# Estimation of an Extent of Sinusoidal Voltage Waveform Distortion Using Parametric and Nonparametric Multiple-Hypothesis Sequential Testing in Devices for Automatic Control of Power Quality Indices

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## Abstract

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## 1. Introduction

- Upsets of production processes, defective products, and economic damages associated with disruption of the normal functioning of essential electrical loads;
- Increases in electricity losses;
- A rise in electricity consumption for the same production processes;
- A reduction in the reliability of both the systems of power supply to industrial consumers and the electrical equipment.

## 2. State-of-the-Art Literature Review

- Mathematical morphology [33];
- Decision trees [34];
- Statistical analysis [39];
- Logistic regression [40];
- Principal component analysis [41];
- K-nearest neighbors method [42];
- Wald’s sequential analysis [43];

- The potential capabilities for classifying PQI deviations from standard values in the event of complex emergency disturbances (distortions of sinusoidal voltage waveforms) and the impact of noise and interference [50];
- The volumes of necessary calculations and their high speed required when implementing PQI control devices based on software and hardware platforms;
- The amount of memory required to store simulation results and other information for making decisions on classification of PQI deviations from standard values [51];
- The organization of special digital processing of current and voltage signals [52];
- The magnitude of the error in classifying various PQI deviations from standard values;
- Other factors.

- Each power supply system of an industrial consumer has its specific relationship between the amount of damage and the depth and duration of the voltage dip [55];
- Voltage dips in external power supply networks, which are random in nature, are often accompanied by PQI deviations from standard values, including distortions of the sinusoidal voltage waveform, the presence of noise and interference.

## 3. Materials and Methods

_{a}, U

_{b}, U

_{c}are the amplitude values of the sinusoidal components of all voltage phases in a three-phase system; k is a discrete time instance; t

_{d}= 1/f

_{d}is a sampling interval; f

_{d}is a sampling frequency; φ is an initial phase; and f is power frequency in the network.

_{a}(k), u

_{b}(k), u

_{c}(k), the space vector is given by Expression (4):

_{p}(k), u

_{q}(k) of the Park–Gorev transform by the vector-matrix relationship in expression (7) [57]:

_{a}= U

_{b}= U

_{c}), the equalities given in Expression (8) hold true:

_{α}(k) and u

_{β}(k) of a balanced three-phase voltage system is a constant value and corresponds to Expression (9):

_{α}(k) and u

_{β}(k), varying in time up to a constant coefficient $\left(\frac{\sqrt{2}}{\sqrt{3}}\right)$, correspond to the orthogonal coordinates of the end of the space vector. The position of the space vector on the complex plane changes in time at a rate proportional to the frequency of the three-phase voltage system. Thus, a complex space vector can be represented by Expression (10):

_{a}$\ne $ U

_{b}$\ne $ U

_{c}) [58], then the complex space vector can be represented by equality (11):

_{α}(k) and u

_{β}(k), which are given in Expressions (12) and (13):

_{a}= U

_{b}= U

_{c}), the complex amplitude of the negative-sequence component of the space vector (expression (11)) becomes zero.

_{d}= 1/t

_{d}= 1 kHz (twenty samples per cycle of power frequency).

_{a}(k), u

_{b}(k), u

_{c}(k) of a three-phase system with a power frequency f = 50 Hz. The dependencies shown in Figure 1 correspond to secondary voltage signals (at the output of the measuring voltage transformer) with an amplitude of 100 V for a specific node in the power supply system of an industrial consumer. This signal is considered to be a reference (undistorted) and, relative to it, we will assess the extent of distortion of the sinusoidal voltage waveform. Figure 1b,c demonstrates the components of a complex space vector and its rotation trajectory on the complex plane, respectively.

## 4. Results and Discussion

- Calculate M likelihood ratio at each step n of the procedure by Expression (18):$${\lambda}_{m}^{n}=\frac{{p}_{m}^{n}\left(x\right)}{{p}_{1}^{n}\left(x\right)},$$
- Determine the two largest of M likelihood ratio values at each step n (${\lambda}_{max1}^{n}$ and ${\lambda}_{max2}^{n}$), and select hypotheses corresponding to these likelihood ratios;
- Determine threshold values ${\lambda}_{m}^{n\ast}$ for each of the selected hypotheses, using Expression (19):$${\lambda}_{m}^{n\ast}=\frac{M}{{4\xb7(1\text{}\u2013\text{}{P}_{mm})}^{2}},$$
- Calculate ratio of ${\lambda}_{max1}^{n}$ to ${\lambda}_{max2}^{n}$ and compare it with the threshold value ${\lambda}_{m}^{n\ast}$:$${\lambda}_{max}^{n}=\frac{{\lambda}_{max1}^{n}}{{\lambda}_{max2}^{n}}\text{},$$
- Make a decision about the validity of hypothesis m using Expression (20) provided that:$${\lambda}_{max}^{n}=\frac{{\lambda}_{max1}^{n}}{{\lambda}_{max2}^{n}}{\ge \lambda}_{m}^{n\ast},$$

- The sequential assessment of the extent of the sinusoidal voltage waveform distortion leads to the adoption of the hypothesis of a 15% distortion of the coefficient μ, which corresponds to an unacceptable amount of damage for an industrial consumer;
- The procedure for multiple-hypothesis sequential testing by Palmer’s algorithm is completed at step 3, which does not require significant time expenditure and has virtually no effect on the performance of the automatic PQI control device;
- The speed of decision making in multiple-hypothesis sequential testing with Palmer’s algorithm depends on the degree of distortion of the sinusoidal voltage waveform, including PQI deviations from standard values.

_{a}(k), u

_{b}(k), u

_{c}(k)

_{,}measured at the analyzed node. At each time instant, module 2 calculates the phase voltage amplitudes U

_{a}, U

_{b}, U

_{c}. The device (Figure 9) does not calculate the space vector, and the coefficient μ is obtained using Expressions (16) and (17).

_{a}(k), U

_{b}(k), U

_{c}(k) are calculated from instantaneous values u

_{a}(k), u

_{b}(k), u

_{c}(k), using “short data window” algorithms, in particular, the two-sample test [71].

_{a}(k) and U

_{aq}(k) are the quadrature (orthogonal) components of the complex vector $\underset{\_}{{U}_{a}}\left(k\right)$.

_{a}, rotating with angular velocity 2π∙f relative to the origin of coordinates.

_{a}(k), U

_{b}(k), U

_{c}(k).

_{a}(k), U

_{b}(k), U

_{c}(k) are used to directly calculate the coefficient μ(k). In this case, it is not necessary to use the Clarke transform. The calculated values of μ(k) from Module 3 arrive at the input of computation unit 4 and memory unit 9.

_{m}[73]. According to this method, for an L-dimensional random variable x

_{l}, hypothesis m (m = 1, 2, …, M) is accepted if the minimum Euclidean distance between x

_{l}and G

_{m,l}is ensured based on the expression:

- Calculate the average value of the random variable x at step n of sequential testing:$${x}^{^}=\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i};$$
- Determine the minimum distance d
_{min}, which for an L-dimensional random variable ${x}^{^}$ can be found by Expression (29):$${d}_{min}=\underset{l}{\mathrm{min}}\sqrt{{\sum}_{l=1}^{L}{\left({x}_{l}^{^}-{G}_{m,l}\right)}^{2}};$$ - Classify the extent of the sinusoidal voltage waveform distortion by comparing it with hypothesis m:$$m=\underset{l}{\mathrm{arg}\mathrm{min}}\sqrt{{\sum}_{l=1}^{L}{\left({x}_{l}^{^}-{G}_{m,l}\right)}^{2}},$$
_{min}with the threshold A_{m}(n), d_{min}≤ A_{m}(n); - Check if the condition d
_{min}≤ A_{m}(n) is met, otherwise continue sequential analysis.

_{m}(n), it is advisable to use the calculation methods employed for parametric algorithms. For example, according to the Armitage algorithm [38], the threshold value is calculated using Expression (30):

_{mq}is the conditional probability of making a decision about the number of hypothesis m regarding the extent of the sinusoidal voltage waveform distortion, provided that the distortions belong to hypothesis q.

_{min}. The number of observations, while maintaining the simplicity of the approach, can be reduced by applying a multiple-hypothesis sequential testing procedure. By analogy with [69], we will establish a set of threshold values that depend on the number of observations n:

^{^}with the mathematical expectations of statistical distributions (Figure 6). At each step n of the multiple-hypothesis sequential testing procedure based on the nearest neighbor method, we calculate the distance for m intervals (Table 1) and compare the obtained distance values with the corresponding thresholds calculated by Expression (31).

**P**to determine the probability of errors and correct decisions when classifying distortions of the sinusoidal voltage waveform of a three-phase system involved in expression (30). For example, given the total number of distortion options M = 5, matrix

**P**is set in the form:

**P**, the threshold values for all options of distortion m (m = 1, …, M) are the same and equal to:

- The sequential assessment of the extent of the sinusoidal voltage waveform distortion based on the nearest neighbor method, as in the case of using Palmer’s algorithm, leads to the acceptance of the hypothesis of a 15% distortion of coefficient μ;
- The multiple-hypothesis sequential testing procedure is completed at Step 3 (Figure 11), which does not require significant time, therefore, there is no need to introduce an adaptive threshold to increase the speed of the algorithm;
- The advantage of the multiple-hypothesis sequential testing based on the nearest neighbor method is that there is no need to use statistics and distributions in the calculation process.

^{^}

_{nor}(k) obtained from the results of averaging the values μ(k) received at its input. Averaging is carried out in accordance with Expression (28), and normalization factors in the normalizing coefficient come from memory unit 9. Distances are calculated using Expression (29) as the square of the difference between μ

^{^}

_{nor}(k) and the normalized values of the range centers (Table 1), which characterize the extent of the sinusoidal voltage waveform distortion.

_{min}(expression (29)) to the input of comparison circuit 7. The threshold value A

_{m}, obtained from expression (30) is received from memory unit 9 at the other input of comparison circuit 7(n). The condition d

_{min}≤ A

_{m}(n) is checked, and if it is met, the multiple-hypothesis sequential testing procedure ends. If the condition d

_{min}≤ A

_{m}(n) is not fulfilled, the procedure continues.

_{min}, for which the condition d

_{min}≤ A

_{m}(n) is met, characterizes the extent of distortion of the sinusoidal voltage waveform of a three-phase system [74,75].

^{−3}/256) = 2.34∙10

^{−4}s at f = 50 Hz.

#### Future Research Directions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Three-phase symmetrical system of discrete voltages: (

**a**) phase voltages u

_{a}(n), u

_{b}(n), u

_{c}(n) of a three-phase system with power frequency f = 50 Hz; (

**b**) components of the complex space vector; (

**c**) space vector rotation trajectory on the complex plane.

**Figure 2.**Distortion of the sinusoidal voltage waveform: (

**a**) phase voltage of a three-phase system with a single-phase short circuit in phase “B”; (

**b**) components of the complex space vector; (

**c**) the space vector rotation trajectory on the complex plane.

**Figure 3.**Time-varying change in the coefficient μ, which characterizes the extent of the sinusoidal voltage waveform distortion.

**Figure 4.**Distorted three-phase system of discrete voltages with simultaneous deviation of several PQI: (

**a**) phase voltages of a three-phase system with a single-phase short circuit in phase “B” and a nonlinear single-phase load in phase “A”; (

**b**) components of the complex space vector; (

**c**) the space vector rotation trajectory on the complex plane.

**Figure 5.**Time-varying change in the coefficient μ: (

**a**) with a single-phase short circuit in phase “B” and distortion by the fifth harmonic in phase “A”; (

**b**) with a voltage dip and distortion by the fifth harmonic in all three phases.

**Figure 6.**An example of probability densities for average values of μ^ corresponding to hypotheses about the extent of sinusoidal voltage waveform distortion.

**Figure 7.**Block diagram of the algorithm that implements the procedure of multiple-hypothesis sequential testing based on Palmer’s algorithm.

**Figure 8.**Implementation of the multiple-hypothesis sequential testing procedure according to Palmer’s algorithm.

**Figure 9.**Flowchart of a device that implements the procedure of multiple-hypothesis sequential testing with Palmer’s algorithm.

**Figure 10.**Block diagram of an algorithm for the procedure of multiple-hypothesis sequential testing based on the nearest neighbor method.

**Figure 11.**Implementation of the multiple-hypothesis sequential testing procedure based on the nearest neighbor method.

**Figure 12.**Flowchart of an automatic PQI control device, which implements multiple-hypothesis sequential testing based on the nearest neighbor method.

Variation Ranges of μ | 0.00–0.05 m = 1 | 0.05–0.10 m = 2 | 0.10–0.15 m = 3 | 0.15–0.20 m = 4 | 0.20–0.25 m = 5 | … |
---|---|---|---|---|---|---|

Average value μ^ | 0.025 | 0.075 | 0.125 | 0.175 | 0.225 | … |

**Table 2.**Sample values of the coefficient μ when classifying the extent of distortion of the sinusoidal voltage waveform of a three-phase system.

Sequential Testing Procedure Step | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | … |
---|---|---|---|---|---|---|---|---|---|

Coefficient value μ | 0.105 | 0.11 | 0.127 | 0.118 | 0.12 | 0.10 | 0.10 | 0.124 | … |

**Table 3.**Calculated values of the multiple-hypothesis sequential testing procedure using Palmer’s algorithm.

Procedure Step | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | … |
---|---|---|---|---|---|---|---|

$\mathrm{Value}\text{}{\lambda}_{max1}^{n}$ | 12.07 | 332 | 66,068 | – | – | – | … |

$\mathrm{Value}\text{}{\lambda}_{max2}^{n}$ | 8.1 | 101 | 2323 | – | – | – | … |

$\mathrm{Ratio}\text{}{\lambda}_{max1}^{n}/{\lambda}_{max2}^{n}$ | 1.49 | 3.29 | 28.44 | – | – | – | … |

$\mathrm{Threshold}\text{}\mathrm{value}\text{}{\lambda}_{m}^{n\ast}$ | 19.8 | 19.8 | 19.8 | 19.8 | 19.8 | 19.8 | … |

Accepted hypothesis m | – | – | m = 3 | – | – | – | … |

**Table 4.**The calculated values of the multiple-hypothesis sequential testing procedure based on the nearest neighbor method.

Procedure Step | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | … |
---|---|---|---|---|---|---|---|

Coefficient μ_{nor} | 31.5 | 33 | 38.1 | – | – | – | ... |

The value of μ^_{nor} | 31.5 | 32.25 | 34.2 | – | – | – | ... |

Distance value for distortion option m | 576 (m = 1) 81 (m = 2) 36 (m = 3) 441 (m = 4) 1296 (m = 5) | 625 (m = 1) 100 (m = 2) 25 (m = 3) 410 (m = 4) 1242.6 (m = 5) | 712.9 m = 1) 136.9 (m = 2) 10.89(m = 3) 334.9 (m = 4) 1108.9 (m = 5) | – | – | – | ... |

$\mathrm{Threshold}\text{}\mathrm{value}\text{}{\lambda}_{m}^{n\ast}$ | 11.9 | 11.9 | 11.9 | 11.9 | 11.9 | 11.9 | ... |

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## Share and Cite

**MDPI and ACS Style**

Kulikov, A.; Ilyushin, P.; Sevostyanov, A.; Filippov, S.; Suslov, K.
Estimation of an Extent of Sinusoidal Voltage Waveform Distortion Using Parametric and Nonparametric Multiple-Hypothesis Sequential Testing in Devices for Automatic Control of Power Quality Indices. *Energies* **2024**, *17*, 1088.
https://doi.org/10.3390/en17051088

**AMA Style**

Kulikov A, Ilyushin P, Sevostyanov A, Filippov S, Suslov K.
Estimation of an Extent of Sinusoidal Voltage Waveform Distortion Using Parametric and Nonparametric Multiple-Hypothesis Sequential Testing in Devices for Automatic Control of Power Quality Indices. *Energies*. 2024; 17(5):1088.
https://doi.org/10.3390/en17051088

**Chicago/Turabian Style**

Kulikov, Aleksandr, Pavel Ilyushin, Aleksandr Sevostyanov, Sergey Filippov, and Konstantin Suslov.
2024. "Estimation of an Extent of Sinusoidal Voltage Waveform Distortion Using Parametric and Nonparametric Multiple-Hypothesis Sequential Testing in Devices for Automatic Control of Power Quality Indices" *Energies* 17, no. 5: 1088.
https://doi.org/10.3390/en17051088