Precise Identification of the WorstCase Voltage Drop Conditions in Power Grid Verification

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Precise Identification of the Worst-Case Voltage
Drop Conditions in Power Grid Verification
Nestoras Evmorfopoulos
Dimitris Karampatzakis
Georgios Stamoulis

• University of Thessaly
• Department of Computer and Communication
  Engineering

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Introduction (1)

• Voltage (IR) drop on non-ideal P/G wires affects
• circuit speed
• noise immunity
• Power grid analysis and verification are
  essential part of the design process
• Verification checking if the grid maintains a
  safe voltage everywhere and at all times
• Dynamic analysis of power grid
• Simulate the supplied digital circuit under
  specific input patterns to determine current
  waveforms drawn by the circuit modules ? current
  sinks
• Solve the linear network representing the power
  grid with the computed waveforms as current
  excitations

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Introduction (2)

• Full dynamic analysis (with time-varying
  excitations) only possible for a small set of
  patterns
• Dynamic analysis usually accompanied by some kind
  of static analysis (with fixed DC excitations)
• One or more static vectors of sink currents are
  used as excitations
• Static vectors selected to represent all input
  patterns
• Resistance-only model is employed for the power
  grid
• Practical experience shows that most significant
  problems and weak spots are identified through
  static analysis
• In most design methodologies the power grid
  signoff is based on the outcome of static analysis

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Motivation (1)

• Common choices for DC excitations in static
  analysis
• Vector of average currents over all input
  patterns
• Vector of maximum currents over all input
  patterns
• Both of the above choices are inefficient
• Vector of average currents is not guaranteed to
  contain the worst-case voltage drop
• Vector of maximum currents substantially
  over-estimates the worst-case voltage drop
  because sinks do not attain their maximum current
  simultaneously

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Motivation (2)

• Proposed approach provides an efficient set of
  static current vectors that give a realistic
  worst-case voltage drop
• Identification of the characteristic subset of
  current vectors inside the set of all possible
  current variations which constitute the
  worst-case conditions for voltage drop
• Approximation of the worst-case subset by a
  number of sampled current vectors, which are
  statistically extrapolated to the proper position
  of the worst-case subset by results from Extreme
  Value Theory (EVT)

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Outline

• Power grid analysis
• Worst-case voltage drop
• Efficient static analysis
• Practical methodology
• Experimental setup
• Procedure
• Results

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Power grid analysis

• Resistive linear network with q non-supply nodes
• n sink nodes (with current sources)
• q-n internal nodes
• Modified Nodal Analysis (MNA)
• GV(t) I(t)
• V(t) voltage drops at internal and sink nodes
• I(t) current excitations at internal and sink
  nodes
• I(t) with q-n zero entries and n nonzero
  entries
• Solution for voltage drops at the n sink nodes
• vk(t) rk1i1(t)rk2i2(t)rknin(t)
  rkTi(t) , 1kn
• with rkj0, ?k,j1,2,,n (since G is an
  M-matrix)

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Worst-case voltage drop (1)

• For verification we are interested in
  maxt??vk(t), 1kn
• Consider each vk(t) as composite function
  vki(t) formed by the composition of
• vk(i) multivar. function or function of vector
  variable
• i(t) vector-valued function, i ???n
• The range D i(?) ? ?n of i(t) (dubbed as
  current space) becomes domain in vk(i) i.e. vk
  D??
• ? maxt??vk(t) maxi?Dvk(i) , 1kn
• In optimization terminology, vk(i) is an
  objective function to be optimized over a
  feasible set D
• ? find maximizing vector i
  vk(i)maxi?Dvk(i)

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Worst-case voltage drop (2)

• vk(i) rkTi , 1kn is linear in i
• ? i??D (?D boundary of D)
• rkj0, ?k,j1,,n ? i?P ? ?D
• P maximal (or noninferior) subset of D w.r.t.
  the partial order in ?n
• P i?D ?i'?D such that ik'ik with ik'gtik
  for at least one 1kn

• Practical objective find a finite set of maximal
  points that approximate the global position of P
  in ?n

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Efficient static analysis (1)
Multi-cycle DC current scheme
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Efficient static analysis (2)

• Possible choices for DC cycle values
• Cycle-mean
• Cycle-peak
• In employed DC current scheme
• Vector of sink currents i ? ibp,bn or i
  O??n O set of pairs of binary input
  patterns bp,bn
• Current space D i(O) ? ?n

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Practical methodology (1)

• Collect a sample S i1,i2,,im of current
  vectors (sample space) for m random pairs bp,bn
• Sample space S has a maximal set PS of its own
• PS scaled down w.r.t. P in each coordinate axis
  1kn

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Practical methodology (2)

• Average degree of down-scaling in each 1kn
• max. of component k in the current space D minus
  max. of component k in the sample space S , i.e.
• dk ?k max(ik1,ik2,,ikm)
• but EVT allows univariate (one-dimensional)
  estimation of ?k for each 1kn on the basis of
  the univariate samples Sk ik1,ik2,,ikm
• Partition each Sk into m/l sub-samples of size l
• Maxima from sub-samples asymptotically (for large
  l) follow an EV distribution with cdf H0(x)
  exp(-exp(-x)) translated and scaled by constants
  ak and bk which are parametrically dependent on
  ?k
• ak,bk (and thus ?k) are determined by ML
  estimation

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Practical methodology (3)

• ? shift PS by dk ?k max(ik1,,ikm) in each
  axis 1kn
• or overall by the difference vector d ?
  max(i1,,im)

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Experimental setup

• Generation of test power grids
• Various uniform grids ranging from 25 to 150
  internal/sink nodes and 1 to 15 voltage nodes
• Random values of intermediate conductances
  (within some process-related limits)
• Randomly placed voltage nodes (assuming C4 pads)
  and sink nodes across the grid area
• Circuits supplied by the grids
• ISCAS85 benchmarks implemented in 0.18µm and/or
  0.13µm technologies
• Each circuit partitioned into a number of
  functional modules representing the current sinks

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Procedure (1)

• Identification of the worst-case static current
  vectors for voltage drop at each sink
• Generate m5000 random pairs for the supplied
  circuit
• Simulate circuit for all generated pairs and
  record DC cycle-mean or cycle-peak current values
  for each pair
• Recorded data make up the sample space S
  i1,i2,,im
• For each univ. sample Sk ik1,ik2,,ikm in S
  estimate maximum ?k of component k in the space D
  via EVT
• A total of m5000 pairs produces 5 statistical
  estimation error for any circuit or current sink
  irrespective of its size
• For each univariate sample Sk determine maximum
  max(ik1,ik2,,ikm) of component k in the space S
• Calculate difference vector d ? max(i1,,im)

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Procedure (2)

• (contd)
• Locate set PS of maximal points in sample space S
• Required comparisons Om(log2m)n-2
• Shift all points/vectors in PS by the difference
  vector d
• Steps of identifying worst-case current vectors
  for voltage drop is independent of the supplying
  grid
• Verification of any grid supplying the digital
  circuit
• Apply shifted points of PS as DC current vectors
  for static analysis
• For each sink determine the maximum value among
  all computed voltage drops from static analyses

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Results (1)
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Results (2)