corresponding latent variable is denoted as I = {I₁, I₂,…,I_T}, and $\lambda =(\pi ,A,\mu ,\Sigma )$ is the model parameter that needs to be acquired. By using the maximum likelihood estimation method, the logarithmic likelihood function $L(\lambda )$ of O with respect to parameter $\lambda$ is maximised.

$$\widehat{\lambda }=\arg \underset{\lambda }{\max }\log L(\lambda )$$

(9)

$$L(\lambda )=\log P(O|\lambda )=\log {\displaystyle \sum _{I}P(O,I|\lambda )}=\log \left({\displaystyle \sum _{I}P(O|I,\lambda )}P(I|\lambda )\right)$$

(10)

Since the Equation (10) contains hidden variables, which makes it difficult to solve directly, the Baum–Welch algorithm is primarily used for the learning of HMM parameters [14]. The calculation process of the algorithm is as follows:

(1)

Set the initial value of the model parameter $\lambda$;

(2)

Step E, construct the Q function so that $L(\lambda )\ge Q(\lambda ,{\lambda }^{(i)})$. Among them, $Q(\lambda ,{\lambda }^{(i)})$ is the expectation of the log-likelihood function $\log P(O,I|\lambda )$ on the conditional probability distribution $P(I|O,{\lambda }^{(i)})$ of the hidden variable I under the known observation data O and the current parameter ${\lambda }^{(i)}$:

$$Q(\lambda ,{\lambda }^{(i)})=E_I\left[\log P(O,I|\lambda )|O,{\lambda }^{(i)}\right]={\displaystyle \sum _I^{}\log P(O,I|\lambda )P(I|O,{\lambda }^{(i)})}$$

(11)

$Q(\lambda ,{\lambda }^{(i)})$ can be represented by ${\eta }_{j,k}(t)$ and ${\zeta }_{i,j}(t)$, where:

$${\eta }_{{}_{j,k}}(t)=P\left(I_t=j,{\varphi }_j(t)={\psi }_{j,k}|O,\lambda \right)$$

(12)

$${\zeta }_{i,j}(t)=P\left(I_t=i,I_{t+1}=j|O,\lambda \right)$$

(13)

Here, ${\eta }_{{}_{j,k}}(t)$ represents the probability that I_t = j, given $\lambda$ and O; ${\zeta }_{i,j}(t)$ is the probability that the concealed state at time t is i, and the concealed state at time t + 1 is j. ${\eta }_{{}_{j,k}}(t)$ and ${\zeta }_{i,j}(t)$ can be calculated using the forward–backward algorithm to facilitate the solution.

(3)

Step M: Find the $\lambda$ value that maximises $Q(\lambda ,{\lambda }^{(i)})$ and update the model parameter $\lambda$ as:

$${\widehat{\pi }}_j={\displaystyle \sum _{k=1}^K{\eta }_{{}_{j,k}}(1)}$$

(14)

$${\widehat{a}}_{i,j}=\frac{{\displaystyle \sum _{t=1}^{T-1}{\zeta }_{i,j}(t)}}{{\displaystyle \sum _{t=1}^{T-1}{\displaystyle \sum _{k=1}^K{\eta }_{{}_{j,k}}(t)}}}$$

(15)

$${\widehat{w}}_{j,k}=\frac{{\displaystyle \sum _{i=1}^T{\eta }_{{}_{j,k}}(t)}}{{\displaystyle \sum _{j=1}^T{\displaystyle \sum _{k=1}^K{\zeta }_{i,j}(t)}}}$$

(16)

$${\widehat{\mu }}_{j,k}=\frac{{\displaystyle \sum _{t=1}^T{\zeta }_{i,j}(t)}{O}_t}{{\displaystyle \sum _{t=1}^T{\zeta }_{i,j}(t)}}$$

(17)

$${\displaystyle \sum _{j,k}=}\frac{{\displaystyle \sum _{t=1}^T{\eta }_{{}_{j,k}}(t)}\left(O_t-{\mu }_{j,k}\right){\left(O_t-{\mu }_{j,k}\right)}^{\mathrm{T}}}{{\displaystyle \sum _{t=1}^T{\eta }_{{}_{j,k}}(t)}}$$

(18)

(4)

After obtaining the updated model parameters, it is determined whether the parameters satisfy the convergence conditions. If convergence is met, the algorithm terminates. If not, the second step is turned around.

2.4. Estimation of PV Output State Sequence

The Viterbi algorithm is an algorithm for dynamic programming. It finds the hidden state sequence most likely to produce the observed sequence. Define the variable δ_t(i): the maximum probability of all paths with state i at time t.

$${\delta }_t(i)=\underset{I_1,I_2,\cdots ,I_{t-1}}{\max }P\left(I_t=i,I_{t-1},\cdots ,I_1,O_t,\cdots ,O_1\mid \lambda \right)$$

(19)

$${\delta }_{t+1}(i)=\underset{1⩽j⩽N}{\max }\left[{\delta }_t(j)a_{ji}\right]b_i\left(O_{t+1}\right)$$

(20)

$${\theta }_t(i)=\arg \underset{1⩽j⩽N}{\max }\left[{\delta }_{t-1}(j)a_{ji}\right]$$

(21)

The calculation process of the Viterbi algorithm is as follows:

(1)

Initialise ${\delta }_1(i)={\pi }_i{b}_i(o_1)$, ${\theta }_1(i)=0$.

(2)

Calculate iteratively according to Equations (20) and (21).

(3)

Determine if it is iterative to time T. If not, proceed to (2); otherwise, proceed to the next step.

(4)

The optimal path is determined by retracing, according to Equation (22).

$$I_t^{}={\theta }_{t+1}(I_{t+1}^{})$$

(22)

3. The Dynamic Reliability Evaluation and Critical Component Identification Method for Power System

3.1. Production of Critical State Sets

In this paper, the reliability of power systems is estimated utilising the SE-based method. Due to the large number of power system components, calculating the system’s reliability using the SE-based method is impractically time-consuming. It is necessary to reduce the power system’s state in order to improve reliability evaluation efficiency.

The system state quick sorting technology [15] can rapidly select S system states with the highest probability of occurrence and avoid ignoring high-order faults with high probability. The system state quick sorting technology involves arranging the fault states of all components in descending order according to the probability coefficient, determining the minimal state set containing the next higher probability state based on the adjacent system state, and determining the next higher probability state with the least amount of calculation. Select until the evaluation precision or quantity requirements are met.

In this paper, the fast sorting method is used to quickly eliminate the first S states of the system with a high probability.

3.2. Polynomial Chaos Expansion

Polynomial chaos expansion (PCE) is an efficient uncertainty quantification theory. It has a solid mathematical foundation, excellent performance, and is widely used in aerospace, machinery, ships, and other fields [16].

The output response of the power system with random variables is uniquely determined by the system’s random variables. If the input variable X = [x₁,x₂,…x_n]^T, it can be represented by normal variables $\xi ={[{\xi }_1,{\xi }_2,\dots ,{\xi }_n]}^{\mathrm{T}}$. The output response Y can then be expressed as follows:

$$Y=c_0+{\displaystyle \sum _{i_1=1}^n{c}_{i_1}}{H}_1\left({\xi }_{i_1}\right)+{\displaystyle \sum _{i_1=1}^n{\displaystyle \sum _{i_i=1}^{i_1}{c}_{i_1{i}_2}}}{H}_2\left({\xi }_{i_1},{\xi }_{i_2}\right)+{\displaystyle \sum _{i_1=1}^n{\displaystyle \sum _{i_2=1}^{i_1}{\displaystyle \sum _{i_3=1}^{i_2}{c}_{i_1{i}_2{i}_3}}}}{H}_3\left({\xi }_{i_1},{\xi }_{i_2},{\xi }_{i_3}\right)+\cdots$$

(23)

where $c_0,c_{i_1},c_{i_1{i}_2},c_{i_1{i}_2{i}_3}$ is the undetermined coefficient, and H_m(·) is the m-order Hermite orthogonal polynomial that can be calculated using Equation (24):

$$H_m\left({\xi }_{i_1},{\xi }_{i_2},\dots ,{\xi }_{i_n}\right)={(-1)}^m{e}^{\frac{1}{2}{\xi }^T\xi }\frac{{\partial }^n}{\partial {\xi }_{i_1}\partial {\xi }_{i_2}\cdots \partial {\xi }_{in}}{e}^{-\frac{1}{2}{\xi }^T\xi }$$

(24)

The first 5-order Hermite orthogonal polynomials can be expanded according to the Equation (25):

$$\begin{array}{l}{H}_0(\xi )=1\\ H_1(\xi )=\xi \\ H_2(\xi )={\xi }^2-1\\ H_3(\xi )={\xi }^3-3\xi \\ H_4(\xi )={\xi }^4-6{\xi }^2+3\\ H_5(\xi )={\xi }^5-10{\xi }^3+15\xi \end{array}$$

(25)

Typically, the stochastic response surface method (SRSM) is used to solve non-embedded coefficient problems [17]. The projection method requires the use of Gaussian quadrature to estimate the expectation, which is more difficult and time-consuming. In this paper, the SRSM method is used to minimise the sum of squared errors between the predicted PCE value and the experimental data in order to determine the coefficient.

The PCE model can take p-order truncation, and the truncated output response Y can be abbreviated as:

$$Y\approx {\displaystyle \sum _{i=0}^P{b}_i}{\mathrm{\Phi }}_i(\xi )$$

(26)

The PCE model needs to produce a series of sample points via experimental design and acquire the corresponding response value via simulation in order to construct the coefficient solution equation. In this paper, the LHS considering the input PDF is used for experimental design. The specific steps are as follows:

(1)

Use LHS to obtain samples $X={[x_1,x_2,\dots ,x_n]}^{{\rm T}}$ [18];

(2)

Use the Nataf method to obtain samples $\xi ={[{\xi }_1,{\xi }_2,\dots ,{\xi }_n]}^{{\rm T}}$ [19];

(3)

Calculate the corresponding response value $g(x^S)={[g\left(x_1^S\right),g\left(x_2^S\right),\dots ,g\left(x_n^S\right)]}^{\mathrm{T}}$;

(4)

Construct the solving equation of the following coefficient b.

$$\left[\begin{array}{cccc}{\Phi }_0\left({\xi }_1^s\right)& {\Phi }_1\left({\xi }_1^s\right)& \cdots & {\Phi }_P\left({\xi }_1^s\right)\\ {\Phi }_0\left({\xi }_2^s\right)& {\Phi }_1\left({\xi }_2^s\right)& \cdots & {\Phi }_P\left({\xi }_2^s\right)\\ ⋮& ⋮& & ⋮\\ {\Phi }_0\left({\xi }_N^s\right)& {\Phi }_1\left({\xi }_N^s\right)& \cdots & {\Phi }_P\left({\xi }_N^s\right)\end{array}\right]\left[\begin{array}{c}{b}_0\\ b_1\\ ⋮\\ b_P\end{array}\right]=\left[\begin{array}{c}g\left(x_1^S\right)\\ g\left(x_2^S\right)\\ ⋮\\ g\left(x_N^S\right)\end{array}\right]$$

(27)

where

$$\psi =\left[\begin{array}{cccc}{\Phi }_0\left({\xi }_1^s\right)& {\Phi }_1\left({\xi }_1^s\right)& \cdots & {\Phi }_P\left({\xi }_1^s\right)\\ {\Phi }_0\left({\xi }_2^s\right)& {\Phi }_1\left({\xi }_2^s\right)& \cdots & {\Phi }_P\left({\xi }_2^s\right)\\ ⋮& ⋮& & ⋮\\ {\Phi }_0\left({\xi }_N^s\right)& {\Phi }_1\left({\xi }_N^s\right)& \cdots & {\Phi }_P\left({\xi }_N^s\right)\end{array}\right],b=\left[\begin{array}{c}{b}_0\\ b_1\\ ⋮\\ b_P\end{array}\right],Y=\left[\begin{array}{c}g\left(x_1^S\right)\\ g\left(x_2^S\right)\\ ⋮\\ g\left(x_N^S\right)\end{array}\right]$$

(28)

The coefficients of the polynomial are

$$b={\left({\psi }^{\mathrm{T}}\psi \right)}^{-1}{\psi }^{\mathrm{T}}Y$$

(29)

3.3. Hyperbolic Truncated Polynomial Chaos Expansion Method

The output response of a general practical engineering system is primarily influenced by the input variables of each dimension and its low-order cross-terms and is less influenced by its high-order cross-terms so that some polynomial terms can be directly removed to reduce the amount of calculation.

When p is the order of a chaotic polynomial model, it must satisfy Equation (30).

$$a|\equiv \Vert a{\Vert }_1\equiv a_1+\cdots a_i+\cdots +a_d⩽p$$

(30)

where a represents the order of the univariate orthogonal polynomial that corresponds to the i-dimensional variable in the multivariate orthogonal polynomial; p represents the order of the polynomial chaos expansion model.

The total number of terms of orthogonal polynomials in Equation (26) is P.

$$P+1=(d+p)!/d!p!$$

(31)

In order to reduce the amount of calculation, the following hyperbolic truncation strategy Equation (30) is considered to eliminate some high-order intersection polynomial terms, thus decreasing the number of chaotic polynomial coefficients and the amount of calculation.

$$I_q^{d,p}=\left\{a\in N^d:\Vert a{\Vert }_q={\left({\displaystyle \sum _{i=1}^d{a}_i^q}\right)}^{\frac{1}{q}}⩽p\right\}$$

(32)

where q is a custom sparse factor; i is the set of a satisfying the inequality.

Taking p = {2, 3} and q = {0.5, 1} as examples, the order combination generated by the above truncation method in two-dimensional space is shown in Figure 2.

The set of terms satisfying the inequality requirement of Equation (32) is represented by the blue circle in Figure 2. The hyperbolic truncation strategy eliminates some of the high-order cross-terms, resulting in a significant reduction in calculation time.

3.4. Load Loss Calculation Model

In this paper, the reliability levels of power systems are characterised by the expected energy not supplied (EENS).

$$EENS=8760\ast {\displaystyle \sum _{i\in S}^{}{\alpha }_i\cdot E_i\cdot P_i}$$

(33)

where P_i is the probability of the system under state i; S is a set of critical states; E_i represents the load loss of the system under state i. And,

$${\alpha }_i=\left\{\begin{array}{l}1,{\mathrm{E}}_i>0\\ 0,{\mathrm{E}}_i\le 0\end{array}\right.$$

(34)

When a power system outage occurs, the optimal power flow (OPF) model is used to schedule to minimise as much as possible the load shedding of the power system under the constraints of power balance and line capacity. The reliability levels of the power system are computed using the quantity of load shedding during a system outage and the probability of its occurrence.

Due to the large number of states that must be calculated during the reliability evaluation of a power generation and transmission system, the optimal power flow model based on the DC power flow is frequently used to reduce the number of calculations. It can be expressed mathematically as Equations (35)–(40).

$$\min {\displaystyle \sum _{i\in ND}^{}{C}_i}$$

(35)

Subject to

$$T\left(S\right)=A\left(s\right)\left(PG-PD+C\right)$$

(36)

$${\displaystyle \sum _{i\in NG}P{G}_i+{\displaystyle \sum _{i\in ND}{C}_i={\displaystyle \sum _{i\in ND}P{D}_i}}}$$

(37)

$$P{G}_i^{\min }\le P{G}_i\le P{G}_i^{\max }\left(i\in NG\right)$$

(38)

$$0\le C_i\le P{D}_i\left(i\in ND\right)$$

(39)

$$\left|T_k\left(S\right)\right|\le T_k^{\max }\left(k\in L\right)$$

(40)

where $T\left(s\right)$ is the line active power flow under state S; $A\left(s\right)$ is the correlation matrix between the line active power vector and the node injection power under state S; $PG$ is the output active power of the generator set; $PD$ is the load power of the load bus; $C$ represents the load loss of load buses; and $P{G}_i$, $P{D}_i$, $C_i$, and $T_k\left(s\right)$ are the elements in $PG$, $PD$, $C$, and $T\left(s\right)$, respectively. $P{G}_i^{\min }$, $P{G}_i^{\max }$, and $T_k^{\max }$ are the maximum values of $P{G}_i$ and $T_k\left(s\right)$, respectively. $NG$, $ND$, and $L$ are the set of power bus, load bus, and branch, respectively.

3.5. The Dynamic Reliability Evaluation Model

The algorithm for dynamically evaluating the operation reliability of a new energy power system proposed in this paper consists primarily of offline modelling and an online calculation model.

The process of offline modelling is as follows:

(1)

The potential distribution function of the PV is derived using GMM-HMM.

(2)

The S system states with the greatest probability are chosen to compose the critical states set.

(3)

For each possible distribution function, all critical states are enumerated, and the analytical function between PV output and reliability levels for each state is determined.

After offline modelling is complete, the system’s reliability levels for each critical state and the analytical function between PVs can be determined. It is not necessary to recalculate the OPF model if the output of PV in the power system varies.

The process of online calculation of power system operation reliability proposed in this paper is as follows:

(1)

For the current PV output, the Viterbi algorithm is utilised to calculate the corresponding hidden state and GMM-HMM parameters.

(2)

The reliability levels of the time are calculated according to the calculated polynomial coefficient.

The mean value of the period’s reliability levels can be determined by sequentially calculating. The dynamic evaluation algorithm of the power system operation reliability is shown as Algorithm 1.

Algorithm 1: Dynamic evaluation algorithm of the power system operation reliability
Input:	PV historical output, PV forecast output, system parameter.
Output:	The mean value of operation reliability levels.
Step1	Set the number of hidden states N.
Step2	Using the GMM-HMM, the potential distribution function of the PV is derived.
Step3	Generating critical state set.
Step4	for i = 1:N do
Step5		for j = 1:S do
Step6			Generate collocation points by sampling the Latin hypercube of the standard normal distribution.
Step7			The Nataf transformation generates the PV output value that corresponds to the collocation point.
Step8			Calculate the OPF to figure out the reliability levels value.
Step9			Compute the coefficients of chaotic polynomials Yij.
Step10		end for
Step11	end for
Step12	for t = 1:T do
Step13		Calculate the hidden state I(t) that corresponds to the actual PV output at time t using the Viterbi algorithm.
Step14		Select the polynomial coefficients according to I(t).
Step15		Transform the output of the PV at time t using the Nataf transformation into the value of the standard normal distribution.
Step16		Calculate the reliability levels at time t.
Step17	end for
Step18	Calculate the mean value of reliability levels.

3.6. The Critical Component Identification Model

After finishing the power system’s reliability evaluation, the dynamic reliability levels of all critical system states can be obtained. Therefore, the critical components of the power system can be quickly identified with the aid of reliability tracking, which provides a basis for reliability-oriented power system dispatch and emergency reinforcement of the reliability of the power system. Reliability tracking requires assigning reliability levels to each component and quantifying the number of reliability levels that each component should contain.

This paper utilises the proportional allocation method for tracking reliability. The method is predicated on two fundamental principles:

(1)

The unreliability of the system is attributable to the faulty components; the normal operation components do not contribute to the allocation of system reliability levels.

(2)

Component allocation is proportional to the system reliability levels.

Based on these two principles, the change in reliability levels caused by the second-order failure event k should be shared by the fault components x₁ and x₂.

$$f\left(k\to 1\right)=\frac{f\left(x_1\right) }{f\left(x_1\right)+f\left(x_2\right)}f\left(k\right)$$

(41)

$$f\left(k\to 2\right)=\frac{f\left(x_2\right) }{f\left(x_1\right)+f\left(x_2\right)}f\left(k\right)$$

(42)

where $f\left(k\to 1\right)$ and $f\left(k\to 2\right)$ represent the reliability levels that should be allocated to the two failure components. Equation (40) depicts the calculation of component reliability parameters.

$$f\left(x_i\right)=\frac{{\lambda }_i}{{\lambda }_i+{\mu }_i}$$

(43)

where ${\lambda }_i$ is the failure rate of component i; ${\mu }_i$ is the repair rate of component i.

4. Example Analysis

4.1. Example Setting

This paper uses the modified IEEE 30-bus power system as an illustration. PV was added at buses 12 and 22. Figure 3 depicts the structure of the power system. And numbers indicate the bus numbers and arrows indicate the load.

The Gurobi solver was used under MATLAB R2018a. The example was carried out on a computer equipped with Intel (R) Core (TM) i5-10500 CPU @ 3.10 GHz and 16.0 GB RAM.

The PV output of two localities in Belgium during the month of March and April is used as historical data, while the output during the first 4 days of May is used as the forecasted value [20]. The PV output during the first 4 days of May is depicted in Figure 4:

In order to demonstrate the validity of the method proposed in this paper, the results are contrasted against the results of the SE method, which enumerates the same critical system states. The calculation process of the SE method presented in this paper is as follows:

(1)

Enumerate the PV output sequence.

(2)

Enumerate the system critical states at time t.

(3)

Calculate the reliability levels of the power system.

The relative error of the reliability levels obtained by the two methods is contrasted in order to demonstrate the validity of the proposed method. The formula for calculating relative error is given in Equation (44):

$$err=\frac{1}{T}{\displaystyle \sum _{t=1}^T\frac{|Y_{t,SE}-Y_{t,PCE}|}{Y_{t,SE}}}\times 100\%$$

(44)

where $Y_{t,SE}$ represents the reliability levels values calculated by the SE method at time t; $Y_{t,PCE}$ represents reliability levels values calculated by the PCE-base method at time t.

$\overline{EENS}$ is used to represent the mean of EENS. The calculation method is as follows:

$$\overline{EENS}=\frac{1}{T}\ast {\displaystyle \sum _{t=1}^{T}EENS(t)}$$

(45)

4.2. The Comparison of the Reliability Evaluation Results

The number of critical states of system S is set to 3000. As shown in

Table 1. Comparison of PCE methods under different hidden states numbers.

Methods	err	$\overline{\mathit{E}\mathit{E}\mathit{N}\mathit{S}}$ (GWh/Year)	Offline Time (s)
PCEN=1	49.85%	132.76	2694.38
PCEN=2	11.63%	182.39	5440.85
PCEN=3	3.47%	176.68	8036.20
PCEN=4	2.29%	172.68	10,918.72
PCEN=5	0.40%	173.82	13,589.80
PCEN=6	0.81%	172.97	20,312.98
SE	-	174.1	34,587.58

, the values of the hidden states number N of various PV outputs are set, and the relative error, the mean value, and the offline calculation time between the PCE methods and the SE method are calculated.

When N = 1, which corresponds to setting the PV output to a Gaussian mixture distribution, Table.1 demonstrates that when N > 1, the calculated relative error is less than when N = 1, proving that the GMM-HMM method utilised in this paper can effectively increase the precision of the analytical modelling. Moreover, it is discovered that the relative error of the calculated EENS decreases as N increases. When N = 6, the accuracy improvement brought by increasing the PV output is not substantial, but the offline running time increases significantly, so this paper sets N = 5 as the number of PV output hidden states.

The truncation order of PCE is set to 2, and q = 0.5 in the HTPCE method.

Table 2. Comparison of the reliability calculation results of the three methods.

Methods	err	$\overline{\mathit{E}\mathit{E}\mathit{N}\mathit{S}}$ (GWh/Year)	Offline Time/s	Online Time/s
PCEN=5	0.40%	173.82	13,589.80	0.130
HTPCEN=5	0.92%	173.58	11,869.97	0.113
SE	-	174.1	34,587.58	360.29

provides a comparison of the mean value and calculation time of the predicted 4-day reliability levels obtained by the SE method, the PCE method, and the HTPCE method.

The dynamic reliability levels at each hour of the forecasted 4 days calculated by the PCE method, SE method, and HTPCE method for the power system are further compared in Figure 5. Figure 5 also shows the predicted 4-day PV output.

Figure 5 shows that the calculation results of the three methods are almost equal. Simultaneously, it can be observed that the higher the PV output, the lower the risk of load loss, and the higher the level of power system reliability. Compared to the SE method, the PCE method has a relative error of only 0.40%, but its offline calculation time is 39.29% of the SE method. This demonstrates that the PCE method can reduce calculation time while maintaining accuracy.

In comparison to the HTPCE and SE methods, the relative error is 0.92%. Even though the error is greater than the PCE method, this method still meets the requirements of engineering practise. Simultaneously, the offline calculation time using the HTPCE method is only 34.31% of the SE method, demonstrating the efficacy of the HTPCE method proposed in this paper.

4.3. The Critical Component Identification Results

In order to facilitate comparisons of the critical components, which are mostly responsible for the load-shedding risks, the components are designated in the order of branches, conventional units, and PVs. The correspondence between components and numbers is shown in

Table 3. Numbering of the power system components.

No.	Componet	No.	Componet	No.	Componet	No.	Componet	No.	Componet
1	branch1–2	11	branch6–9	21	branch16–17	31	branch22–24	41	branch6–28
2	branch1–3	12	branch6–10	22	branch15–18	32	branch23–24	42	unit of bus 1
3	branch2–4	13	branch9–11	23	branch18–19	33	branch24–25	43	unit of bus 2
4	branch3–4	14	branch9–10	24	branch19–20	34	branch25–26	44	unit of bus 5
5	branch2–5	15	branch4–12	25	branch10–20	35	branch25–27	45	unit of bus 8
6	branch2–6	16	branch12–13	26	branch10–17	36	branch28–27	46	unit of bus 11
7	branch4–6	17	branch12–14	27	branch10–21	37	branch27–29	47	unit of bus 13
8	branch5–7	18	branch12–15	28	branch10–22	38	branch27–30	48	PV of bus 12
9	branch6–7	19	branch12–16	29	branch21–22	39	branch29–30	49	PVof bus 22
10	branch6–8	20	branch14–15	30	branch15–23	40	branch8–28

. The proportional allocation method is used to calculate the reliability responsibility of each component, and the reliability responsibility index is computed by dividing each component’s reliability responsibility by the total reliability levels.

Based on the analytical-based method, Figure 6 depicts the outcomes of component reliability liability allocation for the forecasting days’ PV output. The 10 components with the largest share of responsibility are shown in Figure 6.

Figure 6 demonstrates that the reliability responsibility index of the branch between buses 2 and 5 is the highest and that reducing the branch’s failure rate can increase the power system’s reliability. For components with a greater impact on reliability, operators can implement an emergency reinforcement maintenance strategy for the component of the power system. In addition, operators can also formulate reliability-oriented power system dispatch methods based on the dynamic reliability evaluation methods and the critical component identification methods.

5. Conclusions

In this paper, the efficient evaluation of operational reliability and the critical component identification based on the PCE method are investigated for the PV-based new energy power system. The subsequent conclusions are reached:

(1)

The distribution characteristics of PV output are modelled using GMM-HMM, and the model is able to reflect the correlation of PV output more accurately.

(2)

On the basis of SE and PCE methods, the analytical function relationship between the PV output and the operation reliability levels of the power system is established. This eliminates the repeated calculation of the OPF model when the PV output varies and realises the dynamic operation reliability evaluation of the new energy power system as the PV output varies in real time. Compared to the SE method of the same order, the proposed method drastically reduces calculation time under the assumption that the error is insignificant.

(3)

This paper also examines the effectiveness of the HTPCE method in the proposed dynamic reliability evaluation. The HTPCE method can effectively minimise the time required during offline model training.

The method proposed in this paper can be used not only to deal with the uncertainty of photovoltaics but can also be extended to other uncertain variables in new energy power systems, such as wind power and load. When too many uncertain variables are taken into consideration, the method proposed in this paper still has the issue of relatively lengthy polynomial coefficient calculation times. In future research, the sparse polynomial method can be applied to the offline model to reduce the calculation time required offline.