cells are analyzed for the optimization of wire width. Wire widths (e.g.,wxy) are considered as decision variables (qs), which are updated in each iteration by following (4.21). Similar to the minimization of IR drop, the values constriction coefficients (w1, w2, w3) are set within (1,2) and the probabilistic seed value (ε) is evaluated using Algorithm 12. As similar constraints affect both IR drop and metal area, both random boundary handling strategy and constrained- dominance principle are employed for handling the constraints. A number of iterations are carried out on a single cell to evaluate the minimum width of each branch of the cell. Once the minimization is complete, corresponding cell is considered for the evaluation. The branches which are already being evaluated for minimum width (i.e., the overlapping branch between two cells) are not considered for further evaluation. The process of minimization at a single cell is performed for 25times independently and the median of optimal width is considered as global optimum solution. In this way, wire widths corresponding to the affected nodes are optimized and total wire area of the power distribution network is evaluated at the end.
As wire widths are evaluated in the optimal direction, after the evaluation, network area is also reduced in an optimal way. Such evaluation is important for designers as it serves as a guideline for power distribution network optimization.
4.6 Experimental Results
Table 4.4:Comparison of proposed MRFD with SLP and RFD methods on different power distribution benchmarks
Benchmarks SLP [32] RFD [135] MRFD
ϑ′(%)1 A(%)1 Constr.2 Time (sec) ϑ′(%)1 A(%)1 Constr.2 Time (sec) ϑ′(%)1 A(%)1 Constr.2 Time (sec) ibmpg2 0.91 03.90 72 98 (+ 0.23) 6.60 02.02 86 92 (+ 0.02) 1.06 04.00 89 39 (+ 0.02) ibmpg4 1.84 06.00 162 913 (+ 0.50) 4.38 01.30 151 1146 (+ 0.11) 1.08 06.70 176 553 (+ 0.11) ibmpg5 1.80 08.90 139 1221 (+ 0.62) 4.05 01.56 179 1427 (+ 0.30) 0.89 12.40 149 626 (+ 0.30) ibmpg6 2.05 07.12 136 1734 (+ 0.67) 5.21 02.12 121 2201 (+ 0.31) 1.91 08.20 132 989 (+ 0.32) industry1 1.24 03.95 200 258 (+ 0.45) 3.03 00.50 230 397 (+ 0.03) 0.06 06.56 312 189 (+ 0.03) industry2 1.80 07.00 1123 1233 (+ 0.60) 4.03 02.02 1521 1310 (+ 0.32) 0.23 13.96 1035 571 (+ 0.32) industry3 2.61 07.89 1406 3058 (+ 4.30) 6.13 03.34 2010 3732 (+ 1.23) 1.52 10.11 2256 1181 (+ 1.23)
industry4 - - - - 9.25 04.56 2313 7310 (+ 5.23) 1.59 18.23 2751 1634 (+ 5.23)
industry5 - - - - 12.40 06.23 3942 8897 (+ 12.34) 1.78 22.21 5127 2418 (+ 12.34)
industry6 - - - - - - - - 1.98 28.63 5748 4394 (+ 23.41)
1 ϑ′denotes percentage of affected nodes (above threshold) after optimization. ‘A’ denotes percentage of reduction in wire area after optimization.
2 ‘Constr.’ denotes number of constraint violations during optimization. ‘Time’ denotes computational time (algorithm time + process time) required for entire optimization process.
duction in Marea (A) for different power distribution networks (industry and ibmpg). The wire area (Marea) is evaluated using (4.4) by assigning a metal sheet resistance (ρ) of0.01Ω and wire length (l) of 0.01µmfor power distribution networks. A water drop population of 100 is selected to move the sediments (qs) in search for optimum IR drop for a maximum function evaluations of1000 for each node during implementation of MRFD method. Sim- ilarly, 100 water drop population is allowed to flow through altitudes to reach destination during implementation of RFD method. Two different types of benchmarks are analyzed and optimized using RFD and MRFD methods. It can be observed from Table 4.4 that MRFD outperforms RFD method both in minimizing IR drop and wire area reduction while opti- mizingibmpgandindustrybenchmarks. Althoughibmpgbenchmarks have highly irregular structures, number of affected nodes is reduced to 1.06%, 1.08%, 1.89% and1.91%, and a reduction of4.00%, 6.07%, 12.40% and 8.20% wire area is achieved foribmpg2, ibmpg4, ibmpg5 and ibmpg6 benchmarks, respectively, after optimization using MRFD algorithm.
Further, industry1-industry6 benchmarks are analyzed for optimization using MRFD al- gorithm. The percentage of number of affected nodes is reduced to0.06%, 0.23%, 1.52%, 1.59%, 1.78%and 1.98%, and a reduction of 6.56%,13.96%, 10.11%,18.23%, 22.21% and 28.63%wire area is achieved forindustry1-industry6benchmarks, respectively, after opti- mization using MRFD algorithm.
Moreover, to demonstrate the applicability MRFD method, evolution curves (mean and
100 101 102 Function evaluation (FE)
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Figure 4.8: (a) Evolution of IR drop values, (b) mean and (c) standard deviation of five variables of a single node ofibmpg6benchmark at different function evaluations (FEs). FE is considered inlog scale.
standard deviation) of fitness value (IR drop) and decision variables (conductances and cur- rent sink) of a single node are shown in Figure 4.8 and Figure 4.9 foribmpg6andindustry6 benchmarks, respectively. It can be observed from Table 4.4 that with increase in number of nodes in power distribution networks, the reduction in wire area and the affected nodes that exist after optimization using MRFD is random. This may be due to the probabilistic nature (dependence on probability during decision making) of MRFD method. Although the behavior of MRFD is random and is affected by probability, a minimum IR drop is achieved while considering metal area and current as constraints. To demonstrate the effectiveness of MRFD algorithm, a sequential linear programming (SLP) [32] is also implemented (author
4.6 Experimental Results
100 101 102
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Figure 4.9:(a) Evolution of IR drop values, (b) mean and (c) standard deviation of five variables of a single node ofindustry6benchmark at different function evaluations. FE is considered inlogscale.
implementation of corresponding algorithms have been considered during evaluation) and is considered for optimization of similar power distribution benchmarks. It is to be noted that SLP has been used as a popular optimization tool in power distribution network design op- timization. Table 4.4 lists number of affected nodes that exist and amount of reduction in wire area after optimization using RFD and SLP algorithms. It can be observed that SLP algorithm fails to achieve convergence forindustry4toindustry6benchmarks as the com- plexity and performance of SLP algorithm depend on the scale and network architecture [32].
Further, it can be observed that RFD also fails to converge for industry6 benchmark. As the optimization process is iterative, it requires rigorous execution of all methods to analyze all the nodes of power distribution network. Therefore, the computational cost in optimizing
the whole network is also evaluated and listed in Table 4.4 for all the algorithms. As RFD is implemented within similar framework of MRFD method, the process time (read time listed within brackets in Table 4.4) is similar for both the algorithms, while analyzing power distri- bution benchmarks. It can observed that MRFD method outperforms both RFD and SLP in computational cost while optimizing power distribution benchmarks under consideration.