2. Entropy Mathematic Model of Anti-Islanding Protection
The uncertainty associated with each decision made by the protection is directly linked to the probability of that decision being incorrect. Engaging in a priori, frequentist statistical modeling of the problem is essential to determine the probabilities associated with these decisions. Within this modeling framework, the first step involves defining a universe of potential events. This universe comprises a set of events categorized into subsets, such as islanding events and events of no islanding. Each subset represents a distinct outcome or scenario:
Depending on the composition of this set, there will be a priori probability associated with each of the defined events. These probabilities do not depend on the protection but on its environment. Thus, there will be a pair of probabilities for islanding events and events of no islanding, respectively, as determined by Equation (2):
Suppose the behavior of the protection system is statistically evaluated, considering it as a closed black box with a defined universe of events. In that case, it becomes possible to determine a set of probabilities for success and failure for each subset of events. For specific anti-islanding protection with an arbitrarily defined configuration, within the subset of islanding events, there will exist the following probabilities:
In Equation (3), the first term represents the probability of detecting an island when it has actually occurred (success), while the second term represents the probability of not detecting an island when it has occurred (failure). The set of probabilities, (
and
), is associated with the sensitivity of the protection. Similarly, within the subset of events of no islanding, the probabilities are as follows:
In Equation (4), the first term corresponds to the probability of detecting an island when it has not occurred (failure), and the second term corresponds to the probability of not detecting an island when it has not happened (success). The set of probabilities, ( and ) is associated with the stability of the protection.
Figure 1 shows a forward probability diagram depicting the possible paths in the occurrence of an event, whether it is an island or not, and the associated probabilities of detecting the island (triggering the protection) either correctly or mistakenly.
Under this probabilistic model, an ideal anti-islanding protection (with no uncertainty about its decisions) should have a probability of success for detecting islands of one () and a probability of success of one for not detecting false islands ().
According to Shannon’s mathematical theory of communication [
52], the uncertainty of the forward probability set {
;
;
} is determined by the entropy as expressed by Equations (5) and (6):
In the case of the mentioned ideal protection, Equation (6) results in zero when the probabilities and are equal to one. Moreover, entropy will be maximum when the probabilities are equal, thus generating the highest uncertainty regarding the protection behavior when it evaluates an event. On the other hand, a ‘worst’ ideal protection system, in terms of its sensitivity and stability, will have a probability of success of zero for detecting islands when it actually occurs (corresponding to non-sensible protection) or a probability of success of zero for not detecting false islands when other events occur (corresponding to non-stable protection). In these two new ideal situations, there is no uncertainty, similar to the first ideal protection, as there is no doubt about the protection behavior when the protection evaluates a known event (islanding or not). However, the latter is not entirely true, as from the protection’s perspective, when it has operated, it is not known what event triggered it (islanding or not).
To determine the uncertainty over the point of view of the anti-islanding protection, it is necessary to analyze the backward probability scheme depicted in
Figure 2. Two new events related to the first are defined. The first event involves triggering the protection (T), while the second involves not triggering it (NT).
The probabilities associated with these events are known as posterior probabilities, and they can be determined using Equation (7):
The matrix equation (Equation (7)) calculates the probabilities of the events, triggering the protection and not triggering it, based on the confusion matrix. It is to be noted that this matrix is made up of the set of forward probabilities described above: {
;
;
}. In this way, to calculate the backward entropy, it is necessary to determine the set of backward probabilities illustrated in
Figure 2. The backward probabilities can be determined as follows:
In Equations (8) and (9), the probability that an islanding event and an event of no islanding have triggered the protection, i.e.,
and
, are calculated respectively. Each of these probabilities is calculated as the forward probability of triggering the protection by an islanding event and an event of no islanding, respectively, divided by the total probability of triggering it. Similarly, when the protection does not trigger:
In Equations (10) and (11), the probabilities of an islanding event and an event of no islanding not triggering the protection are calculated, respectively.
In Equation (12), the backward entropy is determined similarly to forward entropy.
Finally, in Equation (13), the total entropy (entropy of the protection) is obtained by adding the backward entropy and the forward entropy. The forward entropy depends on a set of probabilities where only two are independent variables. On the other hand, the backward probabilities rely solely on the forward probabilities, as stated in Equation (7). It is possible to express the entropy of the protection in terms of the forward success probabilities,
and
. This allows defining an uncertainty surface based on these probabilities, as shown in
Figure 3.
Once the uncertainty of the protection has been modeled using entropy, it is possible to determine its minimums. In
Figure 3, it is observed that the entropy has two local minimums when both forward success probabilities (
and
) tend to be one or zero. When both success probabilities tend to zero there is no uncertainty about the performance of the protection, since this protection never would act and this constitutes a particular mathematical case. Furthermore, it has two additional local minimums when one probability tends to zero and the other to one. The reaching of the referred local minimums means that the protection has low stability or low sensitivity.
Ideal Anti-Islanding Protection Model
The commonly employed electrical quantities in passive anti-islanding protection include frequency, RoCoF, voltage vector shift (also called vector surge or vector jump), and voltage magnitude. However, it is also possible to utilize other magnitudes measured at the PCC, as suggested in the existing literature on anti-islanding protections. To achieve generalization, the selected signal for island detection will be treated as a ‘feature’ signal, i.e., a general mathematical signal without necessary correspondence to a physical quantity.
Figure 4b evaluates a generic feature signal over time. The protection would trip if a disturbance caused this signal to deviate outside the defined pickup (
) calibration during the time-delay (
) settings. In this work, a mathematical idealization of the anti-islanding protection is employed. Thus, anti-islanding protection is treated as an idealized element that extracts a feature based on phase voltages at the PCC and assesses whether this characteristic complies with the trip criteria defined by the time-delay (
) and pickup (
). A block diagram of the proposed model is presented in
Figure 4a.
The protection setting defines a region that can be interpreted as the minimum detection energy
that the signal must exceed to trigger the anti-islanding protection, as presented in
Figure 4b. Suppose the calibration is adjusted very close to the normal operating values of the feature signal. In that case, the probability of the anti-islanding protection detecting islanding events
will inevitably increase. Nevertheless, as a trade-off, the probability of not detecting false islands
would decrease; thus reducing the protection entropy.
According to the simplification proposed in the anti-islanding protection model, the probability of detecting islands will depend only on the setting made for the protection. Unfortunately, there is no deterministic relationship between the calibration of the protection time-delay
and pickup
setting and its probability of detecting events, and therefore the protection entropy defined by Equation (13). However, it is possible to establish an incremental relationship with the concept of detection energy presented in
Figure 4b. When the detection energy increases, it becomes less likely to detect islands because more significant disturbances in the selected feature will be required for detection. Conversely, if the detection energy decreases, there is a higher probability of triggering incorrect protection responses to other disturbances. Finally, if the pickup and time-delay were made to vary, it would be possible to explore the entropy surface presented in
Figure 3, but depending on the protection settings.
3. Bayesian Methodology for Forward Success Probability Calculation
To calculate the protection entropy, it is initially necessary to know the forward success probabilities and . If a frequentist estimation approach were employed, it would be essential to conduct a substantial number of simulations over a wide range of scenarios by the central limit theorem. Using high sampling rates in this context often results in a significant computational burden. Bayesian inference is more efficient in terms of the amount of data required than the frequentist estimation. This allows estimation of the probabilities without iterating through many simulated cases. To perform a Bayesian inference calculation, defining the statistical experiment employed to estimate the forward probabilities is necessary.
3.1. Definition of the Statistical Experiment for Probabilities Estimation
To define the statistical experiment, islanding detection will be treated as a Bernoulli trial, where the probability of success ‘p’ represents successful detection and the probability of failure ‘q = 1 − p’ denotes unsuccessful detection. These probabilities are intricately linked to the feature selected to detect islands and protection settings. Furthermore, the following key considerations must be taken into account:
There are an infinite quantity of possible cases of islanding and events of no islanding. A random sample of cases is taken with a uniform probability distribution, and the sample size is and cases, respectively.
The experiment will consist of testing the protection performance for each case with a particular fixed setting for time delay () and pickup ). Each case will be considered a one-off trial, like throwing a dice, resulting in success or failure. The probability of success will remain constant from one trial to another, as the parameters of the protection setting will not be modified after each trial.
The experiment will be conducted in two distinct populations: firstly, for the subset of islanding formation cases, and secondly, for the subset of events of no islanding cases, ensuring independence. For the set of islanding events, the population probability of success will be the probability . In the same way, for the set of events of no islanding cases, the population probability of success will be the probability .
Based on the previous considerations, the probability of achieving ‘
x’ successes for specific anti-islanding protection with a success population probability on a sample of
and
island events can be calculated as follows:
Equation (14) represents the binomial model used to calculate the probability of obtaining success in a sample of trials with a success population probability . Similarly, Equation (15) is used for obtaining success on a sample of trials with a success population probability . These statistical probability models will be used to estimate the population success probabilities mentioned above.
3.2. Bayesian Inference for Forward Success Probabilities
Bayesian inference [
53,
54] represents a compelling alternative in scenarios where limitations arise regarding data availability or required computational time. This approach offers valuable prospects for addressing such constraints, allowing for robust statistical inference even under restricted data conditions or when computational resources are limited. The parameter to be estimated is denoted as
and represents, on the statistical experiment defined above, the probability
and
of success in both populations (island events and no-island events). To estimate this parameter, a non-informative prior probability distribution of
will be used:
Equation (16) is the uniform distribution for
, but it actually has a beta distribution with parameters
and
. If the previously described experiment is repeated
times, after observing the successful cases, the probability of their occurrence according to the binomial model presented in Equations (14) and (15) is given as follows:
Equation (17) denotes that the distribution for obtaining a successful set of
in
trials, where
represents the size of the sample obtained from the population
of islands or no-island events, is proportional to a beta distribution with parameters
and
. With the previously defined uniform prior distribution, the posterior distribution will be:
Equation (18) is the posterior distribution of
and it is a beta distribution with parameters
and
. Considering this posterior distribution, the mean is given for:
Equation (19) is the Bayes estimate of under the squared-error loss function. This result allows for point estimation of forward population success probabilities based on a fixed anti-islanding setting.
The methodology for estimating success probabilities and entropy calculation can be summarized using the flowchart in
Figure 5. Datasets of islanding and events of no islanding of size
and
are respectively formed, and simulations are conducted using DIgSILENT Power Factory 2018 SP3. Then, with each simulation, the experiment is performed to verify whether the tested anti-islanding protection model detects the event, counting the successful cases using MATLAB R2021a. This process is repeated until all events in the obtained dataset are verified. Once these results are obtained, Equation (19) can be used to find point estimators for the success probabilities. Finally, it is possible to calculate the entropy of the protection using Equation (13).
3.3. Entropy of the Experiment
The methodology presented for estimating probabilities involves associated uncertainty. It is possible to employ the uncertainty model, as presented in Equation (13), to measure the uncertainty associated with the experiment used to estimate the probabilities. If the tested protection were a perfect ideal protection, it would always be tripping when testing an islanding formation case and never tripping erroneously when testing a no islanding formation case.
Equation (20) represents the maximum achievable probability estimate if the tested protection were perfect. Under this consideration, the average of success
matches the number of trials (sample size resulting:
. When the experiment is performed only once (
), the point estimates of forward success probabilities can be obtained as follows:
Equation (21) presents a set of maximum probabilities that only depend on the sample size, depending on how the statistical experiment is defined. By utilizing the set of probabilities from Equation (21), it becomes possible to determine the minimum achievable entropy through a specific experiment using Equation (13). This conclusion holds tremendous significance when aiming to minimize the uncertainty of the protection, as it represents the lowest conceivable level of uncertainty that can be calculated for a given experiment.