PROCEDIMIENTO PARA LA ENERGIZACIÓN DE LOS ELEMENTOS DE

In large complex domain problems, many factors and relationships among these factors are likely to be defined. As a result, the FCMs designed by the relevant stakeholders could be very complex and include a large number of nodes and connections. In addition, these FCMs may need to be aggregated to obtain stakeholder group FCMs or social FCM. In group or social FCM, the number of nodes could reach hundreds and the number of connections could be up to thousands. In such FCMs, it is a big challenge to understand, analyse or gain insights from them. The FCM condensation process handles this challenge by condensing the large number of nodes (variables) into a small number of higher level categories (groups of variables) and accordingly, small number of condensed connections.

In this section, we propose a novel method for FCM condensation process. It is based on multi-level condensation. The number of condensation levels is determined by the system developer. The developer of the system should take into account several issues when choosing the number of condensation levels, such as the size of FCM, the strength of interdependence between nodes, the smoothness and easiness of the transition process from the lower level of condensation to a higher level, and how to categorize the nodes and connections at the lower level into new nodes (groups) at the higher level without loss or distortion of information in the FCM. The proposed condensation method uses several robust calculations and utilizes the credibility weights of nodes at the lower level in the process of transferring the nodes and their connections into a higher level. Moreover, the values used in this method are represented by β fuzzy values in all computational processes to avoid loss of information. Consequently, this novel method can be considered as a semi- quantitative fuzzy method proposed to overcome the shortage of previous qualitative condensation methods.

There are two major phases of this method. The first phase is subjective; it considers the identification of the groups of nodes at the higher level. The system developer uses his/her knowledge and experience in the problem domain to identify these groups at the higher level of FCM condensation. The second phase is objective; it includes all steps and calculations to secure a valid and accurate transition of data from the lower level to the identified higher level. These steps and calculations are described in Algorithm 3.5. The MATLAB code for the proposed condensation process is detailed in Appendix A.4.1.

Algorithm 3.5. The steps in the condensation process of a large FCM into a small one.

1. Define the number of levels of condensation

2. For each level of condensation do the following:

2.1 Recall βform ofconnection weight values of FCM at the lower level

2.2 Recall the credibility weight values CW of FCM nodes at the lower level

2.3 Define the new nodes at the higher level of condensation (groups of

condensed nodes) and

2.4 Determine for each group its nodes at the lower level and store the groups

and their nodes in a Lookup Table

2.5 Call Algorithm 3.6 (see below) to refine the connection weight values of FCM

at the lower level. This is done to redistribute original weights between nodes

within a group to nodes in other groups (Note: Algorithm 3.6 below and

paragraphs above and below it explain the steps 2.1 to 2.5. After this refinement of weights, the calculation proceeds to step 2.6)

2.6 Initialise an adjacency matrix(G) for the groups of condensed nodes at the

higher level and fill its elements (connection weight values between groups) by zero values

2.7 For each connection weightgijin the matrixGbetween groupgiand group

g do the following:

2.7.1 Obtain from the Lookup Table the nodes at the lower level that belong

tog_iandg_jgroups, respectively, at the higher level

2.7.2 Initialise two dimensional matrix (Grpij),i1to the number of nodes

ing_iand j 1to the number of nodes ing_j

2.7.3 Store inGrpijthe connection values at the lower level from nodes ingi

to nodes ing_j

2.7.4 Select fromGrpijthe nodes in bothgiandgjgroups that have at least

one non-zero connection value and do the following:

i. Reassign new credibility weights New_cw_ki to each node of

where New_cwdi_gi is the new credibility weight of a selected

node di ing_i,di 1D, D is the total number of selected

nodes ing_i, cw_di_giis the lower level credibility weight of the

selected nodedi in gi, and sum(cwDgi)is the sum of credibility

weights at the lower level of all selected nodes in groupgi

ii. Repeat step i for all selected nodes in group g_i

2.7.5 Assign new weight value to the connectiongijbetweengiandgj

groups at the higher level as follows:

) _

( _di _dj _ij

ij sum New cw New cw w

gi 

 (3.14)

where sum(New_cw_di New_cw_dj w_ij)

gi  is the sum of all connection

weights at the lower level between nodes in g_iand nodes ingjafter

multiplying each connection weight between two nodes by their new assigned credibility weights

2.8 Repeat step 2.7 for all connection weights in matrixGat the higher level

2.9 Construct from the matrixGthe condensed FCM at the higher level

2.10 Call Algorithm 3.2 to calculate the CCM for nodes in the condensed FCM

2.11 Call Algorithm 3.4 to calculate the credibility weights for nodes in the

condensed FCM

3. Repeat step 2 for all levels of FCM condensation

A brief overview of steps 2.1 to 2.5 of Algorithm 3.5. The computational phase of condensation starts with the subjective phase, including the determination of the number of levels of condensation and the identification of groups at every higher level. Then and for every level of condensation, the matrix of connection weights as well as the CWof nodes at the lower level are to be identified. Typically, the FCM may have nodes and connections at the lower level, which are condensed to the same group at the higher level. Rather than eliminating or disregarding these connections, it is desirable if these connection weights are refined before proceeding with the above condensation process. The refinement process

) ( / _cw_di cw_di sum cw_D_gi New gi gi  (3.13)

89 redistributes them to nodes in other groups as appropriate. The steps in this process are described in Algorithm 3.6 and the MATLAB code of the calculations is detailed in Appendix A.4.2.

Algorithm 3.6. The steps of the refinement of the FCM before the condensation process

1. For each nodec_iin the group of nodes that require redistribution of connections at

the lower level to nodes in other groups do the following:

2. Obtain from the Lookup Table the group at the higher level to which the nodec_i

belongs:

2.1For each nodecjin the samegroup do the following:

2.1.1 If there is a connection betweenciandcj(wij) , then

i. If there is a connection betweencjand any nodeck(wjk) outside

this group in the FCM, then

a. assign a new connection between c_iandc_kas follows:

jk j ij i ik New cw w New cw w w New_  _   _  (3.15)

whereNew_w_ikis the new connection betweenc_iandc_k, and

New_ and New_cwj are the new credibility weights for ci

andcjnodes, respectively, where New_cwi cwi/(cwicwj)

and New_cw_j cw_j/(cw_icw_j).

b. If the absolute value of New_w_ik is greater than the absolute

value ofw_ik, then w_ik  New_w_ik

ii. Remove the connectionwij

2.2Repeat step 2.1 for other nodes in the group

3. Repeat step 1 for the lower level nodes in other groups in the FCM as required

The objective of the refinement process described in the above method is to remove self- reference in the resulting FCM at the higher level, at each level of condensation. Although this can be easily done by cancelling every connection between any two nodes located in the same group, we propose the above method because such simple cancellation may adversely

90 affect the behaviour of the FCM; for example, when a node influences another node in the same group and the latter in turn influences other nodes in other groups. To overcome this problem, in our novel method described in Algorithm 3.6, the indirect influences of connections between nodes in different groups are converted to direct influences between them. The method calculates the weights for the newly created direct influences by considering the connection weight and credibility weights of nodes that established the indirect influences. Equation 3.15 expresses this calculation. Finally, if the nodes which formed the newly created direct influences have already connection weights between them, then the highest of the absolute value of the newly created direct influence and the existing weight is assigned as the connection weight between these nodes (Algorithm 3.6 and Appendix A.4.2).

After the refinement process, condensation process resumes with the return to step 2.6 in Algorithm 3.5. The process from here involves computing new weights between groups at the higher level. The first step in this process is to initialize to zero values the connection weights between groups at the higher level. Then, for the connection between two groups, we identify at the lower level only the non-zero connections between nodes in these groups. For the identified nodes in each group, we utilize their calculated CW at the lower level to calculate a new CW for them using Equation 3.13. Then, Equation 3.14 uses these new

CW of nodes and their connections at the lower level to calculate a condensed connection between the two groups at the higher level. These steps are repeated until all connections between all groups at the higher level are calculated. As a result, the adjacency matrix of the condensed FCM is created. It is easy from this matrix to draw the condensed FCM and show the connections between its nodes (groups). The CCM and CWvalues for the nodes in the condensed FCM can also be easily calculated using Algorithm 3.2 and Algorithm 3.4 respectively. These values are required for the next level of FCM condensation process. After the completion of all levels of the FCM condensation process, a simpler FCM with few nodes and connections is constructed. In this case, it is easy to gain insight into FCM and analyse its structure as well as apply it for simulating what-if scenarios. It also helps to produce a simple group FCM resulted from combining condensed individual FCMs. Next section presents the proposed aggregation process that can combine FCMs in a group (such as a stakeholder group) as well as combine multiple group FCMs (such as multiple stakeholder groups) into a social FCM.

In document Análisis de transitorios electromagnéticos en la energización de elementos de transmisión utilizando el programa DIGSILENT POWER FACTORY (página 160-165)