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Power Grid Simulation using Matrix Exponential
Method with Rational Krylov Subspaces
Hao Zhuang, Shih-Hung Weng, and Chung-Kuan Cheng Department of Computer Science and Engineering
University of California, San Diego, CA, USA Contact: {zhuangh, ckcheng}@ucsd.edu
Outline • Background of Power Grid Transient Circuit Simulation
– Formulations – Problems
• Matrix Exponential Circuit Simulation (Mexp) – Stiffness Problem
• Rational Matrix Exponential (Rational Mexp) – Rational Krylov Subspace – Skip of Regularization – Flexible Time Stepping
• Experiments – Adaptive Time Stepping Experiment – Standard Mexp vs. Rational Mexp (RC Mesh) – Rational Mexp vs. Trapezoidal Method (PDN Cases)
• Conclusions
2
Power Grid Circuit Power Grid modeled in RLC circuit
• Transient Power Grid formulation where • is the capacitance/inductance matrix • is the conductance matrix • is the voltage/current vector, and is input
sources
3
𝐂𝐱 𝑡 = −𝐆𝐱(𝑡) + 𝐁𝐮(𝑡)
𝐂
𝐆
𝐱 𝐁𝐮(𝑡)
Power Grid Transient Circuit Simulation Transient simulation: Numerical integration
• Low order approximation
– Traditional methods: e.g. Backward Euler, Trapezoidal
– Local truncation error limits the time step
– Power grid simulation contest [TAU’12]
• Trapezoidal method with fixed time-step: only one LU factorization
• Stiffness: smallest time step
• High order approximation
– Matrix exponential based circuit simulation 4
𝐂
ℎ+𝐆
2𝐱 𝑡 + ℎ =
𝐂
ℎ−𝐆
2𝒙 𝑡 +
𝐁𝐮 𝑡 + ℎ − 𝐁𝐮(𝑡)
2
Outline • Background of Power Grid Transient Circuit Simulation
– Formulations – Problems
• Matrix Exponential Circuit Simulation (Mexp) – Stiffness Problem
• Rational Matrix Exponential (Rational Mexp) – Rational Krylov Subspace – Skip of Regularization – Flexible Time Stepping
• Experiments – Adaptive Time Stepping Experiment – Standard Mexp vs. Rational Mexp (RC Mesh) – Rational Mexp vs. Trapezoidal Method (PDN Cases)
• Conclusions
5
Matrix Exponential Method
• Linear differential equation
• Analytical solution
• Case: input is piecewise linear (PWL)
6
𝐂𝐱 𝑡 = −𝐆𝐱(𝑡) + 𝐁𝐮(𝑡) 𝐱 𝑡 = −𝐀𝐱(𝑡) + 𝐛(𝑡)
𝐀 = −𝐂−𝟏𝐆, 𝐛 = −𝐂−𝟏𝐁𝐮(𝐭)
𝐱 𝑡 + ℎ = 𝑒𝐀ℎ𝐱(𝑡) + 𝑒𝐀(ℎ−𝜏)𝐛(𝑡 + 𝜏) 𝑑𝜏ℎ
0
𝐱 𝑡 + ℎ = 𝑒𝐀ℎ𝐱 𝑡 + (𝑒𝐀ℎ−𝐈)𝐀−𝟏𝐛(𝑡) + (𝑒𝐀ℎ−(𝐀ℎ + 𝐈))𝐀−𝟐𝐛(𝑡 + ℎ) − 𝐛(𝑡)
ℎ
Matrix Exponential Computation
• Transform into
• The computation of matrix exponential is expensive (for simplicity, we use 𝐀 to represent 𝐀 , from now on)
Memory and time complexities of O(n3)
7
𝐱 𝑡 + ℎ = 𝐈𝑛 𝟎 𝑒𝐀 ℎ 𝐱(𝑡)
𝐞2
𝐀 =𝐀 𝐖𝟎 𝐉
, 𝐉 =0 10 0
, 𝐞2 =𝟎1,𝐖 =
𝐛 𝑡 + ℎ − 𝐛(𝑡)
ℎ𝐛(𝑡)
𝒆𝐀 = 𝐈 + 𝐀 +𝐀2
2+𝐀3
3!+ ⋯+
𝐀𝑘
𝑘!+ ⋯
Krylov Subspace Approximation
• We derive matrix-vector product:
• Krylov subspace
– Standard Basis Generation
– Orthogonalization (Arnoldi Process):
– Matrix reduction: Hm,m has m=10~30 while size of A can be millions
• Matrix exponential operator
– time stepping, h, via scaling
– Posteriori error estimate [Saad92]
8
𝒆𝐀𝐯
𝑲𝒎 𝐀, 𝐯 = 𝐯,𝐀𝐯, 𝐀𝟐𝐯,… , 𝐀𝒎−𝟏𝐯
𝐀𝐯 = −𝐂−𝟏(𝐆𝐯)
𝐕𝒎 = 𝐯𝟏, 𝐯𝟐, ⋯ , 𝐯𝒎
𝐀𝐕𝒎 = 𝐕𝒎𝐇𝒎,𝒎 + 𝒉𝒎+𝟏,𝒎𝐯𝒎+𝟏𝒆𝒎T 𝐇𝒎,𝒎 = 𝐕𝒎
T𝐀𝐕𝒎
𝒆𝐀ℎ𝐯 ≈ 𝐯 𝟐𝐕𝒎 𝒆𝐇𝒎,𝒎ℎ𝒆𝟏
1
Τ
21, eeehmmErr h
mkrylovmH
Hv
Problems of Standard Krylov Subspace Approximations
Problem of Stiffness:
• When the system is stiff, we need high order approximation so that the solution can converge,
• Standard Krylov subspace tends to capture the eigenvalues of large magnitude
• For transient analysis, the eigenvalues of small real magnitude are wanted to describe the dynamic behavior.
9
𝐀 = −𝐂−𝟏𝐆
𝐱 𝑡 = 𝐀𝐱(𝑡) + 𝐛(𝑡)
𝒆𝐀 = 𝐈 + 𝐀 +𝐀2
2+
𝐀3
3!+⋯+
𝐀𝑘
𝑘!.
Outline • Background of Power Grid Transient Circuit Simulation
– Formulations – Problems
• Matrix Exponential Circuit Simulation (Mexp) – Stiffness Problem
• Rational Matrix Exponential (Rational Mexp) – Rational Krylov Subspace – Skip of Regularization – Flexible Time Stepping
• Experiments – Adaptive Time Stepping Experiment – Standard Mexp vs. Rational Mexp (RC Mesh) – Rational Mexp vs. Trapezoidal Method (PDN Cases)
• Conclusions
10
Rational Krylov Subspace • Spectral Transformation:
– Shift-and-invert matrix A
– Rational Krylov subspace captures slow-decay components
– Use rational Krylov subspace for matrix exponential
11
100
Important eigenvalue: Component that decays slowly. Not so important eigenvalue: Component that decays fast.
𝑲𝒎 𝐀, 𝐯 𝑲𝒎 (𝐈 − 𝛾𝐀)−𝟏, 𝐯
(𝐈 − 𝛾𝐀)−𝟏
Rational Krylov Subspace
Rational Krylov subspace
• Arnoldi process to obtain Vm=[v1 v2 … vm]
• Matrix exponential
– Time stepping by scaling
– No need of new Krylov subspace computation.
• Posterior error to terminate the process
– Larger time step => smaller error
12
𝑲𝒎 (𝐈 − 𝛾𝐀)−𝟏, 𝐯 = 𝐯, (𝐈 − 𝛾𝐀)−𝟏𝐯, (𝐈 − 𝛾𝐀)−𝟐 𝐯,… , (𝐈 − 𝛾𝐀)−𝒎+𝟏𝐯
𝐕𝒎T𝐀𝐕𝒎 ≈
𝐈 − 𝐇𝒎,𝒎−𝟏
𝜸
𝒆𝐀𝒉𝐯 ≈ 𝐯 𝟐𝐕𝒎 𝒆𝒉/𝜸(𝐈−𝐇𝒎,𝒎−𝟏)𝒆𝟏
𝒆𝒓𝒓 𝒎,𝜶 =𝐯 𝟐
𝜸ℎ𝒎+𝟏,𝒎 (𝐈 − 𝛾𝐀)𝐯𝒎+𝟏𝒆𝒎
T𝐇𝒎,𝒎−𝟏𝒆ℎ/𝛾 (𝐈−𝐇𝒎,𝒎
−𝟏)𝒆𝟏
Skip of Regularization
1. No need of regularization for A= 𝑪 −𝟏𝑮 using matrix pencil (𝑮 , 𝑪 )
2. LU decomposition at a fixed 𝛾
• Require LU every time step?
13
𝐯𝒌+𝟏 = (𝐈 − 𝛾𝐀)−𝟏𝐯𝒌 = (𝐂 − 𝛾𝐆 )−𝟏𝐂 𝐯𝒌
𝑳𝑼_𝑫𝒆𝒄𝒐𝒎𝒑 𝐂 − 𝛾𝐆 = 𝐋 𝐔
𝐂 =𝐂 𝟎𝟎 𝐈
, 𝐆 =−𝐆 𝐖
𝟎 𝐉,𝐖 =
𝐁𝐮 𝑡 + ℎ − 𝐁𝐮(𝑡)
ℎ𝐁𝐮(𝑡)
Block LU and Updating Sub-matrix
• The majority of matrix is the same,
• Block LU can be utilized here and the former LU matrices are updated as
• We avoid LU in each time step by reusing and Block LU and updating a small part of U
14
𝑳𝑼_𝑫𝒆𝒄𝒐𝒎𝒑 𝐂 + 𝛾𝐆 = 𝐋𝒔𝒖𝒃 𝐔𝒔𝒖𝒃
𝐋 =𝐋𝒔𝒖𝒃 𝟎𝟎 𝐈
, 𝐔 =𝐔𝒔𝒖𝒃 −𝛾𝐋𝒔𝒖𝒃
−𝟏𝐖
𝟎 𝐈𝐉, 𝐈𝐉 = 𝐈 − 𝛾𝐉
𝑳𝑼_𝑫𝒆𝒄𝒐𝒎𝒑 𝐂 − 𝛾𝐆 = 𝐋 𝐔
Rational MEXP with Adaptive Step Control
15
𝐯 𝟐𝐕𝒎 𝒆𝜶(𝐈−𝐇𝒎,𝒎−𝟏)𝒆𝟏
• large step size with less dimension
Rational Matrix Exponential
16
fix , sweep m and h 1
~
2eeeError
h
hmH
m
AVvv
• large step size with less dimension
Rational Matrix Exponential
17
1
~
2eeeError
h
hmH
m
AVvv fix h, sweep m and
Outline • Background of Power Grid Transient Circuit
Simulation – Formulations – Problems
• Matrix Exponential Circuit Simulation (Mexp) – Matrix Exponential Computation
• Previous Standard Krylov Subspace and Stiffness Problems • Rational Krylov Subspace (Rational Mexp)
– Adaptive Time Stepping in Rational Mexp
• Experiment – Mexp vs. Rational Mexp (RC Mesh) – Rational Mexp vs. Trapezoidal Method (PDN Cases)
• Conclusions 18
Experiment
• Linux workstation
– Intel Core i7-920 2.67GHz CPU
– 12GB memory.
• Test Cases
– Stiff RC mesh network (2500 Nodes)
• Mexp vs. Rational Mexp
– Power Distribution Network (45.7K~7.4M Nodes)
• Rational Mexp vs. Trapezoidal (TR) with fixed time step (avoid LU during the simulation)
19
Experiment (I) • RC mesh network with 2500 nodes. (Time span [0, 1ns] with a fixed step
size 10ps)
stiffness definition:
• Comparisons between average (mavg) and peak dimensions (mpeak) of Krylov subspace using
– Standard Basis:
• mavg = 115 and mpeak=264
– Rational Basis:
• mavg = 3.11, and mpeak=10
• Rational Basis-MEXP achieves 224X speedup for the whole simulation (vs. Standard Basis-MEXP).
20
𝑹𝒆(𝝀𝒎𝒊𝒏)
𝑹𝒆(𝝀𝒎𝒂𝒙)= 𝟐. 𝟏𝟐 × 𝟏𝟎𝟖
Experiment (II) • PDN Cases
– On-chip and off-chip components
– Low-, middle-, and high-frequency responses
– The time span of whole simulation [0, 1ps]
21
Experiment (II)
22
• Mixture of low, mid, and high frequency components.
• 16X speedups over TR.
• Difference of MEXP and HSPICE: 7.33×10-4; TR and HSPICE: 7.47×10-4
Experiment: CPU time
23
Conclusions
• Rational Krylov Subspace solves the stiffness problem.
– No need of regularization
– Small dimensions of basis.
– Flexible time steps.
• Adaptive time stepping is efficient to explore the different frequency responses of power grid transient simulation (considering both on-chip and off-chip components)
– 15X speedup over trapezoidal method.
24
Conclusions: Future Works
• Setting of constant 𝛾
– Theory and practice
• Distributed computation
– Parallel processing
– Limitation of memory
• Nonlinear dynamic system
25
Thanks and Q&A
26