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Comparison of different MCDM Techniques used to Evaluate Optimal Generation Javeed Kittur, Vijaykumar S, Vijeta P.Bellubbi, Vishal P, Shankara M.G Department of Electrical & Electronics Engineering B. V. Bhoomaraddi College of Engineering & Technology Hubli, India [email protected], [email protected]

Abstract—Energy demand as of now is increasing tremendously. Also importing electricity is expensive from load centers which are located far away. Electricity is generated using distributed generators locally and using this electricity is more economical as to solve the increasing energy demand to a certain extent. For a complete day different sources of generation like Combined Heat Power, wind and utility are considered for the study. The first method discussed in this paper is Simple Additive Weighting; second method is Weighted Product and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). These methods are applied to a set of generation data of a particular day to obtain the optimal generation. In order to verify the weights’ assigned to different attributes the method used is Analytic Hierarchy Process (AHP). The obtained results by these multi – criteria decision making techniques applied to a set of data for a complete day helps in identifying the optimal generation. Keywords— Simple Additive Weighting, Optimal Generation, Multi-criteria Decisions, TOPSIS, Analytic Hierarchy Process, Weighted Product

I.

INTRODUCTION

New methods have been emerging in the past relating the decision making process. Decision-making issues generally deal with the solution selection which is best compromise. The selection of the solution that is the decision is made keeping the real criteria and it also depends on the choices made by the decision maker [1][2]. Decision making is a complex process; mathematical models are recommended for simplification and ease in use. In multi – criteria decisions TOPSIS is one of the technique that is used frequently. Apart from this method, several other techniques are available namely PROMETHEE, ELECTRE, etc. In [3], the authors have explained the AHP and TOPSIS technique applied to selection problem of a supplier. The authors have discussed the calculations of weights for different criteria’s using analytic hierarchy process technique and these weights are used as inputs in the TOPSIS technique which is further used to assign ranks to the suppliers. The authors in [4], explain different decision making methods for the potential freight evaluation. The authors in this paper use PROMETHEE and AHP technique for selecting the best solution and for the evaluation of the freight villages. In [5], the authors are making a comparison of analysis of the result

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obtained from different decision making methods like TOPSIS, ELECTRE, VIKOR and PROMETHEE. The main motto of this work is to evaluate optimal generation considering generation from combined heat power, wind and utility for a particular day keeping cost of generation as an important factor. In this paper a logistic estimation of the generation is done using TOPSIS and AHP method. Experts’ objective and subjective opinions can be converted into quantitative form using the Analytic Hierarchy Process [6]. Relative weights for different attributes of the criteria’s of evaluation are determined using Analytic Hierarchy Process. In this paper the assignment of weights of attributes combined heat power, wind, utility and purchasing cost is done based on relative importance of these attributes. The optimal generation is calculated using different methods like SAW, WP ant TOPSIS. The arrangement of this paper is done in seven sections. Following the introduction, the second section briefly describes the Simple Additive Weighting (SAW) method. The third section tells about the procedure of Weighted Product (WP) method. Fourth section discuses how the AHP method can be used to validate the selected weights for different attributes based on relative importance. Fifth section deals with the last technique that is TOPSIS to evaluate the optimal generation. Implementation of all these multi – criteria decision making methods is discussed in next section for the set of data comprising of generation from CHP, utility, wing and cost. The following sections include discussion and results and finally conclusions of the study are presented. II.

SIMPLE ADDITIVE WEIGHTING METHOD

It is one of the simplest methods and is also widely used; weighted sum method is the other name of this method [6]. All the attributes are given weights based on relative importance and the total of these weights must be equal to one. The assessment of every alternative is done with respect to each attribute. The complete performance score of each alternative is obtained using the equation (1), N PSi = wl ( mil ) norm (1) l =1 where, the normalized values of every attribute are represented by (mil)norm and the overall score of each alternative is

¦

172

represented by PSi. The best alternative is the one which has the highest value of the performance score PSi. An attribute may be non – beneficial or beneficial. Normalized values of every attribute are calculated using,

(mil )M (mil ) p where, measure of the Mth alternative’s attribute is denoted by (mil)M and the measure of the Pth alternative’s attribute is denoted by (mil)P and this attribute has the highest value out of the various considered alternatives. This ratio is to be taken only for the attributes that are beneficial. Lower values are of interest for the attributes which are non – beneficial; the calculation of normalized values is done by

• •

• • •

λ −N CI = max N −1

(mil )P

(mij )M III.

WEIGHTED PRODUCT METHOD

Weighted product method is in a way same as the previous method SAW. The major difference being in this method multiplication is done and that in SAW addition was done. The performance score for every alternative is obtained by N (2) PS i = [(mil ) norm ] wl l =1 The calculation of normalized values is similar to as that explained in the section II. As seen in expression (2), the weight assigned to each attribute is raised as a power to the alternative’s normalized value. The best alternative is the one which has the highest value of the performance score PSi.

∏

IV.

ANALYTIC HIERARCHY PROCESS

Analytic Hierarchy Process is a popular technique used to make decisions relating the complex problems [6]. Saaty in 1980, proposed AHP technique which splits a complex problem into different hierarchies relating the alternatives, attributes or criteria and objectives [7]. This method deals effectively with attributes which may be subjective and objective and more certainly in situations there have to be judgments made subjectively from different people which constitute a significant portion in the process of decision making. The following is the procedure followed for AHP method [6][10]: Step i: Determination of objectives and attribute evaluation Step ii: Relative importance determination of all the attributes in view of the objective set.

Using relevance importance scale obtain a matrix by comparing pair – wise, the matrix thus obtained is M1 For each row geometric mean is calculated and then each row’s geometric mean is normalized to obtain each attribute’s normalized weight, the matrix thus obtained is M2 Matrix M3 is obtained by multiplying M1 and M2, matrix M4 is obtained dividing M3 by M2 The matrix M4’s average value is determined which gives an eigen value which is maximum and denoted by Ȝmax In order to get a clear idea about the deviation with the consistency an index called Consistency Index (CI) is calculated using,

• •

(3)

Random Index (RI) is chosen from table 3.2 [6] based on the attribute’s number in the problem used for decision making Consistency Ratio (CR) is then found, and if the CR is 0.1 or less then this will be considered acceptable.

Step iii: Complete performance scores are determined for every alternative. Step iv: After the determination of performance scores for all the alternatives, ranks are given to these scores. The score with the maximum value is selected as the best solution and it is assigned rank 1. V. TECHNIQUE FOR ORDER OF PREFERENCE BY SIMILARITY TO IDEAL SOLUTIONS (TOPSIS) In 1981, Hwang and Yoon proposed this method. This method relates to the concept that the chosen option must have a certain distance which is the shortest from ideal positive solution and farthest from ideal negative solution [6]. There exists a database which includes various solutions and ideal solution is one of the solutions which are hypothetical and for this case the maximum values of attributes occur in database. Similarly the ideal negative solution is also a solution which is hypothetical, for which the minimum values of attributes occur in the database. Hence the TOPSIS method provides a solution which is closest to best hypothetical solution and also is farthest with respect to the worst hypothetical solution [8]. The procedure followed for TOPSIS in selecting an alternative which is the best one is: Step i: Determination and identification of objectives and evaluation of important attributes Step ii:

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Using the information of all the attributes and presenting it in a matrix. Every row represents a different alternative and every column represents a different attribute. This representation of the matrix is given in table 1. Step iii: Normalized values are determined using,

R ij =

xij

(4)

m

¦ xij2

i =1 where, m – Number of alternatives i – ith alternative j – jth attribute

Step iv: Choose the weights as per the relative importance of each attribute with respect to the objective, such that the sum of weights equals one. Step v: The normalized weighted values are determined by multiplying with the related weights corresponding to each attribute with every element of each column. Step vi: Ideal best and worst solutions are determined using,

(

)

(5)

(

)

(6)

Y + = (max Yil ), (min Yil ) ∀ l ∈ X ' , i = 1, 2 ,.... n

= { y1+ , y2+ ,....., y +j ,...... yn+ } Y − = (min Yil ), (max Yil )∀ l ∈ X ' , i = 1, 2 ,.... n

= { y1− , y2− ,....., y −j ,...... yn− } for benefit criteria} for cost criteria}

Step vii: Separation measures are determined. The separation to the ideal solution from the alternative is represented in terms of distance, as per the below equations; •

Ideal Separation

PSi+ = •

174

m

¦ ( yil − yl− )2 , i = 1,2,.......n

(8)

l =1

Step viii: The nearness of ideal solution to individual alternative, is obtained using, PSi− (9) Hi− = PSi+ + PSi− PSi+ + (10) Hi = PSi+ + PSi− This Hi is the overall or composite performance score of alternative Ai.

Table 5 gives the normalized values, which is obtained using equation (2).

where, L = {l = 1, 2, ....n|l X’ = {l = 1, 2, ....n|l

PSi− =

m

¦ ( yil − yl+ )2 , i = 1,2,.......n

l =1

(7)

VI.

IMPLEMENTATION AND RESULTS

This section discusses the implementation of different methods like WP, TOPSIS, AHP and SAW including different attributes like CHP, purchasing cost, wind and utility considering a complete day. The input data for this study is as given in table 1 [3]. The assignment of weights for different attributes was done on the basis of relative importance and the weights found for cost, CHP, wind and utility are 0.45454, 0.27272, 0.0909 and 0.18181 respectively. Attributes utility, CHP, and wind are beneficial attributes and hence their values should be high whereas the attribute purchasing cost is non – beneficial attribute and hence their values should be low. The normalized values for all the four attributes are shown in table 2. TABLE 1. Input data in terms of cost and generation for a day TIME WIND CHP UTILITY PURCHASING COST 0 0.12 1.77 0.88 187.55 1 0.06 1.76 0.88 181.98 2 0.24 1.78 0.90 184.69 3 0.30 1.82 0.94 192.66 4 0.24 1.94 1.05 200.52 5 0.36 2.13 1.22 228.57 6 0.36 2.18 1.24 268.57 7 0.42 2.15 1.21 311.28 8 0.30 1.93 1.01 283.31 9 0.18 2.10 1.15 313.67 10 0.24 2.06 1.12 328.01 11 0.10 2.08 1.15 316.62 12 0.30 2.06 1.14 315.88 13 0.48 2.05 1.12 314.01 14 0.06 2.09 1.17 262.88 15 0.30 2.17 1.24 275.69 16 0.24 2.26 1.33 351.44 17 0.36 2.28 1.34 492.29 18 0.12 2.33 1.41 473.39 19 0.60 2.35 1.41 405.44

Negative – Ideal Separation


20 21 22 23

0.60 0.60 0.54 0.60

2.22 2.10 1.96 1.82

1.29 1.17 1.06 0.93

359.83 307.90 269.28 227.23

TABLE 2. Attributes’ normalized values TIME WIND CHP UTILITY PURCHASING COST 0 0.2 0.7531 0.6241 0.9703 1 0.1 0.7489 0.6241 1 2 0.4 0.7574 0.6382 0.9853 3 0.5 0.7744 0.6666 0.9445 4 0.4 0.8255 0.7446 0.9075 5 0.6 0.9063 0.8652 0.7961 6 0.6 0.9276 0.8794 0.6775 7 0.7 0.9148 0.8581 0.5846 8 0.5 0.8212 0.7163 0.6423 9 0.3 0.8936 0.8156 0.5801 10 0.4 0.8765 0.7943 0.5548 11 0.167 0.8851 0.8156 0.5747 12 0.5 0.8765 0.8085 0.5761 13 0.8 0.8723 0.7943 0.5795 14 0.1 0.8893 0.8297 0.6922 15 0.5 0.9234 0.8794 0.6600 16 0.4 0.9617 0.9432 0.5178 17 0.6 0.9702 0.9503 0.3696 18 0.2 0.9914 1 0.3844 19 1 1 1 0.4488 20 1 0.9446 0.9148 0.5057 21 1 0.8936 0.8297 0.5910 22 0.9 0.8340 0.7517 0.6758 23 1 0.7744 0.6595 0.8008 The normalized values obtained as shown in table 2 are used to calculate performance scores of SAW technique. The highest value alternative gets rank 1 and the subsequent highest values will be given further ranks as presented in table 3. On similar lines, weighted product method is implemented and the performance scores are determined as presented in table 6. AHP method is used to obtain a matrix by comparing pair – wise based on relative importance and this is shown as matrix M1 as presented below and the geometric mean for every attribute is shown in table 5.

ª1 0.25 0 .5 .142 º «4 1 2 0.5 »» [M1] = « « 2 0 .5 1 0.25 » « » 2 4 1 ¼ ¬7 Matrix M1 tells us how the various attributes are important when compared to the other attribute, • Wind attribute is 0.25, 0.5 and 0.142 times more important in comparison of different attributes

• • •

CHP attribute is 4, 2 and 0.5 times more important in important in comparison of different attributes Utility attribute is 2, 0.5 and 0.142 times more important in comparison of different attributes Cost attribute is 7, 2 and 4 times more important in comparison of different attributes

Weight matrix M2 is then determined by dividing the geometric mean of every attribute with its respective total,

ª0.07 º «0.27079» » [M 2 ] = « «0.1354 » » « ¬0.52381¼ TABLE 3 SAW method performance scores TIME PERFORMANCE RANK SCORE 0 0.7781 8 1 0.7813 7 2 0.8068 4 3 0.8072 3 4 0.8094 2 5 0.8209 1 6 0.7754 6 7 0.7349 14 8 0.6916 19 9 0.6829 20 10 0.6720 21 11 0.6660 22 12 0.6933 18 13 0.7184 16 14 0.7171 15 15 0.7572 9 16 0.7055 17 17 0.6599 23 18 0.6451 24 19 0.7494 10 20 0.7447 13 21 0.7541 11 22 0.7531 12 23 0.7860 5

Attrib. Wind Attrib. CHP Attrib. Utility Attrib. Cost

TABLE 4 Matrix M1 Attrib. Attrib. Attrib. Wind CHP Utility 1 0.25 0.5

Attrib. Cost 0.142

4

1

2

0.5

2

0.5

1

0.142

7

2

4

1


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TABLE 5 Every attribute’s geometric mean Attrib. Attrib. Attrib. Attrib. GM Wind CHP Utility Cost 1 0.25 0.5 0.142 0.3655 Attrib. Wind 4 1 2 0.5 1.4142 Attrib. CHP 2 0.5 1 0.25 0.7071 Attrib. Utility 7 2 4 1 2.7355 Attrib. Cost 5.2224 Total TABLE 6 WP method performance scores TIME PERFORMANCE RANK SCORE 0 0.7239 8 1 0.6880 7 2 0.7808 4 3 0.7926 3 4 0.7919 2 5 0.8161 1 6 0.7655 6 7 0.7200 14 8 0.6848 19 9 0.6539 20 10 0.6512 21 11 0.6157 22 12 0.6782 18 13 0.7065 16 14 0.6424 15 15 0.7431 9 16 0.6678 17 17 0.5967 23 18 0.5581 24 19 0.6948 10 20 0.7106 13 21 0.7381 11 22 0.7489 12 23 0.7816 5 Multiplying matrix M1 and M2, matrix M3 is determined and by dividing M3 by M2, matrix M4 is obtained,

ª0.2803º «1.0835 » » [M 3 ] = [ M1 * M 2 ] = « «0.5417» « » ¬2.0970¼ ª4.003477º « » [ M 3 ] «4.001192» = [M 4 ] = [ M 2 ] «4.001192» « » ¬4.003377¼ Matrix M4’s average value = 4.002309

176

The matrix M4’s average values is stored in Ȝmax = 4.002309. The average value obtained must be nearing to the matrix M1’s size which is 4. The calculation of consistency index is done by equation (3) and is equal to 0.000769841 Consistency ratio is calculated using,

Consistency Ratio =

Consistency Index *100 Random Index

(11)

The value of random index is taken from [9], and its value is 0.89 as the number of attributes is four. Consistency index is calculated using equation (11) and its value obtained is 0.086498945%. This calculated value happens to lesser than 0.1% which is exactly what was required [9][10]. As this ratio is in the considerable range, the attributes’ weights assigned are acceptable. In implementing TOPSIS method, weights that were proven acceptable by AHP method are used. The further process in TOPSIS technique is to determine normalized values using step 3 and then weighted normalized values are obtained. From this the ideal best and ideal worst values are chosen. In case of wind, CHP and utility the ideal best values are highest values and ideal worst values are the lowest values, and with respect to purchasing cost, the ideal best values are lowest values and ideal worst values are the highest values as shown in table 7. TABLE 7 ideal best and worst values IDEAL WIND CHP UTILITY COST 0.030 0.063 0.045 0.055 BEST WORST

TIME 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.003

0.047

0.028

0.149

TABLE 8 Performance Indices PI(-) RANK PI(+) RANK 0.733 6 0.266 6 0.723 0.276 7 7 0.764 5 0.235 5 0.779 3 0.220 3 0.781 2 0.218 2 0.801 1 0.198 1 0.705 8 0.294 8 0.592 14 0.407 14 0.628 12 0.371 12 0.546 17 0.453 17 0.512 19 0.487 19 0.525 18 0.474 18 0.555 16 0.444 16 0.582 15 0.417 15 0.649 11 0.350 11 0.675 10 0.324 10 0.467 21 0.532 21 0.211 23 0.788 23 0.206 24 0.793 24


19 20 21 22 23

0.396 0.489 0.616 0.701 0.771

22 19 13 20 4

0.603 0.510 0.383 0.298 0.228

22 19 13 20 4

As per the step vii of TOPSIS method the ideal separation and negative – ideal separation are calculated using equation (5) and (6). Table 8 shows the Performance Indices (PI) determined from step viii (TOPSIS method) using equation (7) and (8). PIs play a significant role in assigning ranks. As seen from table 3, 6 and 8, the optimal generation in a day occurs at 5th hour of the day. At this period 0.36MW is generation amount by the attribute wind, 2.13MW is the generation amount by the attribute CHP and 1.22 is by utility. For 3.71MW of total generation, 228.57£/h is the purchasing cost at 5th hour.

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VII. CONCLUSIONS The work presented in this study discusses different methods: AHP, WP, TOPSIS and SAW to determine for a particular day as to when be the optimal generation occurring. Analytic Hierarchy Process is use to validated the assignment of weights for all the four attributes. 0.086498945% is the determined consistency ratio which happens to be much lesser than 0.1%, which is acceptable and it conveys that the judgment done in the decision making is acceptable. 5th hour was found to be that hour of the day at which the generation was optimal by all the methods. REFERENCES

[1]

Javeed Kittur, et al, “Evaluating optimal generation using different multi-criteria decision making methods”, International Conference on Circuits Power and Computing Technologies, pp. 001 – 005, 2015.

[2]

Javeed Kittur, “Optimal generation evaluation using SAW, WP, AHP and PROMETHEE multi - criteria decision making techniques", International Conference on Technological Advancements in Power and Energy, pp. 304 – 305, 2015.

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Vojislav Tomic, et al, “PROMETHEE method implementation with multi-criteria decisions”, FACTA UNIVERSITATIS, Mechanical Engineering, Vol. 9, No. 2, 2011, pp. 193 – 202.

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