Due to the increasing power consumption of whole society and widely using of new non-linear and asymmetric electrical equipment, power quality assessment problem in the new period has attracted more and more attention. The mathematical essence of comprehensive assessment of power quality is a multi-attribute optimal decision-making problem. In order to solve the key problem of determining the indicator weight in the process of power quality assessment, a rough analytic hierarchy process (AHP) is proposed to aggregate the judgment opinions of multiple experts and eliminate the subjective effects of single expert judgment. Based on the advantage of extension analysis for solving the incompatibility problem, extension analysis method is adopted to assess the power quality. The assessment grades of both total power quality and each assessment indicator are obtained by correlation function. Through a case of 110 kV bus of a converting station in a wind farm of China, the feasibility and effectiveness of the propose method are demonstrated. The result shows that the proposed method can determine the overall power quality of power grid, as well as compare the differences among the performance of assessment indicators and provide the basis for further improving of power quality.
Scientific and reasonable comprehensive assessment of power quality of smart grid is the basis for restraining and urging power companies and power users to jointly maintain the power quality environment of public power grid, and is also the main basis for measuring power quality and formulating electricity price, which has profound theoretical and practical significance [
In the process of evaluation, the above methods are uncertain to a certain extent or too subjective. For example, in reference [
The combination of rough sets theory [
In view of the shortcomings of the above methods, rough sets theory is introduced to improve the classical AHP when determining the weight of indicators in the power quality assessment indicator system and a rough AHP is proposed to aggregate the weight information of multiple experts, so as to eliminate the subjective influence of a single expert. Based on the characteristic of extension analysis for solving the problem of incompatibility of various features of things, extension analysis method is adopted to assess the power quality.
In this paper, six power quality standards of national grid are used as the basis of comprehensive assessment of power quality, and each power quality standard is taken as assessment indicator. The six assessment indicators of power quality are frequency deviation, total harmonic distortion rate of voltage, voltage fluctuation, voltage flicker, voltage deviation and voltage three-phase unbalance degree, which are expressed by
The overall research framework is shown in
The construction of assessment indicator system is given in the above section. The other parts are explained in In the determination of assessment indicator weight, multi-expert comprehensive judgment is adopted for eliminating the adverse effects of single expert judgment, rough sets theory is introduced to aggregate the judgment opinions of all experts, and the weights of the indicators is calculated by AHP. In the assessment of power quality, the classical domain matter-element, joint domain matter-element and matter-element to be assessed are obtained by extension analysis theory. Then based on correlation function both the assessment grade of total power quality and each assessment indicator can be obtained. According to the assessment result, the key factors affecting power quality will be analyzed.
AHP method [
Here,
In
According to the above definition, any ambiguous partition
As can be seen, an ambiguous partition in the domain can be represented by a rough boundary interval containing a rough lower limit and a rough upper limit as follows:
In the proposed rough AHP, the concept of rough boundary interval in rough sets theory is introduced in classical AHP to aggregate the judgment opinions of multiple experts. The detailed steps are as follows:
According to the indicator system shown in
Scale | Meaning |
---|---|
1 | The two factors have the same importance. |
3 | The former factor is slightly more important than the latter factor. |
5 | The former factor is obviously more important than the latter factor. |
7 | The former factor is intensively more important than the latter factor. |
9 | The former factor is more important than the latter factor in the extreme. |
2, 4, 6, 8 | The intermediate value of the above adjacent judgments. |
Reciprocal | If the scale of importance of factor 1 to factor 2 is |
Assuming that there are
For pairwise comparison matrix
Matrix dimensionality | |
---|---|
1 | 0 |
2 | 0 |
3 | 0.52 |
4 | 0.89 |
5 | 1.12 |
6 | 1.26 |
7 | 1.36 |
8 | 1.41 |
9 | 1.46 |
When
Based on pairwise comparison matrices
The rough boundary interval of
Therefore, the rough boundary interval of
Based on the operational rule of rough boundary interval, the average form of
The rough judgment matrix is constructed as follows:
Rough judgment matrix
The eigenvectors corresponding to the maximum eigenvalues of
The weight value of indicator
Lastly the weight vector is obtained as follows:
Extension analysis theory can classify the assessment indicators of power quality from the dynamic and transformation perspective and quantitatively express the quality grade of the indicators by the value of correlation function. The calculation of correlation function is based on the actual measured objective data.
According to the extension analysis theory, the matter-element to be assessed is expressed by
The classical domain refers to the range of values specified by the corresponding characteristics of each assessment grade of the object to be assessed. The classical domain matrix can be expressed as follows:
The joint domain refers to the range of values specified by all assessment grades of the object to be assessed.The joint domain matrix can be expressed as follows:
Correlation function describes the mapping of elements in matter element to real axis, which is specifically expressed as the quantitative relationship between
The module of bounded interval
The distance from point
Assuming that there are two intervals
The properties of correlation function When When When When
Here, let
Based on weight vector
If
With the rapid development of economy and the gradual improvement of people's living standards, the power consumption of the whole society is increasing, and various new non-linear asymmetric electrical equipment have been widely used. A converting station in a wind farm of China is in line with these characteristics and is very representative in China. Therefore its 110 kV bus can be taken as research object to illustrate the feasibility of the proposed method. Its power quality is comprehensively assessed by the proposed method as follows. The power quality is divided into four quality grades: high quality, good, medium and qualified. They are expressed by
The grading limits and measured data of power quality indicators are shown in
Indicator | Measured data | ||||
---|---|---|---|---|---|
[0, 0.05] | [0.05, 0.1] | [0.1, 0.15] | [0.15, 0.2] | 0.04 | |
[0, 0.5] | [0.5, 1] | [1, 1.5] | [1.5, 2] | 0.82 | |
[0, 0.5] | [0.5, 1] | [1, 1.5] | [1.5, 2] | 1.05 | |
[0, 0.2] | [0.2, 0.5] | [0.5, 0.8] | [0.8, 1] | 0.34 | |
[0, 2] | [2, 5] | [5, 8] | [8, 10] | 1.66 | |
[0, 0.5] | [0.5, 1] | [1, 1.5] | [1.5, 2] | 1.07 |
Firstly the weight of indicator is determined by the proposed rough AHP as follows. Assuming that there are three experts, each expert carries out the pairwise comparison of the indicators according to 1–9 scales in
1 | 5 | 2 | 1/2 | 7 | 4 | |
1/5 | 1 | 1/3 | 1/8 | 2 | 2 | |
1/2 | 3 | 1 | 1/4 | 4 | 3 | |
2 | 8 | 4 | 1 | 8 | 6 | |
1/7 | 1/2 | 1/4 | 1/8 | 1 | 1/2 | |
1/4 | 1/2 | 1/3 | 1/6 | 2 | 1 |
1 | 2 | 7 | 2 | 1 | 3 | |
1/2 | 1 | 3 | 2 | 1/2 | 2 | |
1/7 | 1/3 | 1 | 1/3 | 1/8 | 1/3 | |
1/2 | 1/2 | 3 | 1 | 1/3 | 2 | |
1 | 2 | 8 | 3 | 1 | 3 | |
1/3 | 1/2 | 3 | 1/2 | 1/3 | 1 |
1 | 4 | 5 | 2 | 3 | 1/2 | |
1/4 | 1 | 2 | 1/3 | 1/2 | 1/7 | |
1/5 | 1/2 | 1 | 1/3 | 1/2 | 1/9 | |
1/2 | 3 | 3 | 1 | 2 | 1/5 | |
1/3 | 2 | 2 | 1/2 | 1 | 1/6 | |
2 | 7 | 9 | 5 | 6 | 1 |
The consistency index and consistency ratio of the three pairwise comparison matrices is shown in
0.0320 | 0.0254 | |
0.0204 | 0.0162 | |
0.0201 | 0.0159 |
As can be seen, the consistency ratios of the three pairwise comparison matrices all have values less than 0.1, so the consistency of each pairwise comparison matrix is acceptable. Then according to
{1, 1, 1} | {5, 2, 4} | {2, 7, 5} | {1/2, 2, 2} | {7, 1, 3} | {4, 3, 1/2} | |
{1/5, 1/2, 1/4} | {1, 1, 1} | {1/3, 3, 2} | {1/8, 2, 1/3} | {2, 1/2, 1/2} | {2, 2, 1/7} | |
{1/2, 1/7, 1/5} | {3, 1/3, 1/2} | {1, 1, 1} | {1/4, 1/3, 1/3} | {4, 1/8, 1/2} | {3, 1/3, 1/9} | |
{2, 1/2, 1/2} | {8, 1/2, 3} | {4, 3, 3} | {1, 1, 1} | {8, 1/3, 2} | {6, 2, 1/5} | |
{1/7, 1, 1/3} | {1/2, 2, 2} | {1/4, 8, 2} | {1/8, 3, 1/2} | {1, 1, 1} | {1/2, 3, 1/6} | |
{1/4, 1/3, 2} | {1/2, 1/2, 7} | {1/3, 3, 9} | {1/6, 1/2, 5} | {2, 1/3, 6} | {1, 1, 1} |
Next
Then the rough judgment matrix is constructed as shown in
[1, 1] | [2.8889, 4.3889] | [3.3889, 5.8889] | [1.1667, 1.8333] | [2.2222, 5.2222] | [1.5833, 3.3333] | |
[0.2472, 0.3972] | [1, 1] | [1.0926, 2.4259] | [0.3912, 1.3287] | [0.6667, 1.3333] | [0.9683, 1.7937] | |
[0.1984, 0.3770] | [0.6759, 2.0093] | [1, 1] | [0.2870, 0.3241] | [0.6597, 2.5972] | [0.4938, 1.9383] | |
[0.6667, 1.3333] | [2.0278, 5.7778] | [3.1111, 3.5556] | [1, 1] | [1.6481, 5.4815] | [1.3444, 4.2444] | |
[0.2910, 0.7196] | [1.1667, 1.8333] | [1.5972, 5.4722] | [0.5486, 1.9861] | [1, 1] | [0.5741, 1.9907] | |
[0.4676, 1.3426] | [1.2222, 4.1111] | [2.0370, 6.3704] | [0.7963, 3.2130] | [1.4259, 4.2593] | [1, 1] |
Rough judgment matrix
1.0000 | 2.8889 | 3.3889 | 1.1667 | 2.2222 | 1.5833 | |
0.2472 | 1.0000 | 1.0926 | 0.3912 | 0.6667 | 0.9683 | |
0.1984 | 0.6759 | 1.0000 | 0.2870 | 0.6597 | 0.4938 | |
0.6667 | 2.0278 | 3.1111 | 1.0000 | 1.6481 | 1.3444 | |
0.2910 | 1.1667 | 1.5972 | 0.5486 | 1.0000 | 0.5741 | |
0.4676 | 1.2222 | 2.0370 | 0.7963 | 1.4259 | 1.0000 |
1.0000 | 4.3889 | 5.8889 | 1.8333 | 5.2222 | 3.3333 | |
0.3972 | 1.0000 | 2.4259 | 1.3287 | 1.3333 | 1.7937 | |
0.3770 | 2.0093 | 1.0000 | 0.3241 | 2.5972 | 1.9383 | |
1.3333 | 5.7778 | 3.5556 | 1.0000 | 5.4815 | 4.2444 | |
0.7196 | 1.8333 | 5.4722 | 1.9861 | 1.0000 | 1.9907 | |
1.3426 | 4.1111 | 6.3704 | 3.2130 | 4.2593 | 1.0000 |
By Matlab, the eigenvectors corresponding to the maximum eigenvalues of
According to
Based on weight vector
0.2000 | −0.2000 | −0.6000 | −0.7333 | |
−0.2807 | 0.3600 | −0.1800 | −0.4533 | |
−0.3667 | −0.0500 | 0.1000 | −0.3214 | |
−0.2917 | 0.4667 | −0.3200 | −0.5750 | |
0.1700 | −0.1700 | −0.6680 | −0.7925 | |
−0.3800 | −0.0700 | 0.1400 | −0.3162 |
At last, the comprehensive correlation function values are obtained according to
To demonstrate the validity and applicability, the assessment result of the proposed method is compared with the assessment result of D-S evidence theory [
From
According to the nature of correlation function, the assessment result is analyzed as follows. The grade of each indicator, which is shown in
As can be seen in
In this paper, the method of combining extension analysis with rough AHP is used to assess power quality comprehensively. By rough AHP, the adverse effects of single expert judgment can be eliminated. Through the concept of matter-element and correlation function, the relationship between each assessment indicator of power quality and the assessment interval of each quality grade can be quantified. At last the case analysis and application show that the proposed method can determine the overall power quality of power grid, compare the differences among the performance of assessment indicators, find out the key factors affecting power quality, and provide the basis for further improving the power quality of power grid. However, rough AHP is a kind of subjective weighting method and may lead to the lack of effective weighting information. Synthesis of subjective and objective weights is a significant way to overcome this shortcoming. In addition, the definition of assessment grade standard, classical domain and joint domain needs to be further studied in extension analysis.
One assessment indicator,
One quality grade,
A domain which is actually a nonempty finite set of objects
Any object in
Any partition in
The partition of the indistinct relationship
Upper approximation set of
Lower approximation set of
Rough boundary interval of
Rough lower limit of
Rough upper limit of
The number of expert,
Pairwise comparison matrix given by expert
One element in
Consistency ratio of
Consistency index of
Largest eigenvalue of
Random index
Group decision matrix
One element in
Rough boundary interval of
Rough lower limit of
Rough upper limit of
Rough boundary interval of
Average form of
Is the rough lower limit of set
Is the rough upper limit of set
Rough judgment matrix
Rough lower limit matrix
Rough upper limit matrix
Eigenvector corresponding to the maximum eigenvalue of
Eigenvector corresponding to the maximum eigenvalue of
Value of
Value of
Weight value of indicator
Weight vector
Matter-element to be assessed
Unknown assessment grade of
Indicator set which represents the characteristics of
Indicator value vector,
Indicator value on indicator
Classical domain
Indicator value interval corresponding to
Joint domain
Set of all assessment grades of
Indicator value interval corresponding to all assessment grades on index
A bounded interval
Module of bounded interval
Distance from point
Correlation function
Correlation function which describes the quantitative relationship from indicator value
Comprehensive correlation function