Research on Synchronous Transfer Control Technology for Distribution Network Load Based on Imprecise Probability

Abstract

As the penetration rate of distributed power sources increases and distribution network structures grow increasingly complex, the uncertainty in switch action control during load transfer has become a critical issue affecting grid safety and reliability. Traditional control methods based on precise probability-based predictive control are susceptible to bias introduced by prior settings under small-sample conditions, making it difficult to meet the stringent requirements of time-synchronized control. To address this, this study proposes an imprecise probability-based synchronous load transfer control method for distribution networks. By integrating the Imprecise Dirichlet model (IDM) with a Naive Credal Classifier (NCC), it constructs an interval predictive control model for switching action timing. This approach effectively mitigates the prior dependency issue and enhances estimation robustness under small-sample conditions. Combined with a dynamic delay strategy, this approach strictly controls the interval between disconnection and reconnection actions within 20 ms, preventing circulating current risks and ensuring transfer reliability. The simulation and experimental results demonstrate that the proposed method outperforms traditional Bayesian classifiers in both time prediction control accuracy and model robustness, providing a theoretical foundation and a reference for engineering applications for secure action control in distribution networks.

1. Introduction

The increasing penetration of distributed generation (DG) in distribution networks enhances flexibility while introducing uncertainty into system action control. Among these factors, the unpredictability of switching action timing during load transfer has become a critical determinant of grid security and supply reliability. Traditional control methods predominantly rely on precise probabilistic models (e.g., Bayesian classifiers) for timing prediction. However, under conditions of limited samples or insufficient prior information, such approaches are susceptible to subjective bias and struggle to meet stringent requirements for “open-then-close” sequencing and millisecond-level time synchronization control.

Imprecise probability theory characterizes uncertainty through interval probabilities, making it particularly suitable for inference scenarios with small samples or incomplete information [1]. In recent years, it has demonstrated significant advantages in fields such as reliability assessment and fault diagnosis. Against this backdrop, this paper proposes an imprecise probability-based load transfer control method for distribution networks. It integrates the Imprecise Dirichlet Model (IDM) with the Naive Credal Classifier (NCC) [2] to construct an interval prediction model for switch action times. This approach mitigates over-reliance on prior information [3] and enhances estimation robustness under small-sample conditions [4].

This paper aims to systematically address the issue of insufficient time synchronization accuracy during load transfer control. By establishing a prediction model that better aligns with real-world uncertainties and incorporating a dynamic delay strategy, it achieves strict control of the switching time difference within 20 ms. This approach prevents circulating current risks and ensures transfer reliability. This research not only provides new theoretical support and a methodological framework for the safe action of distribution networks but also offers valuable insights for grid control under high penetration of renewable energy sources.

2. Literature Review

2.1. Research Status

With the advancement of new power system construction, the widespread integration of distributed generation (DG), energy storage devices, and flexible loads in distribution grids has made grid operation increasingly complex and dynamic. Traditional deterministic control methods struggle to address load synchronization and transfer issues under high uncertainty and small-sample scenarios. Against this backdrop, imprecise probability (IP) has gained increasing attention in both academic and engineering circles as a modeling tool capable of effectively capturing cognitive uncertainty and data scarcity.

Current research on load transfer control in distribution networks primarily focuses on optimization scheduling, intelligent algorithms, and reliability analysis. Traditional methods, such as control strategies based on Bayesian networks, Markov decision processes, or reinforcement learning, have enhanced system intelligence to some extent. However, under conditions of small sample sizes and high uncertainty, these approaches often suffer from issues like model overfitting and strong prior dependency.

To address these challenges, researchers have increasingly incorporated imprecise probability theory in recent years. Walley [5] systematically proposed a mathematical framework for imprecise probability, expressing uncertainty through upper and lower probability intervals to avoid the excessive reliance on a single prior characteristic of traditional probability models. Bernard [6] further introduced the Imprecise Dirichlet Model (IDM) to estimate probability intervals for multinomial distributions under small-sample conditions, significantly enhancing model robustness.

In classification and decision control, Zaffalon [7] introduced the Naive Credal Classifier (NCC), which employs credal sets to enable multi-class outputs for uncertain samples, effectively reducing misclassification risks. Corani & Zaffalon [8] validated NCC’s superiority in small-sample classification within medical diagnostics, providing theoretical support for its application in power systems.

Regarding model architecture, Cozman [9] introduced Credal Networks, extending Bayesian networks into the domain of imprecise probability. This allows network nodes to represent probability intervals, making them suitable for inference and decision-making in highly uncertain environments. Masegosa & Moral [10] further explored integrating IDM with Credal Networks to learn uncertain probability models from small-sample data, offering novel insights for modeling distribution network control systems.

Although these studies are theoretically mature, their application in load synchronization transfer control for distribution networks remains nascent. Existing research primarily focuses on state estimation, fault diagnosis, or reliability analysis, while studies addressing load transfer path optimization and synchronous control strategy modeling under small-sample conditions remain relatively scarce. Therefore, exploring load synchronous transfer control methods based on imprecise probability not only represents theoretical innovation but also holds promising engineering application prospects.

2.2. Literature Summary

In summary, non-exact probability theory offers a novel research paradigm for load synchronization transfer control in distribution networks under conditions of small sample sizes and high uncertainty. This is achieved through methods such as upper and lower probability intervals, belief sets, and multi-prior fusion. IDM delivers robust probability estimates even under data scarcity, NCC significantly reduces misjudgment risks through multi-class outputs, while Credibility Networks extend traditional Bayesian topology to interval reasoning, endowing models with enhanced generalization capabilities and resilience. Looking ahead, as distributed generation penetration continues to rise, uncertainties across regions and timescales will intensify. Strategies based on imprecise probability—including online learning, rolling optimization, and multi-source information fusion—will become critical enablers for enhancing the resilience of distribution network control. These approaches hold significant theoretical value and wide-ranging engineering prospects for building secure, reliable, and intelligent new power systems.

3. Foundations of Imprecise Probability Theory Control

Based on the imprecise probability theory and its application advantages, this study uses the imprecise Dirichlet model (IDM) with the Naive Credal Classifier (NCC).

3.1. Imprecise Dirichlet Model

The Imprecise Dirichlet Model (IDM) is an advanced Bayesian statistical approach for handling uncertainty, serving as an extension of the Dirichlet distribution for estimating probability distributions under conditions of insufficient information. Consider a multinomial distribution with N possible outcomes, whose Dirichlet prior probability density function (PDF) [11], as shown in Equation (1).

$$f\left(\mathit{\theta }\right)=\mathrm{\Gamma }\left({\displaystyle \sum _{n=1}^N{\alpha }_n}\right){\left[{\displaystyle \prod _{n=1}^N\left({\alpha }_n\right)}\right]}^{-1}{\displaystyle \prod _{n=1}^N{\theta }_n^{{\alpha }_n-1}}$$

(1)

In the formula, θ = (θ₁, θ₂, …, θ_N) represents the probability of the result occurring, so that 0 ≤ θ_n ≤ 1 (n = 1, 2, …, N) and ${{\displaystyle \sum }}_{n=1}^N{\theta }_n$ = 1, α₁, α₂, …, α_N represent the positive parameter of the Dirichlet distribution, and Γ(·) represents the gamma function, which is often used in statistics to represent the probability distribution of random variables.

When the sample observations M are acquired, the original prior Dirichlet probability density function is updated by Bayes’ theorem, and then the posterior Dirichlet probability density function is generated, which reflects the re-evaluation of the parameters by synthesizing the actual observations [12], and the posterior probability density function is shown in Equation (2).

$$f\left(\left.\mathit{\theta }\right|\mathit{M}\right)=\mathrm{\Gamma }\left({\displaystyle \sum _{n=1}^N\left(m_n+{\alpha }_n\right)}\right)\cdot {\left[{\displaystyle \prod _{n=1}^N\mathrm{\Gamma }\left(m_n+{\alpha }_n\right)}\right]}^{-1}\cdot {\displaystyle \prod _{n=1}^N{\theta }_n^{m_n+{\alpha }_n-1}}$$

(2)

In the formula, M = {m₁, m₂, …, m_n} is a sample observation; m_n represents the number of occurrences of the random variable state n.

After obtaining the posterior probability density function, the parameter θ_n is estimated using the posterior distribution expected value [13], as defined in Equation (3).

$${\theta }_n=E\left(\left.{\theta }_n\right|M\right)=\left(m_n+{\alpha }_n\right)/\left({\displaystyle \sum _{n=1}^N\left(m_n+{\alpha }_n\right)}\right)$$

(3)

When analyzing the estimated results of a deterministic Dirichlet model, if observations are lacking, the probability θ_n of the nth result is determined by the parameter α, i.e., ${\theta }_n={\alpha }_n/{\displaystyle \sum }_{n=1}^N{\alpha }_n$, where the parameter α is called the prior weight of the result, often expressed as the parameter s, and is called the equivalent sample size in the Dirichlet distribution. In the probability estimation process, s represents the influence of prior distribution on posterior probability, that is, the larger the value of s, the more observed values are needed to adjust the parameters of prior distribution [14]. When there are fewer available observations, the deterministic Dirichlet model estimates are more affected by the prior distribution, and if the settings are not reasonable, then the estimation results based on the deterministic Dirichlet model may become inaccurate, which may affect the final decision and prediction.

literature [15], the authors, in order to overcome the shortcomings of deterministic Dirichlet models, IDM uses a series of Dirichlet prior distributions instead of a single Dirichlet distribution. In IDM, the prior probability density function is shown in Equation (4).

$$f\left(\mathit{\theta }\right)=\mathrm{\Gamma }\left(s\right){\left[{\displaystyle \prod _{n=1}^N\left(s\cdot r_n\right)}\right]}^{-1}{\displaystyle \prod _{n=1}^N{\theta }_n^{s\cdot r_n-1}},\forall r_n\in \left[0,1\right],{\displaystyle \sum _{n=1}^N{r}_n=1}$$

(4)

In the formula, r_n (n = 1, 2, …, N) is the nth prior weight factor, and in Equation (4), s·r_n has the same effect as α_n. When r_n varies within the interval [0, 1], f(θ) will contain all possible prior PDFS for a given predetermined s, thus avoiding unreasonable effects of prior values.

Then, according to the updating process of Bayes’ rule, a posterior PDF of the IDM relative to the observed value M can be calculated, as shown in Equation (5).

$$f\left(\left.\mathit{\theta }\right|\mathit{M}\right)=\mathrm{\Gamma }\left(s+{\displaystyle \sum _{n=1}^N{m}_n}\right)\cdot {\left[{\displaystyle \prod _{n=1}^N\mathrm{\Gamma }\left(m_n+s\cdot r_n\right)}\right]}^{-1}\cdot {\displaystyle \prod _{n=1}^N{\theta }_n^{m_n+s\cdot r_n-1}},\forall r_n\in \left[0,1\right],{\displaystyle \sum _{n=1}^N{r}_n=1}$$

(5)

In the formula, ${{\displaystyle \sum }}_{n=1}^N{m}_n$ represents the total number of observations.

Thus, a parameter representing the interval valued probabilities of all outcomes in the IDM, ${\tilde{\theta }}_1$,${\tilde{\theta }}_2$, …, ${\tilde{\theta }}_N$ can be estimated from the posterior PDF by calculating the expected value, according to Equation (6).

$${\tilde{\theta }}_n=\left[\underset{¯}{E}\left(\left.{\theta }_n\right|M\right),\overline{E}\left(\left.{\theta }_n\right|M\right)\right]=\left[{\displaystyle \frac{m_n}{{\displaystyle \sum _{n=1}^N{m}_n}+s}},{\displaystyle \frac{m_n+s}{{\displaystyle \sum _{n=1}^N{m}_n}+s}}\right],n=1,2,\dots ,N$$

(6)

The bounds of expectation are calculated based on the bounds of r_n, i.e., 0 and 1. Thus, the imprecise probability of random variable state occurrence in a given case can be estimated based on small-sample data. The IDM statistical model eliminates the adverse effects of unreasonable prior Settings on event probability estimation in the absence of sample size.

3.2. Naive Credal Classifier

Naive Credal Classifier (NCC) is a classifier based on Naive Bayes (NB), which enhances the robustness of the model by introducing imprecise probabilities. The core idea of NCC is to provide more robust classification results by using a set of prior probabilities to model uncertainty in the face of incomplete or small-scale data sets, i.e., multiple possible categories can be returned in the face of uncertain instances. The Bayesian framework learns to update the prior with a profile representing the data evidence to calculate a posterior probability that can be used for decision-making [16]. Formally, a classifier is a function that maps instances of a set of variables (called attributes or features) to the state or class of a class variable.

The credal network utilizes Bayesian network theory for the state value X_c of x_c and then calculates the probability P(x_c|x_E) [16] by looking at the specific values x_E present in the evidence variable X_E, as shown in Equation (7).

$$P\left(x_c|{\mathit{x}}_{\mathit{E}}\right)={\displaystyle \frac{P\left(x_c,{\mathit{x}}_{\mathit{E}}\right)}{P\left({\mathit{x}}_{\mathit{E}}\right)}}={\displaystyle \frac{{\displaystyle {\sum }_{X_M}{\displaystyle \prod _{i=1}^{I}P\left(\left.x_i\right|{\mathit{\pi }}_{\mathit{i}}\right)}}}{{\displaystyle {\sum }_{X_{M,}{X}_c}{\displaystyle \prod _{i=1}^{I}P\left(\left.x_i\right|{\mathit{\pi }}_{\mathit{i}}\right)}}}}$$

(7)

In the formula, I is the number of multi-state random variables in the Bayesian network, P(x_i|π_i) is the conditional probability quality function, x_i is the observation value of the ith random variable X_i, X_i $\in$ X, X represents all random variables in the network, πi is an observation value of Пi, which represents the state of the parent node of X_i, X_M = X\(X_E$\cup$Xc); ${{\displaystyle \sum }}_{X_M}$ represents a full probability operation on different states of variables in the node variable set X_M.

Bayes classifiers perform classification by comparing the calculated posterior probabilities, and the category with the largest posterior probability is the classification result. However, when there is not a sufficient number of samples, Bayesian classifiers may return biased prior-dependent classification results, i.e., depending on the different priors employed, it may identify different classes as the most likely. However, any single a priori choice carries a certain arbitrariness, and these classifications are highly uncertain [17]. The credal network classifier relaxes the classification results of Bayesian classifiers by accepting imprecise probabilistic representations [18]. In a Bayesian classifier, each category of a class variable has a single-valued probability. In contrast, in a credal network classifier, the occurrence probability of each class can be expressed as an interval valued probability, that is, an imprecise precision probability.

In the literature [18], the authors introduced Credal Set (CS) for credal networks in order to deal with the uncertainty of the node random variables. The credal set is used to describe the imprecise probabilistic properties of a node random variable, and mathematically, the credal set K(X_i) is defined as a closed convex set that covers all possible probabilistic mass functions P(X_i) of the random variable X_i. The credible set K(_Xi) is:

$$K\left(X_i\right)=\mathrm{CH}\left\{\begin{array}{l}P\left(X_i\right):x_i\in {\mathsf{\Omega }}_{X_i}\\ P\left(x_i\right)\in \left[\underset{\_}{P}\left(x_i\right),\overline{P}\left(x_i\right)\right]\\ {\displaystyle {\sum }_{{\mathsf{\Omega }}_{X_i}}P\left(x_i\right)}=1\end{array}\right\}$$

(8)

K(X_i) represents the closed convex set consisting of all possible probability mass functions P(X_i) of the random variable X_i, CH represents a convex hull, ${\displaystyle \sum }_{{\Omega }_{Xi}}P\left(X_i\right)=1$ means that the sum of all possible probabilities must equal 1, and Ω_Xi is the range of values for X_i.

As shown in Equation (9), there may be many combinations of prior distribution and observed data, so the credal set contains an infinite number of probability mass functions, but it only contains a finite number of extreme mass functions, which are called the vertices of the credal set, denoted as ext[K(X_i)]. These extremal functions correspond to the vertices that make up the convex hull, and they can be obtained by combining the endpoints of the probability interval. The classification of a credal network classifier consists of calculating the upper and lower bounds of the conditional probability of X_c = x_c given X_E = x_E, a goal that can be achieved by calculating the upper and lower bounds using Equations (9) and (10).

$$\overline{P}\left(\left.X_c=x_c\right|{\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)=\underset{P\left(\mathit{X}\right)\in ext\left[K\left(\mathit{X}\right)\right]}{\max }{\displaystyle \frac{P\left(X_c=x_c,{\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)}{P\left({\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)}}$$

(9)

$$\underset{¯}{P}\left(\left.X_c=x_c\right|{\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)=\underset{P\left(\mathit{X}\right)\in ext\left[K\left(\mathit{X}\right)\right]}{\min }{\displaystyle \frac{P\left(X_c=x_c,{\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)}{P\left({\mathit{X}}_{\mathit{E}}={\mathit{x}}_{\mathit{E}}\right)}}$$

(10)

In the formula, P(X) represents the joint probabilistic mass function of all random variables, K(X) is the convex hull of a set of joint mass functions, i.e., the credal set, ext[K(X)] represents the limiting joint mass function of K(X), and P(X) ∈ ext[K(X)], which means that P(X) should be selected from ext[K(X)].

In this paper, IDM is used to model the prior and then return the imprecise probabilities, which are integrated into the credal network classifier to achieve the organic combination of IDM and credal classifier.

3.3. Naive Credal Classifier Classification Control Standards

Bayesian classifiers determine sample categories based on the principle of maximizing posterior probability in probability theory. Utilizing Bayesian networks, the classifier calculates the probability of each category by applying Bayes’ theorem given known input evidence x. It then compares these probabilities and selects the category with the highest posterior probability as its classification decision, as illustrated in Figure 1. In Figure 1, P(C₁|X), P(C₂|X) on the axis, …, P(C₅|X) is calculated by a Bayesian classifier and the classification result is C₁ because P(C₁|X) has the greatest posterior probability.

Figure 1. A simple diagram of the principle of Bayes classifier.

Figure 2 and Figure 3 illustrate the diagnostic logic and output results of the credal network classifier. As shown in Figure 2, after computation, the Naive Credal Classifier category C₁ as having a lower bound of posterior imprecise probability higher than all other categories. Consequently, C₁ was designated as the sole diagnostic result under evidence condition X. However, in Figure 3, the lower limit of the posterior imprecision probability for C₁ is lower than the upper limit of the posterior imprecision probability for C₂, which indicates that the probability intervals of the two may overlap, as shown by the shaded area in the Figure 3. In this case, the Naive Credal Classifier cannot determine an exact classification result, but instead provides a set of possible categories {C₁, C₂}, indicating that the sample is likely to be classified as C₁ or C₂ based on the conditions of evidence.

Figure 2. The credal network output is a unique class case.

Figure 3. The credal network output is a similar class set case.

As can be seen, compared to Bayesian classifiers, the credal network controller provides a larger probability margin when performing category diagnosis on samples. When samples are unique, the credal network controller can deliver higher judgment reliability [19]. When faced with overlapping regions of maximum a posteriori imprecise probability intervals, the credal network controller can generate sets encompassing multiple possible categories, effectively reducing the risk of misdiagnosis.

4. Construction of a Switching Control Action Time Prediction Model Based on IDM-NCC

4.1. Naive Credal Model for Switching Time Prediction in Distribution Networks

In power system, PCC (Point of Common Coupling) refers to the point where different power systems or power equipment are connected to the main power grid. In this paper, by monitoring the voltage and current at the PCC, the operation status of the distribution network can be monitored effectively. By monitoring the voltage and current at the PCC, the action status of the distribution network can be effectively monitored. Compared with other complex electrical parameters, the three-phase voltage and current are relatively easy to measure and obtain, and can also help to monitor the stability of the power system. And in the distribution network, the distribution network switch needs to coordinate its action time according to the peak voltage and current to ensure that the power can be quickly and accurately cut off in the event of failure, so they become the preferred features for the implementation of switching time prediction and analysis. In this paper, the three phase voltage and three phase current peak value at PCC are taken as nodal evidence variables of the credal network, and the credal network is constructed.

In this study, a typical double-feeder distribution network structure is used as the modeling basis, and its topology is shown in Figure 4. The network contains two feeders (AC, AD) and distributed PV access points, and the PCC (point of common coupling) is located at the key busbar as the key node for monitoring voltage and current. Combining the existing research and practical experience, a total of 12 types of typical operating states are set for distribution networks as single-phase short-circuit (AG, BG, CG), two-phase short-circuit (AB, AC, BC), two-phase short-circuit with ground (ABG, ACG, BCG), three-phase short-circuit (ABC), three-phase short-circuit with ground (ABCG), and non-fault (NF) states [20]. In this paper, the three-phase voltage and current peaks at the PCC are used as input features to the credal network classifier to construct an interval prediction model for switch action time.

Figure 4. Structure of a double-feeder distribution network containing distributed power sources.

The switching time state class and the set of operational state sign attributes of the distribution network are determined according to different operational states, as shown in Table 1 and Table 2. In order to study the operating characteristics of the distribution network in depth, simulation is carried out for different operating conditions of the distribution network, and the three-phase voltages and three-phase currents at the PCC under each operating condition are collected, and in this way, a total of 310 operating conditions is obtained. A total of 248 working conditions (310 × 80% = 248) were randomly selected as the training set sample data, and another 62 working conditions (310 × 20% = 62) were selected as the test set sample data to construct the fault diagnosis model database. Using the imprecise probability estimation method, these sample data were analyzed to achieve the prediction of switching action time in distribution networks.

Table 1. Distribution network switching action time status class.

NO.	Type of Fault	Switch Control Action Time T/ms	NO.	Type of Fault	Switch Control Action Time T/ms
F1	Trouble-free NF	T < 5	F18	BC failure at position 2	26 ≤ T < 27
F2	AG failure at position 1	5 ≤ T < 10	F19	ABG failure at position 2	27 ≤ T < 28
F3	BG failure at position 1	10 ≤ T < 11	F20	ACG failure at position 2	28 ≤ T < 29
F4	CG failure at position 1	11 ≤ T < 12	F21	BCG failure at position 2	29 ≤ T < 30
F5	AB failure at position 1	12 ≤ T < 13	F22	ABC failure at position 2	30 ≤ T < 31
F6	AC failure at position 1	13 ≤ T < 14	F23	ABCG failure at position 2	31 ≤ T < 32
F7	BC failure at position 1	14 ≤ T < 15	F24	AG failure at position 3	32 ≤ T < 33
F8	ABG failure at position 1	15 ≤ T < 16	F25	BG failure at position 3	33 ≤ T < 34
F9	ACG failure at position 1	16 ≤ T < 17	F26	CG failure at position 3	34 ≤ T < 35
F10	BCG failure at position 1	17 ≤ T < 18	F27	AB failure at position 3	35 ≤ T < 36
F11	ABC failure at position 1	18 ≤ T < 19	F28	AC failure at position 3	36 ≤ T < 37
F12	ABCG failure at position 1	20 ≤ T < 21	F29	BC failure at position 3	37 ≤ T < 38
F13	AG failure at position 2	21 ≤ T < 22	F30	ABG failure at position 3	38 ≤ T < 39
F14	BG failure at position 2	22 ≤ T < 23	F31	ACG failure at position 3	39 ≤ T < 40
F15	CG failure at position 2	23 ≤ T < 24	F32	BCG failure at position 3	40 ≤ T < 50
F16	AB failure at position 2	24 ≤ T < 25	F33	ABC failure at position 3	50 ≤ T < 60
F17	AC failure at position 2	25 ≤ T < 26	F34	ABCG failure at position 3	T ≥ 60

Table 2. Distribution network action state symptom attribute set.

Variables	Sign Type
E1	Peak of phase A voltage
E2	Peak of phase B voltage
E3	Peak of phase C voltage
E4	Peak of phase A current
E5	Peak of phase B current
E6	Peak of phase C current

4.2. IDM-Based Conditional Confidence Set Learning Algorithm

Using the theory and methodology of Bayesian networks, a credal network classifier model for estimating the timing of switching control actions in distribution networks is constructed by combining the conditions related to the temporal characteristics of switching actions in distribution networks, as demonstrated in Figure 5, which is capable of analyzing and predicting the temporal characteristics of switching actions in distribution networks.

Figure 5. Distribution network switch action time prediction credal network splitter diagram.

In Figure 5, the parent node is the multi-state random variable of the switch action time of the distribution network, and F₁ ~ F₃₄ is set; The child nodes are the characteristic parameters E₁, E₂, E₃, E₄, E₅ and E₆ of the three-state random variables. For example, E_1,1, E_1,2, and E_1,3 are good, decent, and attention states of phase A current status, respectively. It can be seen that the model in this paper is a 2-layer credal network classifier, and the credal network classifier contains a parent node F, which is a multi-state query variable, that is, a 7-state random variable. The other 6 nodes are evidence variables representing the symptomatic attributes of the operating state of the distribution network.

The credal network structure needs to be calculated from the historical operation data to learn the conditional probability distribution of each variable E_i for a given switching time state F_j. In this paper, the IDM is used to complete this learning process for the prediction of distribution network switching control action time characteristics in the following steps: first, for each switching time state class F_j (j = 1, 2, …, 34), find the conditional confidence set K(E_i|F_j) of the variable E_i under it. For example, to find the conditional confidence set K(E₃|F₂) of the C-phase voltage at the distribution network switching control action time 5 ≤ T < 10 ms, based on the IDM method shown in Equation (4), the imprecise conditional probabilities of the distribution network switching control action time 5 ≤ T < 10 ms are found out from the historical data, Pim(E_3,1|F₂), Pim(E_3,2|F₂), Pim(E_3,3|F₂). To obtain the conditional belief set K(E₃|F₂) for the phase C voltage when the switching control action time is 5 ≤ T < 10 ms, the imprecise conditional probabilities Pim(E_3,1|F₂), Pim(E_3,2|F₂), Pim(E_3,3|F₂) are computed by using the adapted IDM framework as shown in Equation (11):

$$\left\{\begin{array}{l}{P}_{im}\left(\left.E_{3,1}\right|F_2\right)=\left[\underset{\_}{P}\left(\left.E_{3,1}\right|F_2\right),\overline{P}\left(\left.E_{3,1}\right|F_2\right)\right]\\ \underset{\_}{P}\left(\left.E_{3,1}\right|F_2\right)={\displaystyle \frac{n_{3,1}}{N+s}},\overline{P}\left(\left.E_{3,1}\right|F_2\right)={\displaystyle \frac{n_{3,1}+s}{N+s}}\\ P_{im}\left(\left.E_{3,2}\right|F_2\right)=\left[\underset{\_}{P}\left(\left.E_{3,2}\right|F_2\right),\overline{P}\left(\left.E_{3,2}\right|F_2\right)\right]\\ \underset{\_}{P}\left(\left.E_{3,2}\right|F_2\right)={\displaystyle \frac{n_{3,2}}{N+s}},\overline{P}\left(\left.E_{3,2}\right|F_2\right)={\displaystyle \frac{n_{3,2}+s}{N+s}}\\ P_{im}\left(\left.E_{3,3}\right|F_2\right)=\left[\underset{\_}{P}\left(\left.E_{3,3}\right|F_2\right),\overline{P}\left(\left.E_{3,3}\right|F_2\right)\right]\\ \underset{\_}{P}\left(\left.E_{3,3}\right|F_2\right)={\displaystyle \frac{n_{3,3}}{N+s}},\overline{P}\left(\left.E_{3,3}\right|F_2\right)={\displaystyle \frac{n_{3,3}+s}{N+s}}\\ N=n_{3,1}+n_{3,2}+n_{3,3}\end{array}\right.$$

(11)

In the formula, N represents the number of times in which the status of distribution network switch control action time is F₂ within the statistical period of distribution network action; n_3,1, n_3,2, n_3,3 respectively represent the number of times when the C-phase voltage status of the distribution network switch control action time state is good, decent and attention, respectively, under the condition of F₂; s is the parameter set in the IDM, and the value in this paper is 1 [21].

In summary, based on the imprecise probabilities calculated above, the conditional credal set K(E₃|F₂) is formally constructed, as represented by Equation (12).

$$\begin{array}{l}K\left(\left.E_3\right|F_2\right)=\left\{P\left(\left.E_3\right|F_2\right)\right.:\\ P\left(\left.E_{3,1}\right|F_2\right)\in \left[\underset{¯}{P}\left(\left.E_{3,1}\right|F_2\right),\overline{P}\left(\left.E_{3,1}\right|F_2\right)\right],\\ P\left(\left.E_{3,2}\right|F_2\right)\in \left[\underset{¯}{P}\left(\left.E_{3,2}\right|F_2\right),\overline{P}\left(\left.E_{3,2}\right|F_2\right)\right],\\ P\left(\left.E_{3,3}\right|F_2\right)\in \left[\underset{¯}{P}\left(\left.E_{3,3}\right|F_2\right),\overline{P}\left(\left.E_{3,3}\right|F_2\right)\right],\\ P\left(\left.E_{3,1}\right|F_2\right)+P\left(\left.E_{3,2}\right|F_2\right)+P\left(\left.E_{3,3}\right|F_2\right)=\left.1\right\}\end{array}$$

(12)

In the formula, the probability mass function K(E₃|F₂) in the credal set K(E₃|F₂) takes a value in the upper and lower boundary range of its corresponding imprecise probability.

At this point, the vertex ext[K(E₃|F₂)] of the credal set K(E₃|F₂) can be determined by using the upper and lower bounds of the corresponding imprecise probabilities of the credal set K(E₃|F₂), and the vertices of the credence set (extreme points) can be determined based on the probability intervals as shown in Equation (13).

$$\begin{array}{l}ext\left[K\left(\left.E_3\right|F_2\right)\right]=\\ \left\{\underset{¯}{P}\left(\left.E_{3,1}\right|F_2\right),\underset{¯}{P}\left(\left.E_{3,2}\right|F_2\right),\overline{P}\left(\left.E_{3,3}\right|F_2\right)\right\},\\ \left\{\underset{¯}{P}\left(\left.E_{3,1}\right|F_2\right),\overline{P}\left(\left.E_{3,2}\right|F_2\right),\underset{¯}{P}\left(\left.E_{3,3}\right|F_2\right)\right\},\\ \left\{\overline{P}\left(\left.E_{3,1}\right|F_2\right),\underset{¯}{P}\left(\left.E_{3,2}\right|F_2\right),\underset{¯}{P}\left(\left.E_{3,3}\right|F_2\right)\right\}\end{array}$$

(13)

For other conditional confidence sets K(E_i|F_j)(i = 1, 2, 3, …, 6, j = 1, 2, 3, …, 34) The steps of the method of finding are shown in Equation (13).

4.3. Model Solving and Classification Output Process

Then, based on the conditional confidence sets K(E₁|F), K(E₂|F), …, K(E₆|F), the a posteriori inexact probability of each switch action time state of the distribution network can be solved based on the exact inference algorithm of the confidence network shown in Equations (8) and (9), as shown in Equation (14):

$$\left\{\begin{array}{l}\overline{P}\left(\left.F_j\right|E\right)=\max {\displaystyle \frac{{\displaystyle {\prod }_{i=1}^{6}P\left(F_j\right)P\left(E_i\left|F_j\right.\right)}}{{\displaystyle {\sum }_{j=1}^{34}{\displaystyle {\prod }_{i=1}^{6}P\left(F_j\right)P\left(E_i\left|F_j\right.\right)}}}}\\ \underset{¯}{P}\left(\left.F_j\right|E\right)=\min {\displaystyle \frac{{\displaystyle {\prod }_{i=1}^{6}P\left(F_j\right)P\left(E_i\left|F_j\right.\right)}}{{\displaystyle {\sum }_{j=1}^{34}{\displaystyle {\prod }_{i=1}^{6}P\left(F_j\right)P\left(E_i\left|F_j\right.\right)}}}}\\ P\left(E_i\left|F_j\right.\right)\in ext\left[K\left(\left.E_i\right|F_j\right)\right],j=1,2,3,\dots ,34\end{array}\right.$$

(14)

Finally, after obtaining the P_im(F|E_i), according to the classification criterion of the credal network classifier, the predictive results of switching action time characteristics of the distribution network are output.

The flow of imprecise conditional probabilistic control estimation of switch action time for distribution network described in this paper is shown in Figure 6.

Figure 6. Flowchart of switch action time estimation for distribution network based on imprecise probability.

5. Example Analysis

5.1. Simulation Model

In this paper, a microgrid model containing distributed photovoltaic (PV) power generation is constructed as an example of a double-fed distribution network under different operating conditions of the distribution network, and the switching action time prediction method of the DG distribution network is investigated. The structure of this network is depicted in Figure 4. A microgrid model incorporating distributed photovoltaic generation was constructed. The distribution system operates at a reference voltage of 10.5 kV, with all lines configured as overhead lines. Unit resistance and reactance were set as: R = 0.26 Ω/km, X = 0.355 Ω/km. The lengths of AB, BC, and AD are, respectively, 3 km, 3 km, and 10 km. Both Feeder 1 and Feeder 2 terminate with a load capacity of 6 MVA and a power factor of 0.85. The DG output power is adjustable within the range of 0–10 MW. The failure points are f1, f2, f3, and the failure start time is 0.5 s.

A distribution network switch is installed at the PCC to detect the switching on and off times of the distribution network. The simulation is carried out for different working conditions of the distribution network. The specific situation is as follows: the three-phase voltage and three-phase current at PCC under each working condition are collected. The feature data of each operating condition can be formed into a feature vector with a dimension of 1 × 6, considering that there are 310 different operating conditions, these data can be combined into a feature matrix of 310 × 6 dimensions.

In order to eliminate the influence of different dimensions, it is necessary to normalize the eigenmatrix. Using the discrete standardization method, the eigenvalues of S₁–S₃₁₀ can be converted to the valid range of [0, 1]. Normalization as shown in Equation (15).

$$E_i^{\prime }={\displaystyle \frac{E_i-E_{\min }}{E_{\max }-E_{\min }}}$$

(15)

In the formula, E_i and $E_i^{\prime }$ are the values before and after the normalization of the attributes of the i th operating state symptom, respectively; E_max and E_min are the maximum and minimum values of E_i. The same is true for S₂~S₃₁₀. The normalized database is shown in Table 3.

Table 3. Normalized database of eigenmatrix.

	E1	E2	E3	E4	E5	E6
S1	0.6317	0.9733	0.9689	−0.9922	−0.9921	−0.9922
S2	−0.1786	0.9688	0.9778	−0.0907	−0.9890	−0.9905
S3	0.6390	−0.0111	0.9689	−0.9913	−0.0893	−0.9898
S4	0.6317	0.9777	−0.0044	−0.9898	−0.9912	−0.0893
S5	−0.6317	−0.3541	0.9822	0.7220	0.6905	−0.9921
⋮	⋮	⋮	⋮	⋮	⋮	⋮
S310	0.3076	0.5724	0.5733	−0.5693	−0.5682	−0.5682

The normalization of the feature matrix involves six indicators of each operating condition in the database, which together form a feature vector. 310 × 80% = 248 feature vectors were randomly selected from the database as the training data set, and the remaining 310 × 20% = 62 vectors were selected as the test data set. The imprecise probability estimation method is used to predict and analyze these data.

5.2. Case 1: Unique Category Output Results and Analysis

In Case 1, the case where the credal network classifier produces a single unambiguous result at the time when diagnosis is chosen, where the state of the evidence variable is represented as vector S₁. Then, according to this evidence, imprecise probability estimation method and Bayesian classifier are used to predict the operation time (or total action time) of the distribution network switch, respectively, and the prediction results are shown in Table 4 and Table 5.

Table 4. Calculation results of switch control time state class of distribution network based on imprecise probability.

Imprecise Probability	Probability Value	Imprecise Probability	Probability Value
Pim(F1|S1)	[1.60 × 10−5, 5.57 × 10−5]	Pim(F18|S1)	[3.69 × 10−11, 1.50 × 10−10]
Pim(F2|S1)	[5.78 × 10−6, 7.34 × 10−6]	Pim(F19|S1)	[5.22 × 10−12, 1.24 × 10−10]
Pim(F3|S1)	[3.00 × 10−6, 7.34 × 10−6]	Pim(F20|S1)	[1.43 × 10−10, 4.36 × 10−10]
Pim(F4|S1)	[1.54 × 10−6, 1.74 × 10−6]	Pim(F21|S1)	[9.89 × 10−11, 1.04 × 10−10]
Pim(F5|S1)	[5.36 × 10−7, 5.70 × 10−7]	Pim(F22|S1)	[6.53 × 10−11, 7.34 × 10−11]
Pim(F6|S1)	[1.04 × 10−8, 2.29 × 10−7]	Pim(F23|S1)	[4.31 × 10−11, 6.23 × 10−11]
Pim(F7|S1)	[3.87 × 10−6, 7.34 × 10−6]	Pim(F24|S1)	[1.11 × 10−11, 1.74 × 10−11]
Pim(F8|S1)	[6.63 × 10−7, 7.42 × 10−7]	Pim(F25|S1)	[5.28 × 10−11, 5.31 × 10−11]
Pim(F9|S1)	[2.65 × 10−9, 4.80 × 10−9]	Pim(F26|S1)	[2.99 × 10−11, 4.55 × 10−11]
Pim(F10|S1)	[8.14 × 10−9, 1.78 × 10−8]	Pim(F27|S1)	[1.06 × 10−10, 1.50 × 10−10]
Pim(F11|S1)	[6.89 × 10−10, 7.20 × 10−10]	Pim(F28|S1)	[2.37 × 10−10, 4.36 × 10−10]
Pim(F12|S1)	[1.98 × 10−10, 4.36 × 10−10]	Pim(F29|S1)	[2.06 × 10−10, 3.46 × 10−10]
Pim(F13|S1)	[0, 7.20 × 10−10]	Pim(F30|S1)	[1.24 × 10−2, 4.28 × 10−2]
Pim(F14|S1)	[8.51 × 10−10, 1.26 × 10−9]	Pim(F31|S1)	[1.85 × 10−10, 1.26 × 10−9]
Pim(F15|S1)	[3.19 × 10−10, 5.57 × 10−10]	Pim(F32|S1)	[2.91 × 10−8, 3.02 × 10−8]
Pim(F16|S1)	[2.85 × 10−11, 2.77 × 10−10]	Pim(F33|S1)	[2.07 × 10−7, 2.29 × 10−7]
Pim(F17|S1)	[3.93 × 10−10, 4.36 × 10−10]	Pim(F34|S1)	[3.94 × 10−7, 5.70 × 10−7]

Table 5. Bayesian Classifier-Based Calculation of Time-State Classes for Switching Control Actions in Distribution Networks.

Exact Probability	Probability Value	Exact Probability	Probability Value
P(F1|S1)	2.74 × 10−5	P(F18|S1)	6.48 × 10−11
P(F2|S1)	7.01 × 10−6	P(F19|S1)	1.02 × 10−11
P(F3|S1)	4.77 × 10−6	P(F20|S1)	2.39 × 10−10
P(F4|S1)	1.72 × 10−6	P(F21|S1)	1.03 × 10−10
P(F5|S1)	5.68 × 10−7	P(F22|S1)	7.25 × 10−11
P(F6|S1)	2.04 × 10−8	P(F23|S1)	5.64 × 10−11
P(F7|S1)	5.70 × 10−6	P(F24|S1)	1.52 × 10−11
P(F8|S1)	7.34 × 10−7	P(F25|S1)	5.31 × 10−11
P(F9|S1)	3.83 × 10−9	P(F26|S1)	4.01 × 10−11
P(F10|S1)	1.26 × 10−8	P(F27|S1)	1.37 × 10−10
P(F11|S1)	7.19 × 10−10	P(F28|S1)	3.46 × 10−10
P(F12|S1)	3.06 × 10−10	P(F29|S1)	2.89 × 10−10
P(F13|S1)	0	P(F30|S1)	2.12 × 10−2
P(F14|S1)	1.12 × 10−9	P(F31|S1)	3.42 × 10−10
P(F15|S1)	4.55 × 10−10	P(F32|S1)	3.01 × 10−8
P(F16|S1)	5.41 × 10−11	P(F33|S1)	2.27 × 10−7
P(F17|S1)	4.32 × 10−10	P(F34|S1)	5.16 × 10−7

From Table 4, it can be seen that based on the imprecise probability prediction result category F₃₀ The probability interval is [1.24 × 10⁻², 4.28 × 10⁻²], which contains the probability interval of other categories; from Table 5, it can be seen that based on the Bayesian classifier prediction result category F₃₀ has a probability value of 2.12 × 10⁻² and is the largest. This indicates that the imprecise probability is consistent with the Bayesian classifier prediction result category as F₃₀. combined with the switching time of the distribution network in Table 1, it can be seen that the range of switching action time at category F₃₀ is [38, 39) ms.

Through the above analysis, it can be concluded that Case 1 proves that the switching action time prediction control method based on imprecise probability proposed in this paper has the same prediction performance as the traditional prediction methods (e.g., Bayesian classifiers) when dealing with the mapping relationship between the testing data of the distribution network inspection and the operation state with a unique class output.

5.3. Case 2: Similarity Set Output Results and Analysis

In Case 2, the prediction result of the credal network classifier is a similar set of classes, and the results are shown in Table 6.

Table 6. Calculation results of switching time state class of distribution network based on imprecise probability.

Imprecise Probability	Probability Value	Imprecise Probability	Probability Value
Pim(F1|S1)	[1.60 × 10−5, 5.57 × 10−5]	Pim(F18|S1)	[6.31 × 10−12, 1.50 × 10−10]
Pim(F2|S1)	[5.78 × 10−6, 7.34 × 10−6]	Pim(F19|S1)	[4.07 × 10−11, 1.24 × 10−10]
Pim(F3|S1)	[3.00 × 10−6, 7.34 × 10−6]	Pim(F20|S1)	[4.17 × 10−10, 4.36 × 10−10]
Pim(F4|S1)	[1.54 × 10−6, 1.74 × 10−6]	Pim(F21|S1)	[9.21 × 10−11, 1.04 × 10−10]
Pim(F5|S1)	[5.36 × 10−7, 5.70 × 10−7]	Pim(F22|S1)	[3.70 × 10−2, 5.35 × 10−2]
Pim(F6|S1)	[1.04 × 10−8, 2.29 × 10−7]	Pim(F23|S1)	[3.54 × 10−2, 5.53 × 10−2]
Pim(F7|S1)	[3.87 × 10−6, 7.34 × 10−6]	Pim(F24|S1)	[1.73 × 10−11, 1.74 × 10−11]
Pim(F8|S1)	[9.46 × 10−8, 1.06 × 10−7]	Pim(F25|S1)	[3.48 × 10−11, 5.31 × 10−11]
Pim(F9|S1)	[2.65 × 10−9, 4.80 × 10−9]	Pim(F26|S1)	[3.23 × 10−11, 4.55 × 10−11]
Pim(F10|S1)	[8.14 × 10−9, 1.78 × 10−8]	Pim(F27|S1)	[8.16 × 10−11, 1.50 × 10−10]
Pim(F11|S1)	[6.89 × 10−10, 7.20 × 10−10]	Pim(F28|S1)	[2.59 × 10−10, 4.36 × 10−10]
Pim(F12|S1)	[1.98 × 10−10, 4.36 × 10−10]	Pim(F29|S1)	[1.00 × 10−10, 3.46 × 10−10]
Pim(F13|S1)	[4.88 × 10−10, 7.20 × 10−10]	Pim(F30|S1)	[3.29 × 10−11, 2.24 × 10−10]
Pim(F14|S1)	[7.19 × 10−10, 1.26 × 10−9]	Pim(F31|S1)	[1.21 × 10−9, 1.26 × 10−9]
Pim(F15|S1)	[5.73 × 10−11, 5.57 × 10−10]	Pim(F32|S1)	[2.72 × 10−8, 3.02 × 10−8]
Pim(F16|S1)	[2.49 × 10−10, 2.77 × 10−10]	Pim(F33|S1)	[0, 2.29 × 10−7]
Pim(F17|S1)	[1.07 × 10−10, 4.36 × 10−10]	Pim(F34|S1)	[0, 5.70 × 10−7]

In contrast to the imprecise probability methods, this paper also uses Bayesian classifiers for their prediction and the results are shown in Table 7.

Table 7. Bayesian Classifier-Based Calculation of Time-State Classes for Switching Control Actions in Distribution Networks.

Exact Probability	Probability Value	Exact Probability	Probability Value
P(F1|S1)	2.74 × 10−5	P(F18|S1)	1.24 × 10−11
P(F2|S1)	7.01 × 10−6	P(F19|S1)	6.81 × 10−11
P(F3|S1)	4.77 × 10−6	P(F20|S1)	4.36 × 10−10
P(F4|S1)	1.72 × 10−6	P(F21|S1)	1.02 × 10−10
P(F5|S1)	5.68 × 10−7	P(F22|S1)	5.18 × 10−2
P(F6|S1)	2.04 × 10−8	P(F23|S1)	5.36 × 10−2
P(F7|S1)	5.70 × 10−6	P(F24|S1)	1.74 × 10−11
P(F8|S1)	1.05 × 10−7	P(F25|S1)	4.65 × 10−11
P(F9|S1)	3.83 × 10−9	P(F26|S1)	4.55 × 10−11
P(F10|S1)	1.26 × 10−8	P(F27|S1)	1.26 × 10−10
P(F11|S1)	7.19 × 10−10	P(F28|S1)	3.85 × 10−10
P(F12|S1)	3.06 × 10−10	P(F29|S1)	2.34 × 10−10
P(F13|S1)	6.45 × 10−10	P(F30|S1)	1.46 × 10−10
P(F14|S1)	1.03 × 10−9	P(F31|S1)	1.22 × 10−9
P(F15|S1)	1.09 × 10−10	P(F32|S1)	2.77 × 10−8
P(F16|S1)	2.74 × 10−10	P(F33|S1)	2.21 × 10−7
P(F17|S1)	1.88 × 10−10	P(F34|S1)	5.15 × 10−7

As can be seen from Table 6, the probability intervals for the outcome categories F22 and F23 based on the imprecise probability predictions are, respectively, [3.70 × 10⁻², 5.35 × 10⁻²] and [3.54 × 10⁻², 5.53 × 10⁻²], an overlapping situation occurs. According to the diagnostic criteria of the credal network classifier, based on the imprecise probability prediction result outputs a similar set of categories {F₂₃, F₂₃}, and combined with the switching time of the distribution network in Table 1, it can be seen that the predicted time range of the switching control action is [30, 32) ms.

From Table 7, it can be seen that based on the Bayesian classifier prediction results output category F₂₃. combined with Table 1 distribution network switching time, it can be seen that the range of switching action time is [31, 32) ms for category F₂₃. the actual detection of distribution network switching action time is 30 ms.

It can be seen that the imprecise probability adopts the interval mode to portray the probability distribution of the prediction results, analyses the overlap of the intervals, and outputs the similar set, which can effectively avoid the prediction error; the Bayesian classifier-based exact probability has limitations in the prediction of distribution network switch control action time. Therefore, the imprecise probability prediction method enhances the accuracy of distribution network switch action time.

6. Time-Based Model for Whole-Process Control and Experimental Validation

The construction of a time-dependent model for load transfer in distribution networks requires a multidimensional characterization of temporal coupling relationships across all stages. This involves both quantifying the cumulative effects of deterministic processes and analyzing the propagation patterns of random factors. The model’s core objective is to accurately predict the total time control interval from the issuance of the master station command to the final switch operation, while ensuring compliance with “open-then-close” sequencing and time-difference constraints even under extreme scenarios. By integrating mathematical modeling with engineering experience, we can systematically evaluate the boundary conditions of time parameters and their impact mechanisms on the overall process, providing theoretical support for control strategy design. This strategy requires, based on precise time synchronization, that the time interval from complete disconnection to complete reconnection be within 20 milliseconds, disregarding communication time from the master station to the terminal.

6.1. Time Constraints and Problem Modeling

Target: Total time (interval from trip completion to close completion) must satisfy 0 < T_total < 20 ms.

Known Conditions:

Closing time Ton > Opening time Toff, but specific values are unknown.

Master Station Command Sequence:

t0: Issue closing command

t0 + Δt: Issue opening command

Mathematical Model

Opening completion time: toffend = t0 + Toff

Closing completion time: tonend = t0 + Ton+ Δt

Total time constraint: Ttotal = tonend − toffend and 0 < Ttotal < 20 ms

6.2. Control Strategy Design: Determination of Dynamic Delay Time Δt

Constraint Derivation:

Disconnection completion must precede reconnection completion (to prevent parallel circulating currents):

toffend < tonend Δt < Ton − Toff

Total Time Constraint:

0 < Ttotal < 20 ms → Toff − Ton < Δt < Toff − Ton + 20

Safety Margin Determination:

If nominal values of Ton and Toff are known:

Δt = Ton − Toff – 10 ms (take midpoint value, reserve 10 ms margin)

If parameters are unknown, calibration via experimentation is required:

Step 1: Measure typical values of Ton and Toff in the laboratory.

Step 2: Set Δt = Ton − Toff − 10 ms and verify that Ttotal ∈ (0, 20) ms.

6.3. Physical Experiment Verification and Result Analysis

• Test scenario

The experimental setup for time accuracy testing is shown in Figure 7.

Figure 7. Distribution network load synchronous transfer control time accuracy test.

• Test method

Distribution terminals A and B received BeiDou satellite signals and synchronized normally for 10 min;

Set the timed tripping moment for distribution terminal A and the timed closing moment for distribution terminal B via master station commands;

Tested the contact opening moment of circuit breaker A and the contact closing moment of circuit breaker B using a time synchronization tester;

Compared the time difference between the contact opening moment of circuit breaker A and the contact closing moment of circuit breaker B;

Repeat the test 10 times (Table 8).

• Experimental data

Table 8. Accuracy test data for timed action of distribution terminals.

NO.	The Moment When Circuit Breaker A Opens	Closing Time of Circuit Breaker B	Time Difference (ms)
1	18:56:20.241	18:56:20.249	8
2	18:59:50.839	18:59:50.849	10
3	20:03:10.441	20:03:10.443	2
4	20:05:11.544	20:05:11.547	3
5	20:11:11.241	20:11:11.253	12
6	20:14:00.242	20:14:00.247	5
7	20:17:50.841	20:17:50.845	4
8	20:18:50.154	20:18:50.157	3
9	20:23:46.693	20:23:46.698	5
10	20:35:24.471	20:35:24.476	5

Table 8. Accuracy test data for timed action of distribution terminals.

NO.	The Moment When Circuit Breaker A Opens	Closing Time of Circuit Breaker B	Time Difference (ms)
1	18:56:20.241	18:56:20.249	8
2	18:59:50.839	18:59:50.849	10
3	20:03:10.441	20:03:10.443	2
4	20:05:11.544	20:05:11.547	3
5	20:11:11.241	20:11:11.253	12
6	20:14:00.242	20:14:00.247	5
7	20:17:50.841	20:17:50.845	4
8	20:18:50.154	20:18:50.157	3
9	20:23:46.693	20:23:46.698	5
10	20:35:24.471	20:35:24.476	5

• Experimental conclusions

This paper validates the non-exact probability-based synchronous control method through 10 repeated experiments. Experimental results demonstrate that all time differences for disconnection and reconnection operations are controlled within the range of 2–12 ms, fully meeting the 20 ms engineering requirement and effectively avoiding circulating current risks.

The IDM-NCC fusion model employed in this study provides a more reliable time reference for synchronous control by outputting probability intervals for switching action times. Experimental data demonstrate that this model delivers highly credible action time references for control decisions. The “similarity set” output mechanism of the Naive Credal Classifier exhibits strong fault tolerance, effectively mitigating misjudgment risks in complex distribution environments. No control failures due to model misjudgment occurred during experiments, validating the mechanism’s practicality. The imprecise probabilistic model and dynamic delay strategy form an effective closed-loop system. The control setting of the dynamic delay time Δt fully utilizes the probability interval information to achieve adaptive tuning, thereby enabling millisecond-level precision control.

This study empirically demonstrates the application value of imprecise probabilistic theory in distribution network control. It significantly enhances the decision-making robustness of the system under small-sample, high-uncertainty conditions, providing a new technical pathway for constructing highly reliable distribution networks.

6.4. Characterization of Load and Distributed Photovoltaic Generation and Its Impact on Switching Controls

The effectiveness of the synchronous load transfer method based on inexact probabilistic distribution network loads is analyzed on the basis of the operating characteristics of distribution network loads with electricity and distributed PV generation and their uncertainties. Typical daily load profiles and distributed PV output data are shown in Figure 8 and Figure 9. In this section, the practical impact of load fluctuation and PV generation hysteresis on millisecond synchronous transfer control is analyzed in conjunction with the inexact probabilistic framework.

Figure 8. Typical daily load profile.

Figure 9. Typical daily PV power curve. (A) Light day in summer; (B) Light day in spring and autumn; (C) Light day in winter.

(1)

Uncertainty in load electricity consumption mainly arises from factors such as user behavior, seasonal variations, and meteorological conditions. Typical daily loads have cyclical variations, but their fluctuations are relatively limited in magnitude, and they vary gently over short time scales (milliseconds to seconds).

(2)

The output of distributed photovoltaic power generation has obvious diurnal periodicity and weather dependence, showing certain lag and intermittency. PV power generation gradually enhances after sunrise, peaks in the late afternoon, and then gradually declines, with the whole process showing a smooth trend of change rather than sudden change characteristics.

Therefore, the time interval between switching and closing operation of distribution network load synchronous transfer and supply based on imprecise probability is controlled within 20 ms. It fully satisfies the uncertainty of load and distributed generation changes.

7. Conclusions

The core contribution of this study lies in establishing an interval predictive control framework that integrates the Imprecise Dirichlet Model (IDM) with the Naive Credal Classifier (NCC). This framework successfully addresses the challenge of prior dependency in predicting switching operation times for distribution networks under small-sample conditions. Compared to traditional Bayesian control methods, this framework maintains the accuracy of uniqueness judgments while effectively identifying and mitigating misclassification risks in uncertain scenarios by outputting ‘similarity sets’.

The simulation and experimental results demonstrate that the proposed method outperforms traditional Bayesian classifier controllers in both temporal prediction accuracy and model robustness, particularly maintaining high classification accuracy under highly uncertain operational conditions. This research provides a novel theoretical framework and practical technical pathway for load synchronous transfer control in distribution networks, offering valuable insights for enhancing grid operational control capabilities under high renewable energy penetration.