Application of Quadratic Integration methods in Complex power systems. Rohit Atul Jinsiwale

Review of Transient Simulation Methods Trapezoidal integration method has been a dominant method for power transient simulation - Second order accuracy method - A-stable (absolute stable) method - Fictitious oscillations (sample by sample inaccuracies) Fictitious oscillations are problematic for: - Switching systems (malfunction and misfires) - Systems with high order nonlinearities (surge arresters, transformer core magnetization, etc.)

Transient Simulation Methods Control/Suppression of Fictitious Oscillations - Numerical stabilizer methods Change structure of the models Simulation Fidelity is Deteriorated - Critical damping adjustments (CDA) Requires re-computation of algebraic companion models at critical conditions Increased computational burden - Wave digital filter (post-processing) Works well for linear systems

Quadratic Integration Method (QMQI) Basic Concept: Nonlinear equations of systems are reformulated into a set of linear and quadratic equations (quadratization) The resulting equations are integrated assuming that variables vary quadratically over the time period of one time step (quadratic integration) Characteristics and Advantages: A special case of class of methods known as collocation methods Fourth order accurate method Free from artificial numerical oscillations A-stable

Upon evaluation of the integrals and rearranging the following matrix equation is obtained Assuming that the function x(t) varies quadratically in the interval, x(τ)=a+bτ+cτ 2 The parameters a, b, and c are expressed as a function of the three (collocation) points Example: Equation above is integrated form t-h to t and from t-h to t-h/2 Quadratic Integration Method: Integration

Using Lagrange interpolation formula: Butcher Table for the method:

Comparison (Trapezoidal vs. Quadratic) Accuracy Comparison Integration error of (A) quadratic and (B)trapezoidal Simulation fidelity is dependent upon the accuracy of the integration method The shaded parts show the integration error. Solution Accuracy is depended on the integration errors Truncation error: E trapezoidal =O(h 2 ) and E quadratic =O(h 4 ) Simulation fidelity is dependent upon the accuracy of the integration method The shaded parts show the integration error. Solution Accuracy is depended on the integration errors Truncation error: E trapezoidal =O(h 2 ) and E quadratic =O(h 4 ) The dominant error at each step Trapezoidal: proportional to h 2 Quadratic: proportional to h 4 The dominant error at each step Trapezoidal: proportional to h 2 Quadratic: proportional to h 4

Test system: 6 Pulse Converter The system chosen here is a three phase- 6 pulse converter. The switches are thrown in a specified order so as to convert AC-Dc or vice-versa

Comparison(Trapezoidal vs. Quadratic) Fictitious Oscillations Trace (A) is the line-line voltage (A-B) solution at input of six-pulse converter using the trapezoidal integration. Trace (B) is the filtered version of this solution Trace (A) is the line-line voltage (A-B) solution at input of six-pulse converter using the trapezoidal integration. Trace (B) is the filtered version of this solution Consider for an example a first order dynamical system The numerical solution using trapezoidal integration is: If the integration time step is so selected as to z <-2, the numerical solution will oscillate around the true value.

Comparison (Trapezoidal vs. Quadratic) Trace (C) shows the solution, when numerical stabilizers are used in the trapezoidal integration Trace (D) shows the solution by direct application of the quadratic integration with no other controls Trace (C) shows the solution, when numerical stabilizers are used in the trapezoidal integration Trace (D) shows the solution by direct application of the quadratic integration with no other controls The numerical solution using the quadratic integration is: Note that in the numerical solution the coefficient cannot become negative for any selection of the integration time step  the quadratic integration is free of fictitious oscillations Fictitious Oscillations

Cubic Integration Method

Cubic Integration Method - Example

Absolute Comparison Of TI, QMQI, QMCI Exact Solution via Laplace transform

Integration time-step (seconds) Maximum error during simulation period (from 0 to 2 seconds) TrapezoidalQuadraticCubic x x x x x x x x x Cubic Integration Method - Accuracy

Stability Comparison MethodTIQICIQRI Oscillation Free Zone -2 < z ≤2Any z-4.3< z ≤4.3Any z

Recent Literature and Research

Application of QMQI towards system reduction Consider the following subsystem model: The subsystem is integrated using the Quadratic Integration Method. Thus it can be seen to be evolving as a quadratic function of it’s states. Traditional methods towards reducing this model involve developing a static voltage/current source which typi8cally retains a static vale over a time step (Thevenin Equivalent). This model can be interfaced with the rest of the system and helps in reducing system size to be solved and reduces computational burden.

Drawbacks Although unlikely, the system could undergo massive transients during this timestep thereby making the static model meaningless. If switching components like the aforementioned 6-pulse converter are present in the system the toplogy of the system changes continuously which a static model cannopt accommodate. Local truncation error is amplified as the system is solved making the results meaningless.

QD Reduction Approach Since the model using QMQI is quadratic at most it is possible to reduce the model size using Linearization and Kron Elimination. Linearization + Kron Elimination The polynomial basis for the set of equations is quadratic. Hence, it is possible to transform the system back to time-varying equations only relying on interface states. An accurate equivalent can thus e obtained while preserving certain dynamics of the subsystem being reduced. The order of error is theoretically reduced to 2% using this approach.

QD Reduction Approach: Advantages The step size used to integrate the subsystem can be larger (integral multiple of the step size for the rest of the system. Thus, although the mesh on which both subsystems are integrated is different, simultaneous solutions can be obtained. Higher order harmonics can be captured depending on how the system is modeled.

References 1.Quadratic Integration Method, A.P Meliopoulos,G.J. Cokkinides, George.K. Stefopoulos(2002) 2.Quadratic Integration Method for Transient Simulation and Harmonic Analysis; George.K. Stefopoulos, G.J. Cokkinides, A.P Meliopoulos(2008) 3.Application of Electromagnetic Transient-Transient Stability Hybrid Simulation to FIDVR Study, Qiuhua Huang, Student Member, IEEE, and Vijay Vittal, Fellow, IEEE(2015) 4.Improved Numerical Integration Method for Power/Power Electronic Systems Based on Three- Point Collocation, A. P. Meliopoulos, Fellow, IEEE, George J. Cokkinides, Senior member, IEEE, and George K. Stefopoulos, Student member, IEEE (2005) 5.Module-Level Autonomous Setting-less Protection and monitoring of standalone and grid connected photovoltaic array systems using Quadratic Integration Modeling, Aniemi Moffat Umana (Dissertation – 2015)

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