Ranking load in microgrid based on fuzzy analytic hierarchy process and technique for order of preference by similarity to ideal solution algorithm for load shedding problem

ABSTRACT


INTRODUCTION
In the power system, the loads play an important role as electricity consumers. When a power shortage occurs, it is imperative to shed some of the load to ensure power balance and maintain the frequency within the allowable limits. The problem is distribution shedding power to the loads in an optimal way. Therefore, it is necessary to calculate the weights for the loads, this weight helps to rank the loads in the system and it is the weight to decide the amount of power shedding of each load. This weight needs to meet both the economic aspect and the technical aspect to ensure the optimal distribution.
Several investigations have been conducted on power system load shedding, as demonstrated in references [1]- [4]. The primary focus of these studies is on calculating the power that needs to be shed during times of excess demand. The techniques used for power shedding mainly address the technical aspects of the system, and do not take into account the economic costs associated with such shedding. Other studies have examined the problem of load ranking during shedding, as demonstrated in [5]- [7], these studies are interested in economic benefits when ranking loads according to their importance. However, technical issues were not mentioned.

METHOD
In the power system in general and the microgrid in particular, the loads need to be compared and evaluated for their importance. This helps the operator to evaluate and rank the loads. When there is a problem of lack of generating capacity in the grid, The load ranking results will be used in prioritizing power cuts to help the system restore power balance. The problem is what criteria are the basis for comparing and evaluating the loads. In this study, the criteria of LIF, VED and VSI are used as the basis to evaluate and rank the loads. In which, LIF criteria will be evaluated in economic aspect, VED and VSI criteria will be evaluated in technical aspect. The steps of this study are presented in the sequence in Figure 1. LIF is calculated using the fuzzy-AHP algorithm, which was originally developed by Bernasconi et al. [19]. The fuzzy AHP method is used in a similar way to the AHP method, but with the addition of a fuzzy triangulation scale. This scale helps to facilitate the comparison of objects and eliminates the uncertainty inherent in the AHP method. In simpler terms, the fuzzy AHP method is an extension of the AHP method, incorporating a fuzzy scale to increase its effectiveness. The fuzzy AHP method follows these steps: [20], [21] Step 1: Establish a hierarchical model Step 2: Set up the judgment matrix A matrix derives from the opinions of experts when comparing objects with each other. Subjects will be evaluated in pairs according to the ratio-9 method [22]. The form of the judgment matrix is shown in formula 2.1.
Step 3: Calculate the eigenvalues and eigenvectors of the judgment matrix Apply the sum method to perform the calculation steps of step 3 [22]. − Standardize each column in the judgment matrix − Calculate the sum of all elements in each row of the matrix X* This results in obtaining the eigenvector of the judgment matrix X W = [W 1 , W 2 , . . . . , W n ] T (5)

−
The maximal eigenvalue max of the judgment matrix is calculated here, (AW)i refers to the i th element in the vector AW.
Step 4: Evaluate and ensure the consistency of the matrix The judgment matrix proposed by experts only ensures consistency when the consistency ratio (CR) ≤10%. If CR is greater than 10%, it is necessary to re-evaluate and re-propose the judgment matrix [22].
here, CR represents the consistency rate, CI represents the consistency index, and RI is a random index, it depends on the number of order n of the judgment matrix, the RI values are given by Zhu [22]. The CI Consistency Index was established by Saaty and calculated using the (8), Steps 3 and 4 of the process are designed to evaluate the consistency of the judgment matrix, which is a necessary condition for proceeding with the fuzzy in step 5.
Step 5: Set up triangular fuzzy number (TFN) The fuzzy-AHP scale has three values: the lowest value lower (L), the middle value median (M) and the highest value upper (U). These values were selected based on the TFN ratio table [21]. The fuzzy set of triangles is presented in Figure 2. The membership function of the fuzzy triangle is given by (9). , or ℎ (9) Step 6: Determine the weight value of the fuzzy vector The formula (10) is used to calculate the geometric mean of the fuzzy comparison values for each criterion. Here, r ̃ remains in the form of a triangular value [23].
Step 7: Determine fuzzy weights for the criteria − Calculate the vector sum of each ̃ − Obtain the inverse of the vector summation and rearrange the fuzzy triangular numbers in ascending order. − Determine the fuzzy weight ̃ for the criterion Step 8: Defuzzification for weights ̃ = lw + + 3 (12) Step 9: Normalize for the value Mi, the alternative with the highest score will be evaluated for priority for decision making.

Criterion 2: VED
VED is the physical relationship in terms of voltage in the power system [24], [25]. Let ij = ( / )/( / ) represent the voltage drop on the ith load bus relative to the faulty source bus. The VED of i-bus and j-bus is defined by formula (14). To transform VED into weighted values, with m bus loads in the microgrid the VED weight normalization formula is presented as formula (15).
Formula (14) indicates that the DV decreases or αij increases as the distance between the buses becomes closer. In the event of a disconnection from the main grid, voltage fluctuations near the connection point can be significant, resulting in a higher voltage drop at buses with smaller VEDs. Therefore, to ensure 4923 effective voltage recovery of the system, larger load shedding power is required for buses with smaller VEDs and vice versa.

Criterion 2: VSI
VSI is a solution to help operators monitor the voltage drop of the system [25], [26]. VSI is calculated by formula (16). The primary goal of VSI is to determine the distance between the current operating point and the marginally stable point, with the aim of identifying the most sensitive buses in the system. These buses will be given a higher priority for load shedding, with a larger amount of power being shed from them. To transform VSI into weighted values, with m bus loads in the microgrid the VSI weight normalization formula is presented as (16).
here, Vk represents the voltage at bus k, and n denotes the total number of buses in the system (17),

Calculate the aggregate weight of the criteria
Ranking in a multi-criteria environment requires a reasonable calculation, comparison and evaluation about the weights of the criteria. In this section, the fuzzy AHP TOPSIS is used to calculate the aggregate weights among the criteria. From there, the results of the load ranking in the system are determined. Section 2.1.1 showed the approach for implementing fuzzy AHP. a. TOPSIS method Hwang and Yoon [27] were the first to introduce TOPSIS, which is a technique for making multicriteria decisions. The steps to implement TOPSIS are presents in [28] Step 1: An evaluation matrix can be created with m alternatives and n criteria, so there is a matrix (xij)mxn.
Step 4: Using the formula provided below, identify the ideal solution matrix consisting of both positive and negative ideal solutions: where, J+ ={j=1,2,....,n| } is considering the criteria that have a confident influence, J-={j=1,2,....,n| } considering the criteria that have a harmful influence.
Step 5: Determine the distance from the alternatives to the ideal solution: Alternatives distance from confident ideal solution Aw and alternatives distance from harmful ideal solution Ab Step 6: Determine the relative nearness to the ideal result: Step 7: Rank substitutions by Ci + (i=1, 2,…..,m). The C + substitutions are arranged from largest to smallest. The substitutions with the largest C + value is the best result.
Step 8: Normalize the proximity of the substitutions to the ideal result to the weights of the respective loads using the (25) Essentially, the fuzzy AHP-TOPSIS method combines the fuzzy AHP method with the TOPSIS method. The implementation steps of fuzzy AHP-TOPSIS are as: − Apply AHP method to propose hierarchical model, judgment matrix. Then compute the weights and check the consistency of the judgment matrix. Using the formula from (2) to (8). − The judgment matrix is fuzzy according to TFN to easily compare the alternatives and form a decision matrix. − Applying the fuzzy AHP method, the pairwise evaluation between the criteria and the substitutions is calculated to determine the weight vectors of the criteria and the decision substitutions: using formula from (9) to (13). − The TOPSIS technique is used to standardize the decision matrix, determine the positive ideal and harmful ideal results to determine the C + proximity index to the ideal result, and finally rank all the substitutions: using the formula from (18) to (25).

RESULTS AND DISCUSSION
In order to demonstrate the effectiveness of the proposed method, a modified 19-bus microgrid system is used for the calculation cases [29]. The single-line diagram of the system is shown in Figure 3. Apply the fuzzy AHP-TOPSIS method to rank the loads that must be assessed based on the criteria.

Criterion 1: LIF
Hierarchical model for load areas (LA) as illustrate in Figure 4. The judgment matrix of the LA and the judgment matrix of the load units in the area are evaluated by the experts of the system, the matrices are presented below. Apply from formula (2) to formula (8) to calculate the components according to the procedure of AHP algorithm. The values are presented in Table 1. The values of CR are all less than 0.1. Therefore, it is concluded that the opinions of experts in comparing objects in the judgment matrix meet the standard of consistency.  Applying TFN to transform the judgment matrix according to the TFN criteria set, the conversion results are from Tables 2 to 5. This transformation helps to fuzzy the judgment matrices and form fuzzy triangles, which makes the objects compared with each other in more detail. Thereby improving the accuracy of the judgment matrix.   Step 4: Calculate the fuzzy mean using formula (10), the results are shown below. Step 5: Calculate the fuzzy weights using the formula (11), the results are shown below. Step 6: For defuzzification and normalizing the final weights, use formulas (12) After calculating the weight values based on each LA and each load unit in LA, the weight of the load unit based on criterion 1 is presented in Table 6. The weighting results according to criterion 1 show the importance of the loads, the larger the number of weighted loads, the more important they are and vice versa. Then, the results of the weights of the loads according to criterion 1 will be combined with the weights of the loads according to criterion 2 and 3 to get the final weight for the load ranking.

Criterion 2: VED
Calculate the weight of the VED criteria established on the J4 matrix in the Jacobian matrix extracted from the 19 bus microgrid test diagram with the support of Powerworld simulation software. Apply the formula (14) to calculate the VED from the grid-connected bus to the load buses in the microgrid system. Applying formulas (15), the weighted results of the load according to VED (WVED) in Table 7.

Criterion 3: VSI
The voltages of the buses are taken from the system in normal operating mode and extracted from the 19 bus microgrid test diagram with the support of Powerworld simulation software, the voltage values are in pu units. Applying formulas (17), (18), we get VSI results and load weights according to VSI (WVSI), the outcomes are shown in Table 7. In Table 7, the weight values of the two criteria, VED and VSI, are shown. These values represent the technical evaluation criteria of the loads, specifically in terms of the voltage aspect of the system. As the microgrid tested in the study is a distribution grid, the quality of the voltage plays an important role in the operation of the system. After calculating the weights of the loads according to each criterion, using the fuzzy AHP-TOPSIS technique to calculate the final combined weight and use this weight to rank the loads. The comparative judgment matrix between the criteria given by the experts, the results are presented below. Apply formula (2) to formula (8) to calculate the components according to the procedure of AHP algorithm. The results show that the CR=0.048<0.1. This shows that the matrix has a suitable consistency. Applying TFN to transform the judgment matrix of criterions according to the TFN criteria set, the conversion outcomes are shown in Table 8. From the pairwise evaluation matrix data, apply the formula from (10) to (13) to calculate the weighted value of the 3 criteria. The outcomes are shown in Table 9. After determining the importance between the criteria, proceed to build a decision matrix based on the weight values in Tables 7 and 8. The outcomes are shown in Table 10. The standardized conclusion matrix and the standardized conclusion matrix with weighted are built according to the formulas (18) to (20). The results are presented in Tables 11 and 12. The difference in load ratings in this method is that after the weights are normalized, the idea is to measure the distance between the substitutions to the ideal solutions so that the evaluation is more optimal. Applying the formula from (21) to (25), the outcomes are presented in Tables 13  to 15.  In Table 15, the final weight values for each load are presented. These weights are used to calculate the proximity to the best solution, which in turn determines the ranking of the loads based on these values. The relative closeness C + are arranged from largest to smallest. The alternative with the largest C + value is the best resolution. Therefore, based on the values in Table 15, the order of importance from high to low is arranged as follows: L4, L11b, L11a, L17, L18, L10, L9, L12. Loads with a low ranking (i.e., small weight) are the less important loads and will be prioritized for power reduction when it comes to load shedding.

CONCLUSION
The fuzzy algorithm helps to fuzzy the judgment matrix of the AHP method, thereby making it easier to compare objects and eliminating the uncertainty of the AHP technique. The results of the fuzzy AHP method are guaranteed to be accurate, consistent and easy to perform object comparisons thanks to the combination of the advantages of these two methods. The TOPSIS algorithm is used to normalize the conclusion matrix containing the weights of the criteria, determine the confident ideal resolution and the harmful ideal resolution to calculate the comparative nearness to the ideal conclusion and finally rank all alternatives. The confident ideal conclusion is defined as the sum of the best achievable values for each attribute. Therefore, the combination of fuzzy AHP-TOPSIS algorithms is the optimal combination for decision making and ranking problems in a multi-criteria environment. In the next work, the application of data processing techniques in statistics to improve the input data for the AHP technique, thereby improving the certainty and consistency of the results. The cost-by-power functions will be considered to more specifically evaluate the level of economic efficiency.