voltage scaling

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IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 1
Verification of the Power and Ground Grids Under
General and Hierarchical Constraints
Mehmet Avci and Farid N. Najm, Fellow, IEEE
Abstract—As part of power distribution network verification,
one should check if the voltage fluctuations exceed some critical
threshold. The traditional simulation-based solution to this
problem is intractable due to the large number of possible circuit
behaviors. This approach also requires full knowledge of the
details of the underlying circuitry, not allowing one to verify the
power distribution network early in the design flow. Contrary
to previous work on power distribution network verification, we
consider the power and ground (P/G) grids together and describe
an early verification approach under the framework of current
constraints. Then, we present a solution technique in which tight
lower and upper bounds on worst case voltage fluctuations are
computed via linear programs. Experimental results indicate that
the proposed technique results in errors in the range of a few
millivolts. In addition to P/G grid verification techniques, we also
provide very efficient solution technique to power (single) grid
verification under hierarchical current constraints.
Index Terms—Current constraints, hierarchical current
constraints, linear programming (LP), power and ground (P/G)
grid, sparse approximate inverse (SPAI) preconditioning,
verification, voltage fluctuation.
I. INTRODUCTION
THE feature size of modern integrated circuits (ICs) has
been dramatically reduced in order to improve speed,
power, and cost. The scaling of CMOS is expected to continue
for at least another decade and future nanometer circuits will
contain billions of transistors [1]. As CMOS technology is
scaled, the power supply voltage will continue to decrease [1].
With reduced supply voltages and more functions integrated
into ICs, the impact of voltage fluctuation is increasing and
voltage integrity is becoming a big concern for chip designers.
There are many sources of on-chip voltage fluctuations,
such as IR-drop, Ldi/dt drop, and the resonance between the
on-chip grid and the package. Most available grid verification
techniques use some form of circuit simulation to simulate
the grid. Such an approach requires full knowledge of the
current waveforms drawn by underlying transistor circuitry.
Manuscript received May 12, 2014; revised November 13, 2014 and
January 22, 2015; accepted February 13, 2015. This work was supported
in part by the Natural Sciences and Engineering Research Council of
Canada, in part by Semiconductor Research Corporation, and in part by
Intel Corporation.
M. Avci was with the Department of Electrical and Computer Engineering,
University of Toronto, Toronto, ON M5S 3G4, Canada. He is now with
the Toronto Technology Center, Altera Corporation, Toronto, ON M5S 1S4,
Canada (e-mail: mavci@altera.com).
F. N. Najm is with the Department of Electrical and Computer Engineering,
University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:
f.najm@utoronto.ca).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVLSI.2015.2413966
Fig. 1. Five-node grid.
These waveforms would then be used to simulate the grid
and to find the voltage fluctuation at each node. However,
since the number of possible circuit behaviors is very large,
one needs to simulate the grid for a large number of vector
sequences at each node, which is prohibitively expensive.
Another disadvantage of the simulation-based approach is
that it does not allow the designer to perform early grid
verification, when grid modifications can be most easily done.
To overcome these problems, we will adopt the notion of
current constraints [2] to capture the uncertainty about the
circuit details and behaviors. Under these constraints, grid
verification becomes a problem of computing the worst case
voltage fluctuations subject to current constraints.
In the literature, the ground grid has usually been assumed
to be symmetric to the power grid. Popovich et al. [3] claim
that the power and ground (P/G) grids have the same electrical
requirements and therefore, the structures of these grids are
often symmetric, particularly at the initial and intermediate
phases of the design. They show that this symmetry can be
exploited in a way to reduce the complexity of the power
distribution network by an introduction of a virtual ground.
The resulting circuit model contains two-independent sym-
metric grids, and therefore, the analysis of only one circuit
is necessary. However, the assumption that the P/G grids are
symmetric is not reliable, since even in initial stages of the
design, some regions of the P/G grid are removed to make way
for signal routing. This introduces nonsymmetry in the grid,
which might lead to erroneous results if symmetry is assumed.
We note in particular, that the presence of nonsymmetry can
cause the voltage on a given node of the grid to fluctuate
in both directions, i.e., voltage drop and overshoot, even for
an RC grid (for an RC model of the power grid, voltage
levels can normally only be below vdd, under the assumption
that the circuit does not inject current into the power grid).
To see why, consider the simple unsymmetrical five-node grid
shown in Fig. 1. Fig. 2 shows the current waveform assigned
to the current source in the circuit and Fig. 3 shows the node
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2 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS
Fig. 2. Current configuration assigned to the current source.
Fig. 3. Resulting voltage overshoot for node A.
voltage at node A as a result of an HSPICE simulation.
The simulation shows an overshoot where the node voltage
at node A goes above vdd. Therefore, there is a necessity to
verify the P/G grids together (i.e., a P/G grid verification),
which is the main focus of this paper. In the remainder of this
paper, we will refer to P/G networks as P/G grid, and to only
power network with no ground network as power grid.
In addition to P/G grid verification techniques, we also show
that a reduction in the complexity of power grid verification
is possible under hierarchical current constraints. In the liter-
ature, there are proposed methods to tackle the complexity
of the power grid verification problem under hierarchical
constraints [4]–[6]. In this paper, we consider special hier-
archical current constraints, whereby global constraints are
required to be wholly contained in other global constraints, and
we propose a solution based on sorting and deletion, which
is significantly more efficient than standard linear program
solvers.
The remainder of this paper is organized as follows.
In Section II-A, we present the P/G grid model and formulate
the problem under the notion of current constraints.
Section II-B proposes an efficient solution for the lower and
upper bounds on the worst case voltage fluctuations
of the P/G grid. Implementation details are given
in Section II-C, followed by the experimental results
in Section II-D. In Section III, we propose a fast technique to
solve power grid problem under hierarchical current
constraints, and we also show experimental results of the pro-
posed method. Finally, the conclusion is given in Section IV.
II. P/G GRID VOLTAGE INTEGRITY VERIFICATION
A. Problem Formulation
1) P/G Grid Model: We consider an RC model of the
P/G grid where each branch is represented either by a resistor
or by a capacitor. We define the nodes that are on the
Fig. 4. Macroblock model.
Fig. 5. P/G grid model.
power grid as power grid nodes, and similarly, the nodes that
are on the ground grid as ground grid nodes. Resistors are
located between two power grid nodes or between two ground
grid nodes, i.e., no resistor exists between a P/G grid node.
In addition, the capacitors are located only between P/G grid
nodes and, unlike previous work, we assume that a node can
have multiple capacitors.
One common simplification in the literature is to model the
current drawn by the underlying transistor circuitry in a logic
block as a single current source. We usually know the current
drawn by a logic block, but that logic block is usually attached
to multiple nodes in the grid. Therefore, modeling the current
drawn by a logic block as a single current source is not valid,
and yields pessimistic voltage fluctuations. In order to capture
this notion, we introduce the model of a macroblock, which
groups multiple current sources into a single block, as shown
in Fig. 4.
The macroblock model in Fig. 4 captures the true behavior
of a logic block, in the sense that it draws current from
multiple nodes. Note that we do not require having the same
number of current sources for P/G grid nodes. However, for
each macroblock, we have to ensure that the current leaving
power grid nodes must equal the current entering ground
grid nodes. This will be an important equality constraint that
defines the feasibility space of currents.
A simple P/G grid is shown in Fig. 5. Notice that the
macroblock has multiple connections to the grid, and that some
nodes have multiple capacitors attached.

AVCI AND NAJM: VERIFICATION OF THE P/G GRIDS UNDER GENERAL AND HIERARCHICAL CONSTRAINTS 3
2) System Equations: Let the P/G grid consist of
n +α nodes, where nodes 1, 2, . . . , n have no voltage sources
attached, and nodes (n + 1), . . . , (n + α) are connected
to voltage sources. Following the approach in [7], the
Modified Nodal Analysis equation that describes the network
can be written as:
˜G ˜u(t) + ˜C ˙˜u(t) = ˜i(t) + ˜G0vdd(n) (1)
where ˜G and ˜G0 are n × n conductance matrices and ˜C is
an n × n capacitance matrix. ˜u(t) is an n × 1 vector of
node voltages except the nodes that are connected to voltage
sources. ˜i(t) is n × 1 vector of current sources and vdd(n) is
an n × 1 vector whose each entry is equal to the value of
the supply voltage. The node voltages ˜u(t) are with respect to
some reference (datum) node that is part of the ground grid.
Let h be the number of power grid nodes that are not
connected directly to a voltage source, and l be the number
of ground grid nodes that are not ground, so that h + l = n.
We can rewrite (1) by partitioning the vector of node voltages
with respect to P/G grid nodes, and reordering the rows and
columns of ˜G, ˜C, and ˜G0 and the entries of ˜i(t) accordingly
as follows:
Gp 0
0 Gg
up(t)
ug(t)
+
Cp N
NT Cg
˙up(t)
˙ug(t)
=
−ip(t)
ig(t)
+ G0vdd(n). (2)
Since no resistor exists between P/G grid nodes, ˜G can be
partitioned into two submatrices Gp (h × h matrix) and
Gg (l ×l matrix). ip(t) (h × 1 vector) and ig(t) (l × 1 vector)
are nonnegative vectors defining the current sources attached
to P/G grid nodes, respectively. Since capacitors exist
only between P/G grid nodes, Cp (h × h matrix)
and Cg (l × l matrix) are nonnegative diagonal matrices,
and N is an h × l nonpositive matrix.
If we set all current sources to 0, ∀t, then up(t) = vdd(h)
and ug(t) = 0, ∀t, where vdd(h) is an h ×1 vector whose each
entry is equal to the value of the supply voltage. For this case,
the system of equations becomes
Gp 0
0 Gg
vdd(h)
0
= G0vdd(n). (3)
Substituting G0vdd(n) from (3) into (2) and rearranging the
terms, we obtain
Gp 0
0 Gg
up(t) − vdd(h)
ug(t)
+
Cp N
NT Cg
˙up(t)
˙ug(t)
=
−ip(t)
ig(t)
. (4)
Defining vp(t) = vdd(h) − up(t) to be the vector of voltage
drops at power grid nodes, we can write (4) as
Gpvp(t) + Cp ˙vp(t) − N ˙ug(t) = ip(t) (5)
Ggug(t) + Cg ˙ug(t) − NT
˙vp(t) = ig(t). (6)
In matrix notation, (5) and (6) can be combined to yield
Gp 0
0 Gg
vp(t)
ug(t)
+
Cp −N
−NT Cg
˙vp(t)
˙ug(t)
=
ip(t)
ig(t)
.
(7)
Fig. 6. Voltages on the P/G grid.
In this notation, vp(t) is positive when power grid nodes
experience undershoots and ug(t) is positive when ground grid
nodes experience overshoots, as shown in Fig. 6.
Define
ˆG =
Gp 0
0 Gg
, ˆv(t) =
vp(t)
ug(t)
ˆC =
Cp −N
−NT Cg
, î(t) =
ip(t)
ig(t)
.
So that (7) becomes
ˆG ˆv + ˆC ˙ˆv(t) = î(t). (8)
Notice that the circuit described by (8) is the original P/G grid,
but with all the voltage sources set to zero and the directions
of current sources attached to the power grid nodes reversed.
Furthermore, this equation is useful, because the current
vector î(t) is a nonnegative vector and the matrix ˆC consists
of only nonnegative elements.
Assume that only m of n nodes of the P/G grid have current
sources attached. Then, we can reorder the rows and columns
of the matrices and the entries of the vectors in (8) to yield
Gv(t) + C ˙v(t) =
i(t)
0(n−m)
= ¯i(t) (9)
where G and C are matrices of size n × n, which are simply
reordered replicas of ˆG and ˆC. i(t) is the vector of size m
representing the current loads, and 0(n−m) is the zero vector
of size n − m. Finally, using the backward Euler formula,
(9) can be discretized in time as
Av(t) = Bv(t − t) + ¯i(t) (10)
where A = (G + (C/ t)) and B = (C/ t).
3) Current Constraints: We adopt the notion of current
constraints in order to perform verification of the P/G grid.
This approach [2] does not require complete information about
the currents drawn by the underlying circuitry, and may be
called a vectorless approach. The currents are typically hard
to specify for at least two reasons. First, the number of
combinations of possible current waveforms is very large, and
simulation of a large set of waveforms is very time-consuming.
Second, the simulation approach does not allow the designer
to verify the grid early in the chip design. For the simulation,
the details of the underlying circuitry must be already known,

but it might not be available or complete early in the design,
when most of the major changes in grid characteristics can be
most easily incorporated.
We use three types of constraints: 1) local constraints;
2) global constraints; and 3) equality constraints. Local
constraints define upper bounds on individual current sources.
They can be expressed mathematically as
0 ≤ i(t) ≤ iL (11)
where iL is a vector of size m and stands for the peak value
of currents that the current sources can draw. In this paper,
we restrict our work to the case of dc constraints, i.e., the upper
bound is fixed over time. However, note that it is only the
constraints that are dc, the currents themselves are transient.
An alternative is to use transient current constraints, which
are more difficult to use in practice, both from the user’s
standpoint (supplying transient constraints) and the verification
tools that would deal with them.
Local constraints do not completely capture the behavior of
the grid, because it is never the case that all chip components
draw their currents simultaneously. Therefore, we need global
constraints, which are upper bounds on the sums of groups of
current sources. They might represent the maximum current
that the group of current sources in each macroblock can draw
or the peak total power dissipation of a group of macroblocks.
If we assume that we have μ global constraints, then they can
be expressed as
0 ≤ Ui(t) ≤ iG (12)
where U is a μ × n matrix that consists only of 0 and 1 s.
If a 1 is present in a row of U, it indicates that the corre-
sponding current source is included in that global constraint.
Similar to the case of local constraints, iG is a constant
time-independent dc constraint, but the currents themselves
are transient waveforms.
As previously mentioned, we need to ensure that the
currents leaving the power grid are equal to currents entering
the ground grid, which we will call an equality constraint.
If we assume that we have γ macroblocks, then the equality
constraints can be expressed as
Mi(t) = 0 (13)
where M is a γ ×n matrix that consists only of 1, −1, and 0 s.
For each macroblock, 1 s corresponds to current sources that
are attached to the power grid, and −1 s corresponds to current
sources that are attached to the ground grid.
To simplify the notation, we use F to denote the feasible
space of currents, so that i(t) ∈ F if and only if it
satisfies (11)–(13) at all time.
4) Problem Definition: Our problem is to find, for every
node, the worst case node voltage fluctuation over all possible
currents in F. To simplify the notation, let E = A−1 and
D = A−1B, so that we can write (10) as
v(t) = Dv(t − t) + E¯i(t). (14)
We first write the matrix E as follows:
E = [e1, e2, . . . , en] (15)
where ei is the ith column of E. Define
H = [e1, e2, . . . , em] (16)
where H is n ×m matrix formed by the first m columns of E.
Since we know that the last n − m elements in ¯i(t) are 0,
we can write (14) as
v(t) = Dv(t − t) + Hi(t). (17)
Now consider the case in which the grid had no stimulus
for t ≤ 0, which leads to v0 = v(0) = 0. Then, writing at
time t, 2 t, and 3 t, we obtain
v( t) = Dv0 + Hi( t) = Hi( t) (18)
v(2 t) = Dv( t) + Hi(2 t) = DHi( t) + Hi(2 t)
(19)
v(3 t) = Dv(2 t) + Hi(3 t)
= D2
Hi( t) + DHi(2 t) + Hi(3 t). (20)
Repeating this procedure for any future time p t, we have
v(p t) =
p−1
k=0
Dk
Hi((p − k) t). (21)
At every point in time t ∈ [0, p t], the input vector i(t)
must be feasible, i.e., we must have i(t) ∈ F. Under these
conditions, we are interested in the worst case voltage fluc-
tuations attained (separately) by each component of v(p t).
In order to capture this notion, we use the following notation,
introduced in [8].
Suppose f (c) : Rn → Rn is a vector function whose
components are denoted f1(c), . . . , fn(c), and let A ⊂ Rn.
Now, define a vector x ∈ Rn, such that, with i ∈ {1, 2, . . . , n},
and xi is the maximum of fi (c) over all c ∈ A. We denote
this by the following operator:
x = emax
∀c∈A
( f (c)). (22)
Notice that each component xi, ∀i = 1, . . . , n may be found
separately by solving the following maximization problem:
maximize: fi (c)
s.t.: c ∈ A. (23)
Similarly, define a vector y ∈ Rn, such that yi is the minimum
of fi (c) over all c ∈ A. We denote this by the following
operator:
y = emin
∀c∈A
( f (c)) (24)
and each component yi, ∀i = 1, . . . , n may be found
separately by solving the following minimization problem:
minimize: fi (c)
s.t.: c ∈ A. (25)

Using the emax(·) and emin(·) operators, we can express the
worst case voltage fluctuation at all nodes at time p t by
v+
(p t) = emax
∀i(t)∈F
⎛
⎝
p−1
k=0
Dk
Hi((p − k) t)
⎞
⎠ (26)
v−
(p t) = − emin
∀i(t)∈F
⎛
⎝
p−1
k=0
Dk
Hi((p − k) t)
⎞
⎠ (27)
where the notation ∀i(t) ∈ F means that, for every time point
t ∈ [0, p t], the current vector i(t) satisfies all the
(local, global, and equality) constraints. v+(t) is a nonnegative
vector defining the worst case voltage drops on power grid
nodes and the worst case voltage overshoots on ground grid
nodes, and similarly, v−(t) is a nonnegative vector defining
the worst case voltage overshoots on power grid nodes and
the worst case voltage drops on ground grid nodes. We used
a minus sign in front of emin(·) operator in (27) to avoid
confusion about the notion of the lower and upper bounds in
the rest of this paper. Using v+(t) and v−(t), v(t) can be
bounded as
−v−
(t) ≤ v(t) ≤ v+
(t). (28)
Although the RC model is dynamic, i.e., its currents and
voltages vary with time, the constraints are dc and do not
depend on time. Hence, F is the same for each time step. With
this, the components of (26) and (27) can be decoupled [8],
leading to
v+
(p t) =
p−1
k=0
emax
∀i∈F
[Dk
Hi] (29)
v−
(p t) = −
p−1
k=0
emin
∀i∈F
[Dk
Hi] (30)
where i is simply an m×1 vector of variables that satisfies the
(local, global, and equality) constraints, without reference to
any particular point in time. This is an important simplification
of the problem, as it has the advantage that the number of
constraints for each optimization is fixed and does not span
all previous time points. The advantage of using the matrix H
instead of E is clear now, since at each time step, one needs to
compute multiplication of an n ×n matrix by an n ×m matrix
instead of two n × n matrices. Furthermore, the optimization
variables do not include the redundant variables.
Claim 1: emax
∀i∈F
[Dk
Hi] ≥ 0 and emin
∀i∈F
[Dk
Hi] ≤ 0, ∀k.
Proof: Let x∗ = emax
∀i∈F
[Dk
Hi] and y∗ = emin
∀i∈F
[Dk
Hi].
Assume that the claim is not true, i.e., x∗
j < 0 and y∗
j > 0.
Since i† = 0 is feasible, i.e., i† ∈ F, we can choose i†, so that
(Dk Hi†)j = 0 > x∗
j and (Dk Hi†)j = 0 < y∗
j , which is a
contradiction. This completes the proof.
Using (29), (30), and Claim 1, and taking the difference in
two consecutive time steps, we have
v+
(p t) − v+
((p − 1) t) = emax
∀i∈F
[Dp−1
Hi] ≥ 0 (31)
v−
(p t) − v−
((p − 1) t) = − emin
∀i∈F
[Dp−1
Hi] ≥ 0 (32)
meaning that v+(p t) and v−(p t) are monotone
nondecreasing functions of the time point p, for any
integer p ≥ 1.
In practice, we are interested in the steady-state solution,
where the system becomes independent of the initial condition
v(t) = 0, ∀t ≤ 0. Since the RC grid model is a dynamical
system with a limited memory of its past, the steady-state
solution can be obtained by evaluating (29) and (30) at points
far away from the initial condition, i.e., as p → ∞. Thus, the
general solution to the problem is
v+
(∞) = lim
p→∞
p−1
k=0
emax
∀i(t)∈F
[Dk
Hi] (33)
v−
(∞) = − lim
p→∞
p−1
k=0
emin
∀i(t)∈F
[Dk
Hi]. (34)
B. Proposed Solution
Using (33) and (34) is intractable, because they have to be
evaluated for a large number of time steps until convergence is
achieved and the emax(·) and emin(·) operators require linear
programs proportional to the number of nodes in the grid,
which for modern designs is in the order of millions. As an
alternative, we propose an efficient solution to compute lower
and upper bounds on the worst case voltage fluctuations of the
P/G grid.
1) Vector of Lower Bounds: We show that, for specific
initial conditions, both v+(t) and v−(t) will be monotone
nondecreasing functions of time t. We have found that the
dc verification solution of the grid is a good initial condition
that satisfies the monotonicity property.
a) Nonzero initial conditions: We start by investigating
the impact of starting the verification with different (nonzero)
initial conditions on the worst case voltage fluctuations.
If we have the initial condition v0 at time t = 0, then
the voltage on the grid at any future time p t can be
expressed as
v(p t) = Dp
v0 +
p−1
k=0
Dk
Hi((p − k) t). (35)
Because the RC grid is a stable linear system and because the
backward difference approximation used in (10) is absolutely
stable [9], it follows that for i(t) = 0, ∀t and any bounded
initial condition v0, (35) converges to 0 as t → ∞. For
i(t) = 0, ∀t, the voltage on the grid at any time p t can
be written as
v(p t) = Dp
v0. (36)
Writing (36) as p → ∞, we get
v(∞) = lim
p→∞
v(p t) = lim
p→∞
Dp
v0 = 0. (37)
Since (37) is valid for any bounded initial condition, it is clear
that Dp → 0 as p → ∞. We use the following theorem [10]
to conclude that this actually means ρ(D) < 1, where ρ(D) is
the magnitude of the largest eigenvalue of D, also called the
spectral radius of D.

Theorem 1: Let D be a square matrix. Then, the
sequence Dk, for k = 0, 1, . . . , converges to zero if and only
if ρ(D) < 1.
It is easy to see that at steady state, i.e., as p → ∞,
choosing a different initial condition other than v0 = 0 does
not have any impact on v(∞), because of the fact that Dp
converges to zero as p → ∞. Therefore, (21) and (35) become
equivalent as p → ∞. Using the notation in Section II-A-4
and using the fact that F is the same for each time step, we
can express the worst case voltage fluctuations at time point
p t with the initial condition v0 as
v+
(p t) = Dp
v0 +
p−1
k=0
emax
∀i∈F
[Dk
Hi] (38)
v−
(p t) = Dp
v0 −
p−1
k=0
emin
∀i∈F
[Dk
Hi]. (39)
b) Monotone nondecreasing v+(t): Under the dc model
of the P/G grid, (9) becomes
Gv = ¯i. (40)
Let L = G−1 and let K be a n × m matrix, which is obtained
as the first m columns of L, such that
K = [l1,l2, . . . ,lm] (41)
where li defines ith column of L. Using the matrix K, we can
write v = Ki. Assume that we have the dc solution of the
system as the initial condition, leading to
v0 = Ki0. (42)
We define the voltage vector v0 to be feasible, if it satisfies (42)
for a current vector i0 ∈ F.
Claim 2: If v0 is feasible, then v+(p t) given in (38) is a
monotone nondecreasing function of the time point p, for any
integer p ≥ 1.
Proof: The claim is true if we can show
that v+(p t) ≥ v+((p − 1) t), for any integer p ≥ 1.
Substituting v0 from (42) into (38), we obtain
v+
(p t) = Dp
Ki0 +
p−1
k=0
emax
∀i∈F
[Dk
Hi]. (43)
Substituting Dp K from (86) (Appendix) into (43), we obtain
v+
(p t) =
⎛
⎝K −
p−1
k=0
Dk
H
⎞
⎠i0 +
p−1
k=0
emax
∀i∈F
[Dk
Hi]. (44)
Taking the difference in two consecutive time steps, we have
v+
(p t) − v+
((p − 1) t) = emax
∀i∈F
[Dp−1
Hi] − Dp−1
Hi0.
(45)
We see in (45) that the emax(·) operator assigns the
maximum value to the first term on the right-hand side over
all i ∈ F, whereas the second term on the right-hand side
has the variables i0 ∈ F, which may not result in the optimal
solution. Therefore, we conclude that if v0 is feasible, then
v+(p t) ≥ v+((p − 1) t), for any integer p ≥ 1.
This completes the proof.
c) DC initial condition: Using the notation (X)j to define
jth row of a matrix X and incorporating (38), (39), and (42),
we can express the worst case voltage fluctuation for the
jth node at time p t as
v+
j (p t) = (Dp
K)ji0 +
p−1
k=0
max
∀i∈F
(Dk
H)ji (46)
v−
j (p t) = (Dp
K)ji0 −
p−1
k=0
min
∀i∈F
(Dk
H)ji. (47)
A good choice of i0 for a given node would be the current
combination that leads to the worst case voltage fluctuation
for that node under the dc model of the P/G grid. The worst
case voltage fluctuation for the jth node under the dc model
is given by
v+
j = max
∀i∈F
(K)ji (48)
v−
j = − min
∀i∈F
(K)ji. (49)
Denote the optimal value of i of the maximization problem
in (48) as i+
j (m × 1 vector) and that of the minimization
problem in (49) as i−
j (m × 1 vector). Since G is an
M-matrix [8], its inverse consists only of nonnegative
elements, which means that K is a nonnegative matrix.
Therefore, the result of the minimization problem in (49)
under nonnegative current constraints will be 0, which leads
to i−
j = 0. Using i+
j and i−
j as the initial current at t = 0 for
the jth node, (46) and (47) become
v+
j (p t) = (Dp
K)ji+
j +
p−1
k=0
max
∀i∈F
(Dk
H)ji (50)
v−
j (p t) = −
p−1
k=0
min
∀i∈F
(Dk
H)ji. (51)
d) Lower bound: Algorithm 1 describes the computation
of the lower bound vector in detail, based on (50) and (51).
Using Claim 1 and Claim 2, we can see that v+
j (p t) in (50)
and v−
j (p t) in (51) are monotone nondecreasing functions
of the time point p, for any integer p ≥ 1. Since we stop the
main loop of Algorithm 1 for a finite p, when
v+(p t)−v+((p −1) t) and v−(p t)−v−((p −1) t) are
less than a threshold , then v+
lb(∞) and v−
lb(∞) are clearly
lower bounds on v+(∞) and v−(∞), respectively.
To see the advantage of starting the verification with a
dc initial condition, consider Figs. 7 and 8. Fig. 7 shows
v+
j (p t) for a particular node j on a P/G grid with the initial
condition v0 = 0, whereas Fig. 8 shows v+
j (p t) for the same
node with the initial condition v0 = Ki+
j . Clearly, starting the
verification with dc initial condition requires less number of
times steps to converge to steady-state value than starting the
verification with zero initial conditions.
2) Vector of Upper Bounds: Following an approach similar
to [11], we compute an upper bound on the worst case voltage
fluctuations of the P/G grid. Although the upper bound in [11]
was derived for an RLC model of the power grid, it can be
shown that it is also valid for the P/G grid model presented

Algorithm 1 Lower Bound Algorithm
Fig. 7. v+
j (p t) with zero initial condition.
Fig. 8. v+
j (p t) with dc initial condition.
in this paper. Due to space considerations, and because it is
mostly an adaptation of [11] to the P/G grid case, we will not
show the background work associated with the derivation of
the upper bound. Instead, we simply describe the computation
of the upper bound on the worst case voltage fluctuations
Algorithm 2 Upper Bound Algorithm
in Algorithm 2, in which we use the notation |X| to denote the
matrix of the element-wise absolute values of the entries of a
matrix X. Our main contribution in Algorithm 2 is the addition
of the convergence criterion Q F ≥ 1, which significantly
reduces the number of execution of the main loop. Further
details can be found in [11].
C. Implementation
1) Inverse Approximation Method: It is obvious from
Section II-B that the inverse of the system matrix A is needed
for the computation of the lower and upper bounds, because
D = A−1 B. For the lower bound, we need to invert the
conductance matrix G as well as A. We now explain how
this can be efficiently done.
Power distribution networks have a mesh structure, where
a node has a small number of neighboring nodes. Such a
structure results in a matrix (i.e., A or G) that is sparse,
symmetric, positive definite, and banded. In the literature, it is
well known that the inverse of a nonsingular sparse matrix
is dense. In particular, a matrix that results from a mesh
structure has the inverse that is almost full. However, it is
also well known in the literature that the inverse of a sparse,
symmetric, positive definite, and banded matrix has entries
whose values decay exponentially as one moves away from the
diagonal [12]. This fact is the main idea of constructing sparse
approximate inverse (SPAI) preconditioners to precondition
large sparse linear systems when using an iterative method,
such as conjugate gradient method. SPAI preconditioners try
to find a good SPAI M, such that M A ≈ I, where I denotes
the identity matrix.
There are numerous SPAI techniques in the literature.
One of them is SPAI [13] that is based on Frobenius norm
minimization. SPAI starts with an arbitrary initial matrix M
and iteratively refines its columns by minimizing the Frobenius
norm M A − I F . This technique has been applied to power

distribution network verification in [8]. Another technique is
called AINV [14], which is based on the conjugate
Gram–Schmidt (or A-orthogonalization) process. AINV has
the advantage of using the fact that the matrix whose inverse
is to be approximated is symmetric positive definite, while
SPAI is a general sparse approximate preconditioner for
unsymmetric matrices.
In what follows, we briefly explain the AINV method given
in [14]. Assume that A is an n ×n symmetric positive definite
matrix. It is shown in [14] that the factorization of A−1 can
be obtained from a set of conjugate directions z1, z2, . . . , zn
for A. By writing a set of conjugate directions in matrix form,
we have
Z = [z1, z2, . . . , zn] (52)
where Z is the matrix whose ith column is zi . Knowing a set
of conjugate directions for A, we can write
ZT
AZ = D =
⎡
⎢
⎢
⎢
⎣
d1 0 · · · 0
0 d2 · · · 0
...
...
...
...
0 0 · · · dn
⎤
⎥
⎥
⎥
⎦
(53)
where di = zT
i Azi . To obtain a factorization of A−1, we write
A−1
= Z D−1
ZT
. (54)
Notice in (54) that the inverse of A can be obtained easily if we
know a set of conjugate directions z1, z2, . . . , zn. In [14], a set
of conjugate directions is constructed by means of a conjugate
Gram–Schmidt (or A-orthogonalization) process applied to
any set of linearly independent vectors v1, v2, . . . , vn. The
Gram–Schmidt process is a method for orthogonalizing a
set of vectors in an inner product space of size n. It takes
a finite, linearly independent set of vectors v1, v2, . . . , vn
and generates an orthogonal set u1, u2, . . . , un that spans
the same inner product space n. For further details on the
Gram–Schmidt process, the reader is referred to [15].
Since the Gram–Schmidt process can be applied to any
set of linearly independent vectors, it is convenient to
choose vi = ei, where ei is the ith unit vector. From the
Cholesky factorization of A, we have
A = LDLT
(55)
where L is unit lower triangular that leads to Z = L−T. Since
the inverse of a unit lower triangular matrix is a unit lower
triangular matrix, it follows that Z is unit upper triangular.
For more details about the algorithm, the reader is referred
to [14]. For a dense matrix, this algorithm requires roughly
twice as much work as Cholesky factorization [14]. Although
for a sparse matrix the cost can be significantly reduced,
the method is still expensive because the resulting matrix Z
tends to be dense. However, the sparsity can be preserved by
reducing the amount of fill-in occurring in the computation
of z vectors. Reducing the amount of fill-in can be achieved
by ignoring all fill-in outside selected positions in Z or by
discarding fill-ins whose magnitude is less than a tolerance δ.
Zhang [16] proposes several strategies and special data
structures to implement AINV algorithm efficiently. We have
used many aspects of their implementation and we have made
some modifications in their data structures to be able to access
entries in a matrix that is in compressed column storage
format. Since we do not know the sparsity pattern of the
inverse upfront, we have used only the strategy in which
we discarded fill-in occurring in the computation of z-vectors
whose magnitude was less than δ = e−6.
Experimental results [17] show that AINV is more
effective and faster than the other approximate inverse
methods. Therefore, we have adopted it for our
implementation.
2) Network Simplex Method: We show that the linear
programs in our formulation can be efficiently solved with the
help of a network simplex method. Using the notation given
in Section II-A-3, we have the following linear program for
each node:
maximize/minimize: f (i)
s.t.: 0 ≤ Ui ≤ iG
Mi = 0, 0 ≤ i ≤ iL (56)
where f (i) : Rn → R is the linear objective function of i.
To simplify the notation, we augment the matrices U and M
into the matrix T , and we augment the vectors iG and the
zero vector of size γ into the vector a
T =
U
M
, a =
iG
0
where T is a matrix of size (μ + γ ) × n and a is a vector of
size (μ + γ ) × 1. With this notation, the linear program (56)
can be rewritten as
maximize/minimize: f (i)
s.t.: 0 ≤ Ti ≤ a
0 ≤ i ≤ iL. (57)
The constraint matrix T consists of entries that are
1, −1, or 0 s. It resembles the node-arc incidence
matrix (NAIM) of a network, in the sense that NAIM also
has entries that are 1, −1, or 0 s. This is the key observation
that allows us to formulate the optimization problem (57) as
a network flow problem. In our optimization problem, the
equality constraints define flow conservation constraints of
the network flow problem, whereas local constraints define
capacity constraints on the flow along the edges in the network.
Furthermore, global constraints define side constraints on
the sum of the flows along the edges. For a more detailed
discussion of network flow problems, the reader is referred
to [18].
Network flow problems can be efficiently solved with
the help of the network simplex method. Empirical results
have shown that the method is significantly faster than the
standard simplex method, when applied to the same network
problem [19]. Furthermore, it is shown in [20] that significant
computational speed up can be achieved if closely related
instances of network flow problems are solved sequentially.
This observation is quite beneficial for our problem in the
sense that the feasibility space of currents remains the same
at each instance of the optimization problem. Thus, we have

TABLE I
RUNTIME AND ACCURACY OF THE PROPOSED TECHNIQUE
the same optimization problem for each node except different
objective functions.
D. Experimental Results
To test our method, we have implemented Algorithms 1–3
in C++. We have set = e−5 to stop the main loop
of Algorithm 1. To solve the required linear programs,
Algorithms 1 and 2 use the network simplex method of
the MOSEK optimization package [21]. We have used the
hot-start option of the MOSEK network simplex solver for
fast objective function switching. Several experiments were
conducted on a set of test grids that were generated from
user specifications, which include grid dimensions,
metal layers (M1–M9), pitch and width per layer, and
C4 and current source distribution. Minimum spacing, sheet
resistance and via resistance were specified according to
a 65-nm technology. A global constraint is specified for each
macroblock, and additional global constraints were specified
covering the entirety of the grid area. The computations were
carried on a 64-bit Linux machine with 8-GB memory.
Table I shows the speed and the accuracy of our proposed
solution technique for the computation of the vectors of lower
and upper bounds on the worst case voltage fluctuations.
The results are compared with each other and the maximum
absolute difference between the upper and the lower bound
vector is reported in column 6, where the absolute error is
defined as (vub(∞) − vlb(∞)). The results show that our
solution technique resulted in a maximum absolute error
of 2.72 mV across all nodes of all test grids. The number of
time steps shown in columns 3 and 5 reports the number of
time steps for which the lower and upper bound algorithms
converge. The runtime for each one of two methods is also
shown in Table I.
Fig. 9 shows a scatter plot with the lower bound on the
worst case voltage fluctuation on one axis and the absolute
error on the other axis, for a 4285-node grid. The figure
shows that the absolute error between the upper and the lower
bound is very small, meaning that the proposed method is very
accurate. For the same grid, Fig. 10 shows a scatter plot with
the lower bound on the worst case voltage fluctuation on one
axis and the relative error on the other axis, where the relative
error is defined as (vub(∞)−vlb(∞))/vlb(∞). The figure also
Fig. 9. Absolute error comparison for all nodes of a 4285-node grid.
Fig. 10. Accuracy of the proposed technique.
shows the curve corresponding to 3 mV absolute error, where
a point on the curve represents a node that has 3 mV difference
between its upper and lower bound. Note that the relative error
can be high, but only for small values of voltage fluctuations,
and the absolute error does not exceed 3 mV.
Table II compares the worst case voltage fluctuations
of (only) power grid verification and the (unsymmetrical)
P/G grid verification, in which the P/G grid contains the
same power grid as in the power grid verification and an
unsymmetrical ground grid. The worst case voltage fluctuation
for both scenarios is reported in columns 3 and 4, respectively,
and the absolute difference between columns 3 and 4 is

TABLE II
COMPARISON BETWEEN (ONLY) POWER GRID VERIFICATION AND THE (UNSYMMETRICAL) P/G GRID VERIFICATION
Fig. 11. Comparison between (only) power grid verification and the
(unsymmetrical) P/G grid verification.
reported in column 5. The results show that the absolute
difference can be as high as 35 mV. The results also show that
in some cases (only) power grid verification results in larger
worst case voltage fluctuation, whereas in some cases P/G grid
verification results in larger worst case voltage fluctuation.
This result is expected since the unsymmetry between the
P/G grids can impact worst case voltage fluctuation. Fig. 11
shows a bar chart for the comparison between (only) power
grid verification and the (unsymmetrical) P/G grid verification
based on Table II.
Looking at the runtime results given in Table I, we notice
that the computation of the lower and upper bound vectors
of a 18472-node grid takes ∼4 h (most of this runtime is
dominated by approximate inverse computation). Such a grid
is of course small compared with full-chip grids containing
millions of nodes. However, the proposed method is applicable
for early grid verification, where the size of the grids is
normally not as large. The power of our approach is that it
finds tight upper and lower bounds on the worst case voltage
fluctuations under all feasible current combinations. It is a
unique approach that offers this type of guarantee.
III. POWER GRID VERIFICATION UNDER HIERARCHICAL
CURRENT CONSTRAINTS
In this section, we will first show that the linear programs
in power grid verification can be efficiently solved, if
we assume that the global constraint matrix U has no
overlapping constraints, i.e., each column of U has only
one nonzero element. We start with a base case, in which we
assume that there is only one global constraint, and we present
the solution to the problem in which there are more than one
global constraint that are not overlapping. Then, we present the
optimal solution to linear programs with tree-structured global
constraint matrices (whereby global constraints are required to
be wholly contained in other global constraints) based on [22].
Due to space considerations, we will not show the problem
formulation of power grid verification problem in this paper.
For background information on power grid verification with
current constraint matrices, refer to [8]. The proposed methods
in this section are only applicable to power grids, not to
P/G grids. This is due to additional equality constraints in
P/G grid problem formulation; these equality constraints are
not part of the power grid problem formulation.
A. Verification Under Nonoverlapping Global Constraints
1) Problem Definition: Define x as a vector variable of size
m representing the current sources, and f (x) : Rm → R as a
vector function of x. Assuming that we have only one global
constraint (and since we do not have any equality constraints
in power grid model as apposed to P/G grid model), we have
the following linear program:
maximize: f (x) = aT
x
s.t.: cT
x ≤ b
0 ≤ x ≤ d (58)
where a is a vector of size m that defines the coefficients
in the objective function and c is a vector of size m whose
each entry is equal to 1. b is a positive number that is
the upper bound defining the global constraint and d is the
local constraint vector of size m. The linear program (58)
resembles the fractional knapsack problem, except the fact
that the coefficients in the constraint matrix of a fractional
knapsack problem can be any nonnegative value [23]. Similar
to the optimal solution of the fractional knapsack problem, we
will show that the linear program (58) is solvable by a greedy
strategy.
Assume without loss of generality that
a1 ≥ a2 ≥ · · · ≥ am ≥ 0. (59)
Thus a is sorted in a descending order and it consists of only
nonnegative entries. Assume that cT d > b, because otherwise

x = d is optimal. Let k be the index satisfying
k−1
i=1
di < b and
k
i=1
di ≥ b. (60)
Define
η = b −
k−1
i=1
di. (61)
We first state the following theorem [24], which will be useful
for the proof of Claim 3.
Theorem 2 (Strong Duality): If x∗ is a feasible solution to
the primal problem max{aT x | Ax ≤ b, x ≥ 0} and u∗ is a
feasible solution to the dual problem min{bT u | AT u ≥ a,
u ≥ 0}, then x∗, u∗ are primal-dual optimal if and only
if aT x∗ = bT u∗.
Now define
J =
cT
I
, ζ =
b
d
(62)
where I is an m × m identity matrix. Notice that J is
an (m + 1) × m matrix, and ζ is a vector of size m + 1.
With the help of (62), we modify the linear program (58) to
be in the format of the primal problem in Theorem 2 as
maximize: f (x) = aT
x
s.t.: Jx ≤ ζ
x ≥ 0. (63)
Claim 3: The optimal solution to the linear program (63)
is given by
x∗
i =
⎧
⎨
⎩
di, i = 1, . . ., k − 1
η, i = k
0, i = k + 1, . . . , m.
(64)
Proof: The dual of the linear program (63) is given by
minimize: ζT
u
s.t.: JT
u ≥ a
u ≥ 0. (65)
Define y = u1 and zi = ui+1 for i = 1, . . . , m. Then, we can
rewrite (65) as
minimize: by + dT
z
s.t.: y + zi ≥ ai, i = 1, . . . , m
y, z ≥ 0. (66)
Now choose the variables y∗ and z∗ as
u∗
1 = y∗
= ak (67)
u∗
i+1 = z∗
i =
ai − ak, i = 1, . . . , k − 1
0, i = k, . . . , m.
(68)
Notice that the solution u∗ given by (67) and (68) is a feasible
solution to the dual problem (66) and that the solution x∗ given
by (64) is a feasible solution to the primal problem (63). If we
can show that x∗ and u∗ satisfy the condition aT x∗ = ζT u∗,
then they are primal-dual optimal by Theorem 2. Using x∗
given by (64), we write
aT
x∗
=
k−1
i=1
aidi + akη
=
k−1
i=1
aidi + ak b −
k−1
i=1
di
= akb +
k−1
i=1
aidi − ak
k−1
i=1
di. (69)
Using u∗ given by (67) and (68), we write
ζT
u∗
= by∗
+ dT
z∗
= akb +
k−1
i=1
di(ai − ak)
= akb +
k−1
i=1
aidi − ak
k−1
i=1
di. (70)
Thus, aT x∗ = ζT u∗ for x∗ given by (64) and for u∗ given
by (67) and (68). Therefore, they are primal-dual optimal by
Theorem 2. This completes the proof.
In (63), we assumed that the coefficients of the objective
function are nonnegative. However, the objective function
can have negative coefficients under the RLC model of the
power grid [11]. Now, we show that the optimization variables
that correspond to the negative coefficients of the objective
function in the linear program (63) will always be equal to 0 in
the optimal solution. We first state the following theorem [24].
Theorem 3 (Complementary Slackness): If x∗ is a feasible
solution to the primal problem max{aT x | Ax ≤ b, x ≥ 0},
where A is an l × n matrix and u∗ is a feasible solution to
the dual problem min{bT u | AT u ≥ a, u ≥ 0}, then x∗, u∗ are
primal-dual optimal if and only if the following hold.
1) For i = 1, 2, . . . ,l, if u∗
i > 0, then A[i,:]x∗ = bi.
2) For j = 1, 2, . . . , m, if x∗
j > 0, then (u∗)T A[:, j] = aj.
Here, A[i,:] corresponds to ith row of A and A[:, j] corresponds
to jth column of A.
Now assume without loss of generality that
a1 ≥ a2 ≥ · · · ≥ am. (71)
Let h be the index satisfying
ah ≥ 0 and ah+1 < 0. (72)
Claim 4: The entries x∗
h+1, . . . , x∗
m of the optimal
solution x∗ to the linear program (63) are equal to 0.
Proof: Assume that the optimal solution to the dual
problem (66) is given by u∗. Applying condition 2) of
Theorem 3 to (63), we have
For j = 1, 2, . . . , m, if x∗
j > 0, then u∗
1 + u∗
j = aj .
Since aj is negative for j > h and since u ≥ 0, meaning
u∗
1 + u∗
j ≥ 0, we conclude that x∗
j > 0 for j > h, leading to
x∗
j = 0 for j > h. This completes the proof.
Thus, the optimization variables that have negative
coefficients in the objective function can be simply ignored.

Let us have a look into the general problem in which we
have more than one global constraint that is not overlapping.
Suppose that there are p nonoverlapping global constraints.
Denote the variables belonging to ith global constraint by x(i),
their part of the objective function by fi (x(i)) and their feasible
space due to ith global constraint and local constraints of
x(i) by Fi. It is shown in [25] that if the variables x(i)
do not appear in common constraints, they are independent
and the optimal value can be obtained by simply solving
p-independent subproblems and summing the results of each
p subproblem. Define
ei = max
∀x(i)∈Fi
fi (x(i)
) (73)
where ei is the optimal solution to ith global constraint.
The optimal solution e of the maximization problem due to
p nonoverlapping global constraints can be written as the sum
of the optimal solutions due to p independent maximization
problems as follows:
e = max
∀x∈F
f (x) = e1 + e2 + · · · + ep. (74)
Notice that the optimal solution to each subproblem in (74)
is given by (64). Since every minimization problem can be
converted into a maximization problem by simply negating
the coefficients in the objective function, (74) is valid for
minimization problems as well.
B. Verification Under Tree-Structured Global Constraints
Faaland [22] solved a simple class of linear programs with
tree-structured constraint matrices. The special linear
programming (LP) structure is in the following form:
maximize:
j∈J
cj x j
s.t.:
j∈J(i)
aj x j ≤ bi, i = 1, . . . , m
0 ≤ x j ≤ u j , j ∈ J (75)
where bi, cj , and aj are positive scalars, J = ∪m
i=1 J(i),
and the sets J(i) are nested meaning that if i = k, either
J(i) and J(k) are disjoint, or one set is properly contained
in the other. We may assume that bi < bk, whenever J(i) is
a proper subset of J(k). When m = 1, (75) reduces to the
fractional knapsack problem with upper bounds on variables.
In each column of the constraint matrix, all nonzero
entries are equal and positive. This is also the case for the
tree-structured linear programs in our formulation, because
all nonzero entries in the global constraint matrix are 1 s.
The only difference between (75) and the linear programs in
our formulation is the lower bound for the global constraints.
However, it is shown in [26] that the results obtained in [22]
are valid for linear programs with nonnegative lower bounds
for the constraint matrices.
The solution method shown in [22] separates the problem
into a sequence of fractional knapsack problems. Each of
these fractional knapsack problems requires linear time to
solve, for total solution time no worse than proportional to
the number of nonzero entries in the original constraint matrix.
TABLE III
EFFECTIVENESS OF THE SORTING METHOD COMPARED WITH LP SOLVER
The optimal solution is obtained by giving each variable their
minimum value from the list obtained by solving the sequence
of fractional knapsack problems.
Tree-structured global constraints are more realistic
than nonoverlapping global constraints in modern designs.
We believe that modern designs are very hierarchical, and
consist of functional blocks; IPs and subsystems, all of which
can have their current constraints expressed as tree-structured
global constraints. These constraints can come from
high-level power constraints and functional block
power/current constraints [4]. For more information on
the application of this type of constraints, the reader is
referred to [4].
C. Implementation and Experimental Results
We have implemented the solution method given in [22]
in C++. To sort the coefficients of the objective function, we
used quicksort algorithm.
To see the runtime behavior of our implementation, we
have conducted several experiments to solve linear programs
resulting from power grid verification with tree-structured hier-
archical constraints. We compared the results of our implemen-
tation to the results obtained from solving the linear programs
using MOSEK [21]. The computations were performed on the
same machine as mentioned in Section II-D. The comparison
of the results is presented in Table III. Column 1 shows the size
of the power grid, columns 2 and 3 reports the runtime to solve
the linear program for all the nodes using our implementation
of the method given in [22] and using LP solver, respectively,
and column 4 reports the achieved speed up. Notice that the
runtime numbers do not include the amount of time needed
to compute the matrix inversion. It can be seen from Table III
that the proposed method is significantly fast, solving the large
linear programs in significantly shorter time. In addition, note
that both solutions are exact solutions to the LP problem, thus
yielding the same results in both cases.
IV. CONCLUSION
Voltage fluctuations of the power distribution network are
a key concern for modern chip design. In this paper, we
presented an early grid verification technique that takes both
P/G grids into account. We first formulated the grid verifi-
cation problem as an optimization problem to compute the
exact worst case voltage fluctuations at each node subject to
current constraints, which is seen to be too expensive. As an
alternative, we proposed a solution approach that formulates
both upper and lower bounds on the worst case voltage fluc-
tuations. Experimental results show that the proposed bounds
have errors in the range of a few millivolts, and that the

verification of both P/G grids is required to capture realistic
voltage fluctuations on the power distribution network. We also
showed that, under hierarchical current constraints, power grid
problem can be efficiently solved without any loss of accuracy,
resulting in significant run-time savings.
The P/G grid model presented in this paper is an RC model
of the grid. The grid inductance can contribute significantly to
the voltage fluctuation. Our approach can be easily extended
to take inductive effects into account for P/G grid verification.
In addition, designers may be interested in the voltage
difference between P/G grid nodes, rather than individual node
voltages, and therefore, the future work should extend the
mathematical formulation to calculate the worst case voltage
difference. Besides, the current constraints used in this paper
are dc constraints. The P/G grid verification problem should
also be extended and solved under transient current constraints.
The future work should also include the application of the
improvements in power grid verification under hierarchical
constraints to P/G grid verification.
APPENDIX
We will prove additional claims that are useful in the context
of Section II-B1. We start with the following claim.
Claim 5: I − Dp is invertible for any integer p.
Proof: Denoting the set of all eigenvalues of D by σ(D)
and using Theorem 1, we can write
max
∀λ∈σ(D)
|λ| < 1. (76)
From the spectral mapping theorem [27], we know that if k is
an integer, we have the following relationship:
σ(Dk
) = {λk
: λ ∈ σ(D)}. (77)
Thus, ρ(Dp) < 1, for an integer p. We also know that
the series ∞
q=0 Xq for a square matrix X is known to
converge [10] if and only if ρ(X) < 1, under which condition
the series limit is (I − X)−1, meaning that I − Dp is invertible
for an integer p. This completes the proof.
With the help of Claim 5, we will prove the following claim.
Claim 6: Suppose W(p) is a function of p for p ≥ 1
defined as
W(p) = (I − Dp
)−1
p−1
k=0
Dk
H. (78)
Then W(p) = W(1), ∀p ≥ 1.
Proof: The case for p = 1 is satisfied trivially. The claim
is true by induction if we prove the following, ∀p ≥ 2:
W(p − 1) = W(1) ⇒ W(p) = W(1). (79)
Left multiplying both sides of (78) with (I−Dp), and breaking
the sum
p−1
k=0 Dk H into
p−2
k=0 Dk H and Dp−1 H, we obtain
(I − Dp
)W(p) =
p−2
k=0
Dk
H + Dp−1
H. (80)
From Claim 5, we know that I − Dp is invertible
for an integer p. Left multiplying both sides of (80)
with (I − Dp−1)−1, we obtain
(I − Dp−1
)−1
(I − Dp
)W(p)
= (I − Dp−1
)−1
p−2
k=0
Dk
H + (I − Dp−1
)−1
Dp−1
H. (81)
Since (I − Dp−1)−1 p−2
k=0 Dk H = W(p −1), we can replace
it with W(1). Rearranging the terms of (81), we have
(I − Dp−1
)−1
((I − Dp
)W(p) − Dp−1
H) = W(1). (82)
Left multiplying both sides of (82) with (I − Dp−1) leads to
W(p) − Dp
W(p) − Dp−1
= W(1) − Dp−1
W(1). (83)
Adding Dp−1W(1) to both sides of (83) and rearranging the
terms, we obtain
W(p) − W(1) = (I − Dp
)−1
Dp−1
(H − (I − D)W(1)).
(84)
Replacing W(1) on the right-hand side of (84)
with (I − D)−1 H leads to
W(p) − W(1) = (I − Dp
)−1
Dp−1
(H − H) = 0. (85)
Meaning that W(p) = W(1). This completes the proof.
Finally, our main result is captured in the following claim.
Claim 7: For any integer p ≥ 1
Dp
K = K −
p−1
k=0
Dk
H. (86)
Proof: Since W(p) = W(1) by Claim 6, meaning that
(I − Dp
)−1
p−1
k=0
Dk
H = (I − D)−1
H. (87)
Left multiplying both sides of (87) with (I − Dp), we obtain
p−1
k=0
Dk
H = (I − Dp
)(I − D)−1
H. (88)
Since D = A−1 B and B = A − G, we have
D = A−1(A − G) = I − A−1G, leading to I − D = A−1G.
Since K and H are matrices obtained from the first m columns
of G−1 and A−1, respectively, we can write K = (I −D)−1 H,
leading to
p−1
k=0
Dk
H = (I − Dp
)K. (89)
Rearranging the terms of (89), we get
Dp
K = K −
p−1
k=0
Dk
H (90)
which completes the proof.

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Mehmet Avci received the B.S. (Hons.) degree in
electrical and electronics engineering from Middle
East Technical University, Ankara, Turkey, in 2008,
and the M.A.Sc. degree in electrical and computer
engineering from the University of Toronto, Toronto,
ON, Canada, in 2010.
He is currently with the Toronto Technology
Centre, Altera Corporation, Toronto, as a Senior
Design Engineer, where he is involved in timing
modeling and analysis. His current research inter-
ests include computer-aided design for integrated
circuits, with a focus on timing modeling and analysis, power modeling, and
grid integrity verification.
Farid N. Najm (S’85–M’89–SM’96–F’03) received
the B.E. degree in electrical engineering from the
American University of Beirut, Beirut, Lebanon, in
1983, and the Ph.D. degree in electrical and com-
puter engineering from the University of Illinois at
Urbana—Champaign (UIUC), Champaign, IL, USA,
in 1989.
He was with Texas Instruments, Dallas, TX, USA,
from 1989 to 1992. He joined the Department of
Electrical and Computer Engineering, UIUC, as an
Assistant Professor, where he became an Associate
Professor in 1997. He joined the Department of Electrical and Computer
Engineering, University of Toronto, Toronto, ON, Canada, in 1999, where
he is currently a Professor and the Chair. He is a Professor with the
Edward S. Rogers Sr. Department of Electrical and Computer Engineering,
University of Toronto. He has authored a book entitled Circuit Simulation
(New York: John Wiley and Sons, 2010). His current research interests include
computer-aided design for VLSI, with an emphasis on circuit level issues
related to power, timing, variability, and reliability.
Dr. Najm is a fellow of the Canadian Academy of Engineering. He has
received the IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN Best
Paper Award, the National Science Foundation (NSF) Research Initiation
Award, the NSF CAREER Award, and the Design Automation Confer-
ence (DAC) Prolific Author Award. He was an Associate Editor of the
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED
CIRCUITS AND SYSTEMS from 2001 to 2009, and the IEEE TRANSACTIONS
ON VERY LARGE SCALE INTEGRATION SYSTEMS from 1997 to 2002.
He served on the Executive Committee of the International Symposium on
Low-Power Electronics and Design (ISLPED) from 1999 to 2013, and served
as the Technical Program Committee Chair and General Chair of ISLPED.
He has also served on the Technical Committees of various conferences,
including the International Conference on Computer-Aided Design, DAC,
the Custom Integrated Circuits Conference, the International Symposium on
Quality Electronic Design, and ISLPED.

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