1. Introduction
The electric power quality (PQ) in a ship system is described by the set of parameters characterizing a process of generation, distribution, and utilization of electrical energy in all operation states of the ship (maneuvering, sea voyage, staying in port) and its impact on the operation and safety of the ship as a whole. This set of parameters under consideration covers two aspects [
1]:
parameters describing the risk of loss of power supply continuity, and
parameters of voltage and currents at all the points of the analyzed system.
Parameters of the first group are essential, but the second group of parameters is significantly better recognized in the area under consideration. Nevertheless, electrical energy must first and foremost be delivered to the consumers, and then its parameters can be evaluated. Bearing in mind the aforementioned assumption, parameters of the first group are mainly associated with correct distribution of active and reactive loads among generating sets working in parallel. A main goal of their control is to avoid the “black-out” phenomenon, resulting from apparent overloading of the ship power station. Parameters of the second group are mainly expressed by the coefficients of rms (root mean square) voltage value and its frequency deviations, coefficients of voltage asymmetry, and coefficients characterizing the shape of voltage and current waveforms, which characterize the distortion of supply voltage from the sinusoidal wave. There are phenomena occurring in ship electrical power systems that can barely be detected by measuring devices in current usage. The suitable estimation of the properties of voltage in the system under consideration requires a wide range of data of steady- and non-steady-state disturbances. The occurrence of various kind of interference is strictly related to different stages of the ship’s exploitation. The variations of rms value of voltage and its frequency over switching from shaft generator to diesel propelled generator on a ro-ro ship (a roll-on/roll-off ship used to carry wheeled cargo) are depicted in
Figure 1. For example,
Table 1 presents the results of statistical analysis of parameters obtained from measurements on a ferry during maneuvering, in a network with a nominal voltage of 380 V (instantaneous and rms voltage values were obtained for a time interval of 2048 samples within 40 min). The corresponding probability density functions of assuming instantaneous values are shown in
Figure 2. In addition to non-steady phenomena, there are steady-state phenomena such as harmonics (
Figure 3). The usually description of harmonic distortion is done by means of total harmonic distortion (THD) or/and factors of respective harmonics content (
Table 2). Therefore, tests are needed for a relatively long period of the ship’s exploitation processes in various electrical power plant configurations. According to the recently updated International Association of Classification Societies (IACS) requirements [
2], newly built ships are to be equipped with devices to continuously monitor the levels of harmonic distortion, while the PQ factors should be measured on existing ships annually under seagoing conditions. The related measurements, therefore, result in the need to collect a large amount of digital data, which in turn leads to a huge memory occupancy and burden in data processing.
A promising solution could be the implementation of a compressing sensing (CS) technique for data acquisition, and further, the use of appropriate algorithms for data reconstruction.
The main idea of CS is to combine sampling and compression of the signals that are sparse or compressible, either in their original domain or in a certain transformed domain. Differently from the typical approach, a new CS technique provides an estimate of the examined signal from a small number of linear incoherent measurements [
5,
6,
7,
8]. The linear projection values of the original signal are acquired by a measurement matrix directly with a lower sampling rate than the Nyquist frequency. The sampling frequency does not depend on the signal bandwidth, but on the structure and contents of the information in the signal. The essential assumption in the CS approach is that most of the signals in real applications have a sparse representation in a transform domain, which means a few coefficients are significant, while the rest are negligible or zero. Many of the signals appearing in real applications have a sparse representation in the discrete Fourier transform (DFT) domain [
9,
10,
11,
12]. Another relevant condition of the CS technique is the incoherence between measurement (observation) basis and domain in which the signal has a sparse representation.
Recent research on the CS technique indicates the possibility of accurate signal frequency analysis [
13,
14]. CS is widely used in diverse fields, e.g., biomedical applications, communication systems, and pattern recognition, as well as electrical PQ estimation [
15,
16]. Various new techniques for identifying and estimating harmonic sources in electricity supply systems have been presented in the literature [
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28]. The papers [
17,
18,
19,
20,
21] concern the distributed monitoring of harmonic and interharmonic pollution in electrical power delivery systems. In order to reduce implementation costs, the authors propose a distributed architecture, based on cost-effective nodes that use the CS strategy. Some papers [
22,
23,
24,
25,
26,
27,
28] consider the different methods of compressive sampling and innovative CS algorithms for reconstruction of the PQ interference signal.
There are many CS reconstruction algorithms in the literature that present different approaches for finding a sparse estimation of the original input signal, based on a minimum number of measurements, and characterized by interference resistance, speed, complexity, good performance, etc. One type of reconstruction method is based on the Bayesian framework, where the posterior density function of the sparse solution is estimated [
29]. This approach to CS constitutes the reconstruction as a Bayesian inference problem and is applicable for the input signals that fit some known probability distribution. Bayesian compressive sensing (BCS) can be used to estimate the time–frequency spectrum of a nonstationary signal [
30]. A certain type of BCS approach is the Monte Carlo Bayesian compressive sensing (MC-BCS) method, which is distinguished by the fact that it numerically evaluates the posterior sparse solution [
31].
In many cases, reconstruction algorithms perform moderately at low measurement rates and are computationally expensive. In practice, the purpose of measurement is not always to perfectly reconstruct the input signal, but to determine some of its parameters. Recent advances in the areas of CS have shown that effective inference is possible directly from the compressive measurements, without reconstruction, using correlational features [
32]. This idea is currently developed in computational imaging [
33,
34,
35]. A good research direction might be to consider the possibility of using this approach for harmonic detection. However, the contribution of the present paper is the CS reconstruction algorithm, which allows obtaining a satisfactory accuracy of harmonic detection based on a much smaller number of measurements than in the classic approach.
This paper proposes the application of a fast reconstruction procedure based on the CS technique for detecting harmonics in a tested signal. The procedure uses random projections as measurements. The measurement matrix, generated from Bernoulli’s random variables, allows recovering the signal with high accuracy. The ℓ1-minimization problem in the CS signal reconstruction is solved by means of discrete Radon transform (DRT) techniques with the use of the K-rank-order filter in the signal’s sparse domain to accelerate the solution convergence.
The organization of the remainder of this paper is as follows.
Section 2 discusses the CS framework in three aspects: the sparsity of the signal, the sensing process, and the reconstruction condition.
Section 3 explains the algorithmic implementation of the reconstruction procedure based on DRT techniques.
Section 4 shows the preliminary results of simulation obtained for the selected multitone signals. A brief discussion is carried out in
Section 5. Finally, concluding remarks are formulated in
Section 6.
4. Numerical Simulations
The simulations were performed using a virtual instrument designed based on an accessible application in the LabVIEW environment [
32]. As an example, a multi-tone signal with fundamental harmonic 50 Hz was simulated, according to the parameter sets shown in
Table 3. In presented exemplary results of simulations, sparsity level
K was set to 7. The sampling frequency was equal to 10 kHz and the length of the time window was equal to 1000 samples. The time waveform and sparse representation of the tested signal in the Fourier domain is presented in
Figure 4.
First, the influence of the value of the ones probability
p on the level of quality of a signal reconstruction and the number of iterations (measurements) was examined. For a fixed
t-threshold level of 99%, the value
p was changed in the range of 0.01 to 0.9 (
Figure 5). The simulation results show that depending on the value of
p the algorithm converges for a different number of iterations. With an assumed
t-threshold, the loop stops more quickly at the lowest
p-values, with an approximate accuracy of 99.67%. The algorithm achieves the best reconstruction accuracy for a
p-value of 0.5 but with the largest number of measurements. Comparing the segments of time waveforms of the original signal and reconstructed signal (
Figure 5) for the parameters probability
p, accuracy, and the number of iterations, we can preliminarily conclude that the most optimal case occurs at
p = 0.3, for which, at 490 iterations, the accuracy is in the order of 99.8%.
In the case of generating the tones with the slightly different levels of amplitude (set 1), the algorithm identifies harmonics with accuracy above 97%. The most accurate signal reconstruction (99.32%) is obtained for 210 sampled measurement signals, acquired with the ones probability
p equal to 0.3 (
Figure 6).
For the multi-tone signal with different levels of amplitude (set 2), the reconstruction based on 280 measurements and the same features of the measurement matrix does not allow the correct detection of all harmonics (
Figure 7).
The proper reconstruction requires twice as many measurements, i.e., 400. The scenario with the dominant fundamental harmonic (set 3) shows that 500 sampled measurements are not sufficient to identify all components in the frequency domain (
Figure 8). The algorithm incorrectly detects 7th harmonics. Analyzing the results of simulations carried out for set1, set2 and set3, respectively, shown in
Figure 6,
Figure 7 and
Figure 8, and bearing in mind the data listed in
Table 1, we may conclude that increasing the number of measurements results in better efficiency of harmonic identification. In this case, the number of random samples reaches 800.
To study the effect of noise on the efficiency of signal reconstruction, white Gaussian noise is added to obtain a signal to noise ratio (S/N) of 20 dB.
Figure 9 shows the result of a sparse reconstruction of a multi-tone waveform with insignificant differences of amplitude levels (set 1) for 210 iterations (measurements). The presence of noise adversely affects the correct identification of harmonics. In the case of set 2, in which one tone is characterized by a much higher amplitude level, adding noise does not affect the detection of harmonics (
Figure 10). Noise interference distorts the precision of the spectral analysis of a multi-tone signal when there is a strong dominant component in the signal (set 3).
Performing a spare signal reconstruction allows for more accurate detection of harmonics in the signal (
Figure 11). Taking into account the results of simulation with the additive noise corresponding to the appropriate data sets shown in
Figure 9,
Figure 10 and
Figure 11, we may formulate a preliminary opinion that accurate detection of harmonics is only possible with a very large number of iterations in the measurement algorithm, but the number of measurements is still smaller than when using the classic approach.
6. Conclusions
In ship operations, effective resource management is crucial to its proper functioning. In view of the multitude of measurements on a ship, the search for effective means for their implementation is of great importance. One approach to reduce the load on memory and data processing systems is to use the CS technique.
Preliminary results of the simulations carried out confirmed that the key advantage of the proposed approach is the high speed of analysis, which uses software algorithms widely available in programming environments. Furthermore, the data processing algorithm is quite simple, and therefore memory consumption and burden in data processing are relatively low.
The future task of the planned research is to improve and develop the procedures presented with an extension to a real object investigation in a ship environment.