ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Lecture 22: SVD; Numeric Solution of Differential Equations Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 11/13/2014

Announcements HW 7 is due Thursday, November 20 Final exam on Monday Dec 15 from 1:30 to 4:30pm in this room (ECEB-4026) Closed book, closed notes; you can bring in two note sheets (one new note sheet and exam 1 note sheet), along with simple calculators

Singular Value Decomposition An extremely useful matrix analysis technique is the singular value decomposition (SVD), which takes an m by n real matrix A and represents it as where U is an m by n orthogonal matrix (UTU = I), S is an n by n diagonal matrix whose elements are the non-negative singular values of A, and V is an n by n orthogonal matrix Note, there is an other formulation with U as m by m, and S as m by n Computational order is O(n2m); ok if n is small

Matrix Singular Values The singular values of a matrix A are the square roots of the eignenvalues of ATA The singular values are real, nonnegative numbers, usually listed in decreasing order Each singular value s satisfies where u (dimension m) and v (dimension n) are both unit length and called respectively the left-singular and right-singular vectors for singular value s

SVD Applications SVD applications come from the property that A can be written as where each one of the matrices is known as a mode More of the essence of the matrix is contained in the modes associated with the larger singular values An immediate application is data compression in which A represents an image; often a quite good representation of the image is available from just a small percentage of the modes

SVD Image Compression Example Image source http://fourier.eng.hmc.edu/e161/lectures/svdcompression.html

SVD Applications Another application is removing noise. If the columns of A are signals, since noise often affects more the smaller singular values, then noise can be removed by taking the SVD, and reconstructing A without the small singular-value modes Noise strength is uniform across all modes Another application is principal component analysis (PCA) in which the idea is to take a data set with a number of variables, and reduce the data and determine the data associations The principal components correspond to the largest singular values when data is appropriately normalized

Pseudo-inverse of a Matrix The pseudo-inverse of a matrix generalizes concept of a matrix inverse to an m by n matrix, in which m >= n Specifically talking about a Moore-Penrose Matrix Inverse Notation for the pseudo-inverse of A is A+ Satisfies AA+A = A If A is a square matrix, then A+ = A-1 Quite useful for solving the least squares problem since the least squares solution of Ax = b is x = A+ b

Pseudo-inverse and SVD pseudo-inverse can be directly determined from the SVD in which S+ is formed by replacing the non-zero diagonal elements by its inverse, and leaving the zeros Numerically small values in S are assumed zero V is n by n S+ is n by n UT is n by m A+ is therefore n by m Computationally doing the SVD dominates

Simple Least Squares Example Assume we which to fix a line (mx + b = y) to three data points: (1,1), (2,4), (6,4) Two unknowns, m and b; hence x = [m b]T Setup in form of Ax = b

Simple Least Squares Example Doing the SVD Computing the pseudo-inverse

Simple Least Squares Example Computing x = [m b]T gives With the pseudo-inverse approach we immediately see the sensitivity of the elements of x to the elements of b

Switching to Dynamic Systems The analysis we've done so far in the class has been associated with power system static analysis Determining characteristics of the power system quasi-steady state equilibrium Now we're going to do a brief coverage of techniques for analysis of power system dynamics, with fuller coverage detailed in ECE 576 Appropriate models depend on time period of interest Faster dynamics can be represented as algebraic constraints Slower dynamics can be represented as constants

Power System Time Frames Lightning Propagation Switching Surges Stator Transients and Subsynchronous Resonance Transient Stability Governor and Load Frequency Control Boiler/Long-Term Dynamics 10-7 10-5 10-3 0.1 10 103 105 Time (Seconds) Voltage Stability Power Flow Image source: P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, 1997, Fig 1.2, modified

Power Grid Disturbance Example Figures show the frequency change as a result of the sudden loss of a large amount of generation in the Southern WECC Time in Seconds Frequency Contour

Frequency Response for Gen. Loss In response to rapid loss of generation, in the initial seconds the system frequency will decrease as energy stored in the rotating masses is transformed into electric energy Solar PV has no inertia, and for most new wind turbines the inertia is not seen by the system Within seconds governors respond, increasing power output of controllable generation Solar PV and wind are usually operated at maximum power so they have no reserves to contribute

Solution Considerations In ECE 530 we introduce several solution methods that are more fully considered in ECE 576 A wide variety of different solution methods are possible, with different classes of problems (such as power system transient stability) having customized solutions There is a balance between the problem to be solved and the solution method Can we bound the dynamics considered, with fast dynamics represented as algebraic constraints, and slow as constants

Differential Algebraic Equations Many problems, including many in the power area, can be formulated as a set of differential algebraic equations (DAE) of the form A power example is transient stability, in which f represents (primarily) the generator dynamics, and g (primarily) the bus power balance equations We'll initially consider the simpler problem of just

Ordinary Differential Equations (ODEs) Assume we have a problem of the form This is known as an initial value problem, since the initial value of x is given at some time t0 We need to determine x(t) for future time Initial value, x0, must be either be given or determined by solving for an equilibrium point, f(x) = 0 Higher-order systems can be put into this first order form Except for special cases, such as linear systems, an analytic solution is usually not possible – numerical methods must be used

Equilibrium Points An equilibrium point x* satisfies An equilibrium point is stable if the response to a small disturbance remains small This is known as Lyapunov stability Formally, if for every e > 0, there exists a d = d(e) > 0 such that if ||x(0) – x*|| < d, then ||x(t) – x*|| < e for t  0 An equilibrium point has asymptotic stability if there exists a d > 0 such that if ||x(0) – x*|| < d, then

Power System Application A typical power system application is to assume the power flow solution represents an equilibrium point Back solve to determine the initial state variables, x(0) At some point a contingency occurs, perturbing the state away from the equilibrium point Time domain simulation is used to determine whether the system returns to the equilibrium point

Initial value Problem Examples

Numerical Solution Methods Numerical solution methods do not generate exact solutions; they practically always introduce some error Methods assume time advances in discrete increments, called a stepsize (or time step), Dt Speed accuracy tradeoff: a smaller Dt usually gives a better solution, but it takes longer to compute Numeric roundoff error due to finite computer word size Key issue is the derivative of x, f(x) depends on x, the value we are trying to determine A solution exists as long as f(x) is continuously differentiable

Numerical Solution Methods There are a wide variety of different solution approaches, we will only touch on several One-step methods: require information about solution just at one point, x(t) Forward Euler Runge-Kutta Multi-step methods: make use of information at more than one point, x(t), x(t-Dt), x(t-D2t)… Adams-Bashforth Predictor-Corrector Methods: implicit Backward Euler

Error Propagation At each time step the total round-off error is the sum of the local round-off at time and the propagated error from steps 1, 2 , … , k − 1 An algorithm with the desirable property that local round-off error decays with increasing number of steps is said to be numerically stable Otherwise, the algorithm is numerically unstable Numerically unstable algorithms can nevertheless give quite good performance if appropriate time steps are used This is particularly true when coupled with algebraic equations

Forward Euler’s Method The simplest technique for numerically integrating such equations is known as the Euler's Method (sometimes the Forward Euler's Method) Key idea is to approximate In general, the smaller the Dt, the more accurate the solution, but it also takes more time steps

Euler’s Method Algorithm

Euler’s Method Example 1

Euler’s Method Example 1, cont’d xactual(t) x(t) Dt=0.1 x(t) Dt=0.05 10 0.1 9.048 9 9.02 0.2 8.187 8.10 8.15 0.3 7.408 7.29 7.35 … 1.0 3.678 3.49 3.58 2.0 1.353 1.22 1.29

Euler’s Method Example 2

Euler's Method Example 2, cont'd

Euler's Method Example 2, cont'd x1actual(t) x1(t) Dt=0.25 1 0.25 0.9689 0.50 0.8776 0.9375 0.75 0.7317 0.8125 1.00 0.5403 0.6289 … 10.0 -0.8391 -3.129 100.0 0.8623 -151,983 Since we know from the exact solution that x1 is bounded between -1 and 1, clearly the method is numerically unstable

Euler's Method Example 2, cont'd Below is a comparison of the solution values for x1(t) at time t = 10 seconds Dt x1(10) actual -0.8391 0.25 -3.129 0.10 -1.4088 0.01 -0.8823 0.001 -0.8423