2.1. Currents’ Physical Component Theory Overview
The Currents’ Physical Component based power theory deals with the calculations of various powers in a single and three phase systems. The concept is applied usually for the calculation and definition of non-active powers in the presence of non-sinusoidal currents and voltages. The spectra of each waveform are used to define the waveform origin by means of CPC theory. CPC theory was developed by L. S. Czarnecki [
21]. Since it was first publicized, there have been developments and adaptations of this power theory [
22,
23,
24,
25]. In this section, a short overview of the method is presented with emphasis on the components and the powers which are used in this study as features for uniquely identifying an electric load.
Fourier series is the most general way to present time domain non-sinusoidal harmonic voltage
where
is the DC component (which is ignored in the rest of the paper since it usually does not exist) and
is the complex RMS value of the voltage at the
nth harmonic, where
, and
N is an integer. The harmonic
admittance for the
nth harmonic can be expressed by the complex expression,
Multiplying the voltage and admittance yields the resulting current harmonic Fourier series
as follows
The active power of such a system is derived from the scalar product of Equations (
1) and (
3) and therefore only the harmonic components which have the same index in both the voltage and the current will contribute to this power
,
where
and
are the voltage and current amplitudes at the
nth harmonic, and
is the angle between the voltage and the current at the same harmonic. After the active power is calculated, the equivalent active conductance of the system can be defined as
where
is the RMS value of the voltage, which can be expressed as
The active current
can now be calculated as
Subtracting this
active current from the general current yields
The imaginary part of Equation (
8) can be considered as
, the
reactive current, which is
shifted relative to the same voltage harmonic [
26,
27],
The remaining part of Equation (
8) is called the
scattered current
. This current appears if
and it is a measure at voltage
of the source current increase due to a scattering of conductance
around the equivalent conductance
[
26,
27] as shown next
Nonlinear loads can be considered as harmonic generating loads. In the presence of such loads the direction of energy flow between the source and the load can be decided by investigating the angle between the current and the voltage of the nth harmonic. If then there is an average energy flow from the source to the load and, if , then the energy flow is from the load towards the source. In this case, the harmonics can be sorted into two groups: (1) is the set of harmonics which are originated at the source; and (2) is the set of harmonics which are originated at the load. Therefore, the system can be divided into two subnetworks at the measuring point. The first is the source network, denoted by D and include the set of harmonics. The second is the load subnetwork, C, which includes the set of harmonics. Each subnetwork can be considered to have its own current and voltage, as noted below.
The current which is associated with the
group is called the
load generated current,
. The total current can now be written as
This current can be rewritten with respect to the
and
disaggregation as follows:
where
N is the set of all harmonic content in the decomposed total current. This set is divided into the source and load sets
and
, accordingly [
28]:
Note that the load generated current can be also disaggregated to the , , and components; however, in this work, we limit the amount of features and take into consideration the aggregated .
It can be shown that all components in CPC theory are orthogonal and therefore
The same procedure can be implemented to the voltage. Thus, the expression for the voltage, taking into account the direction of the energy flow, is:
In [
28], a detailed discussion is presented regarding the decomposition and analysis of various powers in a non-sinosoidal system. There, the apparent power
S can be expressed as:
where
and
are the apparent powers of the subnetworks
D and
C when they are considered as sourceless loads, and
is the forced apparent power, which is the voltage of each subnetwork (
D and
C) combined multiplied by the current of the other subnetwork. Namely, the voltage
is multiplied by
and vice versa. Now, the powers
and
can be decomposed into the active, reactive and scattered powers.
can be written as:
where
is the active power,
is the reactive power and
is the scattered power of subnetwork
D. For the purpose of this paper, we do not decompose
to limit the number of features. Therefore, the total apparent power can be expressed as:
There has been a debate about the applicability of this method for measuring power, and whether these components are mathematical or physical [
24,
29]. However, this paper focuses on the classification and recognition method, and, for these purposes, CPC is a promising method for extracting a fixed, small amount of descriptive and quantity features from the signals, as opposed to using the FFT harmonics, which will have a different number of features for each signal depending on its nonlinearity extent.
2.2. Artificial Neural Network
The artificial neural network (ANN) is a technique inspired by the neural structure of the brain that mimics the learning capability from experiences. It means that, if a neural network is trained using past data, it will be able to generate outputs based on the knowledge extracted from the training. Neural networks, a machine-learning method, have been used to solve a wide variety of tasks that are hard to solve using ordinary rule-based programming. When there is a sufficient amount of data to train the ANN, classification produces good results. As mentioned in [
30], there are many advantages to using ANN for the power and energy field. In this study, we used the back-propagation algorithm for the neural network [
31]. The network contains three layers: input layer, referred to by the index
i; hidden layer, referred to by the index
j; and output layer, referred to by the index
k.
In the training phase, as shown in
Figure 1, the training data, features with known labels, are fed into the input layer. In this stage, appropriate weights between the nodes are adjusted iteratively for improving the network until it can perform the task for which it is being trained. After the weights are updated, the network best maps a set of inputs to their correct output.
One of the central ideas in ANN is the transfer function, which is a monotonically increasing, continuous function, applied to the weighted input of a neuron to produce its output. A well-known transfer function is the sigmoid which defined as
An exponential function of the derivative is simply
Another exponential function with simple derivative is softmax, which is defined as
Due to the simplicity of derivative, these functions were chosen as the transfer function of the nodes in the training network. The sigmoid is mainly used in the hidden layer while the softmax is mainly used in the output layer.
Next, simplified back-propagation algorithm is describing shortly: