Conclusions

This chapters studies the mathematical background of MM and introduces the development of the basic morphological operators, including brief proofs of some significant characteristics of the operators. Basic morphological oper- ators, containing dilation, erosion, opening and closing, can be defined in both binary and grey-scale situations. The grey-scale morphological operators are considered in this thesis and several examples have also been provided to make a clearer understanding of these operators.

A set of advanced morphological operators based on the basic ones are then presented, including MG, MMG, envelope extraction MF, LFO extraction MF and CMF. MG are adept in detecting sudden changes in signals, which can be applied for detecting disturbances of transient signals. MMG, as its name suggests, is a higher version of MG and has better performance when a noisy environment is considered. The envelope extraction MF makes an achievement in successfully extracting both the upper and lower envelopes of the target signal and provides group search optimiser (GSO) algorithm with enough envelope information for calculating parameters of the continuous LFO component and reconstructing it. The LFO extraction MF, can be applied for direct extraction of the damped LFO components. CMF, combination of MF and convolution process, extracts the desired frequency component with narrow bandwidth and has benefit in eliminating other frequency components. The advanced morphological operators proposed and applied will be further illustrated in the following chapters, respectively.

Dectection and Classification of

Power Quality Disturbances

Power quality disturbances have always been of great importance to both electricity consumers and electrical utilities. Some disturbances might only briefly interfere with the most highly sensitive equipment. While others, as a result of extensive damage to the electric delivery systems, could lead to total loss of power for days. Therefore, it is necessary to minimise the influences of power quality disturbances by carefully detecting and classifying them for further analysis.

Multi-resolution decomposition scheme is a robust and significant technique for signal processing. Inspired by the application of MMG in [51], this chap- ter proposes a refined MMG algorithm based on MG to detect typical power quality disturbances. Additionally, as an improvement of the MMG approach, trapezoid SEs are applied. Utilising the proposed method to detect different types of typical power quality disturbances, the simulation results are satis- factory even in consideration of noisy environments. Furthermore, a closing operator is also used for the classification of the detected disturbances and the simulation outcomes of the proposed method are also approving.

3.1

Multi-Resolution Morphological Gradient

(MMG)

As introduced formerly, the MMG algorithm is derived from MG which is based on basic morphological operators. Morphological operators are a set of transformations which are effective in processing signals or extracting desired features. Let S and G denote the input set and SE, respectively. Two basic operators of MM, dilation and erosion can be defined as follows respectively:

δG(S) = S ⊕ G = [ g∈G (S + g), (3.1.1) εG(S) = S G = \ g∈G (S − g). (3.1.2)

MG is defined as the subtraction of erosion from dilation. The MG is denoted as:

ρG= δG(S) − εG(S). (3.1.3)

As mentioned in the previous chapter, MG can detect sudden changes of wave- forms. In this thesis, two scalable trapezoid SEs, G1 and G2, are applied to

extract the rising edges and the falling edges of the input signal respectively. They are defined as:

G1 = {g1, 0, · · · , 0, g2}, (3.1.4)

G2 = {g1, 0, · · · , 0, 0}, (3.1.5)

where g1 is set to be the peak value in the original signal, and g2 is set to be a

threshold, which can be the maximum value of the disturbance signal existed in the sinusoidal signal or higher than that. By introducing these two SEs, the computational complexity of the MMG is further reduced. According to [51], the length of SE is calculated by l = 21−a_l

G, where a is the level of SE and lG

is the primary length of the SE of G. According to the definitions of MG and SE, the dyadic MMG can be defined as:

ρa_G2 = δG2(ρa−1) − εG2(ρa−1), (3.1.7)

ρa= ρa_G1 + ρa_G2. (3.1.8) The technique of quadratic MMG (level a = 2) is processed to extract the features of typical power quality disturbances in the signal. The extracted results can be used to detect and locate the disturbances. The Difference between other approaches applying MMG is that the method proposed in this chapter applies trapezoid SEs and does not need to initially use an MF to remove the noise in the input signal. In other words, this improved version of MMG algorithm is noisy resistant to some extent.

(a) Time (s) 0.485 0.49 0.495 0.5 0.505 Original signal (kV) -0.5 0 0.5 (b) Time (s) 0.485 0.49 0.495 0.5 0.505 ρ a G 1 (kV) 0.5 1 1.5 2 (c) Time (s) 0.485 0.49 0.495 0.5 0.505 ρ a G 2 (kV) -2 -1.5 -1 -0.5 (d) Time (s) 0.485 0.49 0.495 0.5 0.505 Extracted Signal (kV) -0.5 0 0.5

Figure 3.1: Identification process using MMG: (a) original signal; (b) result of ρa

G1; (c) result of ρaG2; (d) extraction result of MMG

Figure 3.1 illustrates the process of how MMG works on the target signal. The rising and falling edges are both detected and extracted by using MMG.

Additionally, in the selection of SEs, G1is able to remove the sharp parts of the

sinusoidal signal, and to maintain the disturbance in the signal. G2 is able to

detect the falling and rising edge of the signal. Therefore, the final extraction contains all the information related to the disturbance. Moreover, by adjusting the threshold value g2 in the SE G1, the performance of MMG can be further

improved.