In order to accurately estimate the harmonic responsibility of harmonic sources, experts and scholars have conducted a lot of research [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17]. These methods can be divided into direct algorithms and indirect algorithms. The idea of the direct algorithm is to use linear regression to estimate the harmonic responsibility directly, which only needs the amplitude information of the harmonic voltage and harmonic current. But this algorithm cannot estimate harmonic parameters such as harmonic impedance. The idea of the indirect algorithm is to estimate the harmonic impedance first, then calculate the harmonic contribution voltage of each harmonic source, and then calculate the harmonic responsibility. Although this method can estimate multiple harmonic parameters, the method fails when the power quality monitor cannot accurately measure the phase value information. The method of manually constructing the phase can avoid the failure of the indirect algorithm, but it will introduce uncontrollable errors [
21].
The following sections will analyze the classic algorithms of the two methods in detail.
2.2.1. Direct Method for Estimating Harmonic Responsibility
In the direct algorithm, [
3] established a model for assessing the harmonic responsibility of multiple harmonics, and solved it by linear regression. Take the direct algorithm proposed in [
3] as an example for analysis. Based on the phasor relationship in
Figure 4, it can be obtained as
where
where
is the harmonic contribution impedance of the feeder
i in the power system. It is the parallel value of other harmonic impedances, except feeder i.
The harmonic voltage (
) at the PCC point is considered as the dependent variable. The harmonic current of each feeder (
) is considered an independent variable. Linear regression is performed on equation (3) to get the regression coefficient (
). The calculation method of the harmonic responsibility of each harmonic source is as follows:
According to the above analysis, the estimated characteristics of the direct method are as follows:
- (1)
From the perspective of information input, the direct algorithm only needs the amplitude information of the harmonic voltage and harmonic current.
- (2)
From the perspective of information output, the direct algorithm can only estimate the harmonic liability, but cannot estimate other harmonic parameters.
- (3)
The direct algorithm does not use the phase difference information of the harmonic voltage and harmonic current measured by the power quality monitor so that the algorithm cannot estimate other harmonic parameters.
2.2.2. Indirect Method for Estimating Harmonic Responsibility
In the indirect algorithm, paper [
7] first estimates the harmonic impedance parameters and then calculates the harmonic responsibility of each harmonic source. Take the indirect algorithm proposed in [
7] as an example of the analysis. The phasor relationship in
Figure 4 can be expressed as:
When the phase values of the harmonic voltage and harmonic current are known, the real and imaginary parts of equation (6) can be expanded as:
The subscript x of the variable represents the real part, and the subscript y represents the imaginary part. The harmonic voltage is considered as the dependent variable, and the harmonic current is considered as the independent variable. Linear regression is performed on Equation (7) to obtain the harmonic impedance (
). The harmonic contribution voltage (
) of the harmonic source can be expressed as:
The harmonic responsibility can be calculated as:
In summary, the estimation characteristics of the indirect algorithm are as follows:
- (1)
From the perspective of information input, the amplitude information and phase information of the harmonic voltage and harmonic current are required.
- (2)
From the perspective of information output, harmonic parameters including harmonic contribution impedance, harmonic contribution voltage, and harmonic responsibility can be estimated.
- (3)
When the phase information of the harmonic voltage and harmonic current is missing, the phase needs to be constructed artificially to avoid indirect algorithm failure, as this will inevitably cause errors.
2.2.3. The Basic Principle of the Proposed Algorithm
According to the equivalent circuit in
Figure 3, the phasor relationship in
Figure 4 can be expressed as:
Note that the harmonic current (
) in equation (10) is the harmonic current of the equivalent harmonic source, and the harmonic impedance (
) is the parallel value of all harmonic impedances in the power system. The harmonic current (
) in equation (6) is the harmonic current of the feeder, and the harmonic impedance (
) is the parallel value of all harmonic impedances in the system except the feeder. The harmonic impedance in Equation (10) is called the total harmonic impedance, and it can be expressed as:
In order to further distinguish the physical meaning of the total harmonic impedance and the harmonic contribution impedance, the equivalent circuit when a single harmonic source acts alone is shown in
Figure 5.
When the harmonic source acts alone, the theoretical harmonic current (
) on the feeder can be expressed as:
In practical power systems, the harmonic impedance on the user side is much larger than the harmonic impedance on the power supply side (
). Equation (12) can be approximated as:
The actual harmonic current on the feeder is approximately equal to the theoretical harmonic current [
3], which is expressed as:
Equation (10) can be rewritten as:
We conjugate the two ends of equation (15) and multiply the harmonic voltage (
) at both ends of the equation. The resulting equation is expressed as
where
and
can be expressed as:
where
represents the apparent power of feeder
i. and
represent the active power and reactive power of feeder i, respectively. Superscript * indicates the conjugate of a variable.
In Equation (16), the algebraic formula (
) can be regarded as a constant (
). Take the real part of equation (16), and it can be rewritten as:
In Equation (18), the active (
) and reactive power (
) of feeder i are used as independent variables, and the harmonic voltage (
) at the PCC is used as the dependent variable. The total harmonic impedance can be estimated by linear regression. Further, the harmonic contribution voltage of feeder i can be calculated, which can be expressed as:
Since the general power quality monitor can only measure the phase difference of the harmonic voltage and harmonic current instead of their phase values, the phase difference is taken as the phase value of the harmonic current in Equation (19). The phase of the calculated harmonic contribution voltage is the phase difference between the harmonic contribution voltage () and the harmonic voltage () at the PCC. This processing method does not affect the calculation of harmonic responsibility of feeder i. Harmonic responsibility can still be estimated by Equation (9).
In summary, the harmonic parameters estimated in this paper include the total harmonic impedance of the system, the harmonic contribution voltage of each feeder, and the harmonic responsibility of each feeder. The total harmonic impedance can be estimated by linear regression. And the harmonic contribution voltage of each feeder can be estimated by Equation (19). After estimating the harmonic contribution voltage of each feeder, the harmonic responsibilities of each feeder can be estimated by Equation (9).
The characteristics of the algorithm in this paper are as follows:
- (1)
The harmonic parameters, including the total harmonic impedance of the feeder i, the harmonic contribution voltage of the feeder i, and the harmonic responsibility of the feeder i can be estimated by the algorithm.
- (2)
During the estimation process, the harmonic voltage at the PCC, and the power of each feeder are required. The calculation of power parameters no longer requires the phase values of the harmonic voltage and harmonic current, only their phase difference. This makes it possible to estimate harmonic parameters such as harmonic impedance with a general power quality monitor.
Compared with the above-mentioned classic algorithm, this algorithm has the following advantages:
- (1)
Compared with the direct algorithm, the algorithm can estimate the harmonic parameters such as total harmonic impedance, in addition to the harmonic responsibility.
- (2)
Compared with indirect algorithms, the parameters required for the algorithm can be measured with a general power quality monitor.