tination, the algorithm is repeated by inserting new drops at the initial nodes which in turn multiplies the erosion of potential paths. Sometimes due to excessive erosion, the altitude of nodes may become close to zero making the gradient imperceptible. This may affect the already formed paths in due process and may lead to inefficient solutions. To avoid such situ- ations, the altitude of nodes are increased by depositing sediments which helps in unblocking alternate paths from initial nodes. In the end, all possible potential paths are analyzed to select the optimum one.
When there are multiple drops flowing from several initial nodes, it is not necessary that RFD would provide a single best possible solution. It can be considered that instead of a single main stream, a group of tributaries tends to follow several individual paths to reach the destination from each initial node. In this case, it is not imperative that each individual path from origin to destination is the best one, i.e. shortest path. In particular, this behavior of forming multiple individual paths to reach destination can be applied suitably to perform steady state power grid analysis. Water drops can follow individual paths from each node to reach anyVDD nodes (destinations or PADs) to complete individual flows by eroding tra- versed nodes. The amount of erosion at a node is calculated by the gradient between the current and the next node. However, it can also be evaluated by considering gradients of all possible choices at the current node instead of taking a single gradient, which results in a straightforward adaptation to perform power distribution network analysis.
Graph Representation River Formation
a
b
c d
e f
g h
Sediment
Figure 3.5: Graph representation of RFD scheme.
3.4 River Formation Dynamics (RFD)
3.4.1 Application of RFD in the analysis of VLSI power grid networks
For implementation of RFD algorithm, a resistive model of VDD grid is taken as shown in Figure 3.1(a) for steady state analysis and a RC-modeled power distribution network is considered as shown in Figure 3.2(a) for transient analysis. As it is already shown in literature that a power distribution network can be expressed in the form of linear system of equations, the potential of nodes can be obtained by finding best paths traversed by each drop from source to destination. A generic way to approach power distribution network analysis is to allow a number of drops to take different paths and use the average gradient evaluated in each path as an approximated solution. If the gradient value is averaged by taking a sufficient number of water drops, then an acceptably accurate solution can be obtained. A simple graph representation of RFD is shown in Figure 3.5.
3.4.1.1 Graphical Approach
Let Θbe a connected graph havingN nodes and E edges. If a drop is allowed to flow in graph Θ, starting at an initial node n1, then the probability of drop at nodenk after kth transition will be d(n1
k), where d denotes the dimension of the graphΘ. Let a drop is set to flow on a power distribution network having system matrix,ArGGATrGof size(n−1)×(n−1) as described in (3.5). LetTpbe a matrix of transition probabilities such that,
Tp = (Pninj)ni,nj∈N =GAG, (3.19) whereGis the diagonal matrix andAGis the adjacency matrix corresponding to conductance type branches in power distribution network. The rule of transition can be expressed by [110],
Pnk+1 =TpTPnk, (3.20)
Pnk = (TpT)kP0, (3.21)
wherePnk denotes probability distribution afterkth step. If a drop starts flowing from node ni, then it reaches nodenjinkthstep and the value ofPninj can be calculated by looking into
the(ij)thentry of matrixTk. Alternatively, harmonic functions [110] can be used to show the adaptability of RFD algorithm to express potential of the nodes across a power grid network.
A function f : V → R is said to be a harmonic function with set of poles M if it can be expressed as [110],
Pni∑
nj∈deg(ni)f(nj) = f(ni) ni ∈/ M. (3.22) Let f(ni) denotes the probability that a drop flowing from node ni would reach node nj before nodenk, then,f is said to be a harmonic function having polesnj andnk. IfΘcan be considered as a graph to power distribution network having unit conductance value along the edges, thenf(ni)denotes the potential at node ni with current (drop) flowing from nodenj to nodenk.
3.4.1.2 Electrical Circuit Theory Approach
Steady state analysis of power distribution network requires modeling of the grid as an electrical network composed of resistors, source voltages and currents. Figure 3.1(a) repre- sents a single nodexof theVDDgrid. Application of Kirchhoff’s current law at nodexresults in an algebraic equation (3.23),
∑k y=1
Iy =Ix, (3.23)
wherekis the number of neighbors ofx;Iydenotes the current flowing throughythconnected node andIx is the current drawn from nodex. Further, eachIycurrent can be represented by,
∑k y=1
Iy =
∑k y=1
gy(Vy−Vx), (3.24)
wheregy andVy denote the conductance and potential ofythadjacent node, respectively, and Vxrepresents the potential at nodex. If we relate the flow of a drop from origin to destination with solution of a node potential in a power distribution network,VddPADs would correspond to the destinations (sea). Similarly, potential of each node would correspond to altitude of a node after erosion process and weight (cost) of an edge can be correlated to resistance. Ifpx,y represents probability of a drop flowing in any directionystarting fromx, then (3.17) can be
3.4 River Formation Dynamics (RFD)
formulated as,
px,y = ∇Dr(x, y)
∑k
y=1∇Dr(x, y), (3.25)
and the gradient between nodexand nodeycan be represented as,
∇Dr(x, y) =gy(Vx−Vy) (3.26) Comparing (3.26) and (3.24) and taking their absolute values, the probability that a drop traverses from nodexto nodeycan be represented as,
px,y = Iy
Ix = gy(Vy−Vx)
∑k
y=1gy(Vy −Vx) (3.27)
Similarly, during transient analysis of power distribution network, the probabilitypx,ycan be represented as,
px,y = Iy(t)
Ix(t) = gy(Vy(t)−Vx(t))
∑k
y=1gy(Vy(t)−Vx(t))− Chx(Vx(t)−Vx(t−h)) (3.28)
3.4.2 Implementation Details
In this subsection, implementation framework of RFD scheme to compute voltage fluctu- ation for each node is shown in detail to analyze the power distribution network. RFD takes natural advantage of grid locality to improve efficiency in speeding up the analysis to pro- vide way for location estimation of hotspots. One of the reasons for this improved efficiency in terms of memory may be the avoidance of any explicit representation of system matrices corresponding to power distribution networks. In RFD, water drops search for new paths ac- cording to altitude gradients of power grid network edges. The search for new edge from each node of power grid network comes with the computation of probability from altitude gradients as described in (3.26). After following an edge, the altitude or potential of the corresponding node is updated by taking into account the amount of erosion. For efficient applicability of RFD to power grid analysis, the amount of erosion at a particular node is considered as the potential across current sink branch of that node (exceptVdd nodes), which is computed by
using (3.29), i.e.,
Vx =− Ix
Gtotalx , (3.29)
whereVxdenotes the potential across current sink branch of nodex,Ixrepresents the amount of current flowing through current sink branch of node xandGtotalx denotes sum of conduc- tances of each edge connected to nodex. The same procedure is repeated until all the drops reachVdd nodes. Once a drop reaches anyVDD node across the power distribution network, the altitude or potential of start node is computed by summing up the amount of erosion at visited nodes with the potential of destination. The process of estimating the potential across different nodes of a power grid network is performed until convergence is achieved, i.e., stop- ping criteria. Here stopping criteria is represented as,
−β < Vx−Vx′ < β, (3.30) whereVx′denotes estimate of solutionVxof an initial nodexandβrepresents error threshold.
During both steady state and transient analyses, similar procedure is repeated with an excep- tion that the drop can end its traversal in any of the virtual node with probability as described in (3.12) and the amount of erosion corresponding to the drop is estimated asVx(t−h)during transient analysis. The procedure to perform power distribution network analysis using RFD is described in Algorithm 6.
Although RFD is performed to estimate the location of hotspots by considering all rel- evant parameters during implementation, the accuracy is affected by few error components.
However, error in solution is sustained so that the potentials at each node can be approximated to the closest of millivolts with small inaccuracy. The error is evaluated for each solutionVx by estimating the varianceσx2forNxnumber of drops having path gain ofDxfrom nodex.
σx2 = 1 Nx−1
Nx
∑
n=1
(Dxn−Vx′)2 (3.31)
The error is not optimal if search of a drop follows a path that leads to a loop, i.e., local traps.
Often these traps happen in case of complex power grid networks having irregular structures.