Example: Modal analysis using DIgSILENT PowerFactory.

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Example: Modal analysis using DIgSILENT PowerFactory. This example shows the application modal analysis with Power Factory to P.M. Anderson Test System

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  • Excelente material. Un amigo me presto el libro do Prof. Francisco Gonzales (PowerFactory Aplications for Power System Analysis). Infelizemnte el perdió los archivos *pfd que acompañan el libro. Alguien podria informarme donde puedo bajarlos?
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Example: Modal analysis using DIgSILENT PowerFactory.

  1. 1. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 1/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve Francisco M. Gonzalez-Longatt, Dr.Sc Manchester, UK, November, 2009 Tutorial: Introduction to Modal Analysis
  2. 2. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 2/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve This tutorial is a simple theoretical introduction to modal analysis Tutorial: Introduction to Modal Analysis Francisco M. Gonzalez-Longatt, Dr.Sc fglogatt@fglongatt.org.ve Manchester, 28th October 2009
  3. 3. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 3/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 1. Introduction Introduction to Modal Analysis in Power System
  4. 4. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 4/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 1. Introduction: Small Signal Stability • Small signal stability is the ability of the power system to maintain synchronism when subjected to small disturbances [1]. • A disturbance is considered to be small if the equations that describe the resulting response of power system may be linearized for the purpose of analysis. • Instability that result can be of two forms [1]: – Steady increase in generator rotor angle due to lack synchronizing torque, – Rotor oscillation of increasing amplitude due to lack of sufficient damping torque. [1] P. Kundur, Power System Stability and Control. New York: McGraw- Hill, 1994.
  5. 5. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 5/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 1. Introduction: Small Signal Stability Instable • TS Negative • TDPositive δ∆ ST∆ DT∆eT∆ ω∆ δ∆ P 0 t δ∆ P 0 t Stable • TS Positive • TDPositive δ∆ ST∆ DT∆ eT∆ ω∆ (a) With constant field voltage Non-oscillatory
  6. 6. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 6/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 1. Introduction: Small Signal Stability δ∆ P 0 t Stable • TS Positive • TDPositive δ∆ ST∆ DT∆ eT∆ ω∆ δ∆ P 0 t Instable • TS Positive • TD Negative δ∆ST∆ DT∆ eT∆ ω∆ (b) With excitation control
  7. 7. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 7/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 1. Introduction: Small Signal Stability • ln today's practical power systems, the small-signal stability problem is usually one of insufficient damping of system oscillations. • Small signal inherent analysis using linear techniques provides valuable information about the dynamic characteristics of the power system and assists its design.
  8. 8. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 8/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2. Fundamental Concepts of Small Signal Stability
  9. 9. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 9/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2. State-Space Representation • The behaviour of a dynamic system, such as a power system, may be described by a set of n first order non- linear ordinary differential equations. • This can be written using vector-matrix notation [1]: where: ( )t,,uxfx =             = nx x x  2 1 x             = ru u u  2 1 u             = nf f f  2 1 f
  10. 10. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 10/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2. State Space Representation • The column vector x is deferred to as the state vector. • xi are referred as state variables. The column vector u is the vector of inputs to the system. • u are the external signal that influence the performance of the system. • t denote time • is the derivate of a state variable respect to time.             = nx x x  2 1 x             = ru u u  2 1 u             = nf f f  2 1 f x
  11. 11. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 11/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2. State Space Representation • The system is called autonomous if the derivatives of the stated variables are not explicit functions of time. • The output variables can be observed in the system. • These may be expressed in terms of the state variables and the inputs variables [1]: • where: ( )uxfx ,= ( )uxgy ,=             = my y y  2 1 y             = rg g g  2 1 g
  12. 12. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 12/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2. State Space Representation • The column vector y is the vector of outputs • g is a vector of non-linear factions relating state and input variables to output variables.             = my y y  2 1 y             = rg g g  2 1 g
  13. 13. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 13/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.1. State Concept • The concept of state is fundamental to the state-space approach. • The state of a system represents the minimum amount of information about the system at any instant in time t0 that is necessary so that its future behaviour can be determined without the input before t0. • Any set of n linearly independent system variables may be used to describe the state of the system –state variables [1]. • State variables form a minimal set of dynamic variables that, along with the inputs to the system, provide a complete description of the system behaviour
  14. 14. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 14/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.1. State-Space Concept • The system state may be represented in a n-dimensional Euclidean space, called state space.
  15. 15. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 15/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.2. Equilibrium (or singular) points • Those points where all derivatives of state variables are simultaneously zero [1]. • They define the points on the trajectory with zero velocity. • This system is at rest since all the variables are constant and unvarying with time.             = = = = 0 0 0 2 1 nx x x     x
  16. 16. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 16/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.2. Equilibrium (or singular) points • The equilibrium or singular point must therefore satisfy the equation: where x0 is the state vecto x at the equilibrium point. ( ) 00 0 === xfx
  17. 17. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 17/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.3. Stability of a Dynamic System • Linear system: stability is entirely independent of the input. • State of a stable system with zero input will always return to the origin of the state space, independent of the finite initial state. • Non-linear system: Stability depends on the type and magnitude of input and the initial state.
  18. 18. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 18/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.4. Classification of Stability • Classification of stability of non-linear system, depending on the region of state space in which the state vector ranges: – Local stability or stability in the small. – Finite stability. – Global stability or stability large.
  19. 19. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 19/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.4.a. Local Stability • The system is locally stable about equilibrium point if, when subjected to small perturbation, it remain within a small region surrounding the equilibrium point. • If, a t increase, the system return to the original state, it is said to be asymptotically stable in the small.
  20. 20. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 20/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.4.b. Finite Stability • If the state of a system remains within a finite region R, it is said to be stable within R. • If, further, the state of the system returns to the original equilibrium point form any point within R, it is asymptotically stable within the finite region R.
  21. 21. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 21/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 2.4.c. Global Stability • The system is said to be globally stable if R include the entire finite space.
  22. 22. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 22/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linealization
  23. 23. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 23/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • Let x0 be the initial state vector and u0 the input vector corresponding to the equilibrium point [2] ,[3]. • Let include a perturbation from the above state where the prefix Δ denote a small deviation. • As the perturbations are assumed to be small, the nonlinear functions f(x,u) can be expressed in terms of Taylor’s series expansion. ( ) 0, 000 == uxfx xxx ∆+= 0 uuu ∆+= 0
  24. 24. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 24/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • The original equations are [1]-[3]: • The lienarized forms are: – Δx is the state vector of dimension n – Δy Is the output vector of dimension m – Δu is the input vector of dimension r – A is state of plant matrix of size nxn – B is the control or input matrix of size nxr – C is the output matrix of size mxn – D is the (feedforward) matrix which defines the proportion of inputs which appears directly in the output, size nxr ( )t,,uxfx = ( )uxgy ,= uDxCy uBxAx ΔΔΔ ΔΔΔ += +=
  25. 25. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 25/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • Taking the Laplace transform the state equations in the frequency domain are obtained: uDxCy uBxAx ΔΔΔ ΔΔΔ += += ( ) ( ) ( ) ( ) ( ) ( )sΔsΔsΔ sΔsΔΔsΔ uDxCy uBxAxx += +=− 0
  26. 26. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 26/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • Block diagram of the state-space representation [1]-[3] B D I s 1 A ΣΣ C u∆ y∆ x∆x∆ + + + +
  27. 27. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 27/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • The initial conditions are Δx(0) assumed zero. • The solution of the state equations can be obtained [1]: • The Laplace transform of Δx and Δy are seen to have two components: (i) Dependent on the initial conditions and (ii) Dependent on the inputs. • These are the Laplace transforms of the free and zero- state components of the state and output vectors ( ) ( ) ( ) ( ) ( )[ ] ( )sΔsΔΔ s sadj sΔ uDuBx AI AI Cy ++ − − = 0 det
  28. 28. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 28/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 3. Linearization • The poles of Δx and Δy are the roots of the equation: • The values of s which satisfy this conditions are known as eigenvalues of matrix A [1]-[3]. • The equation [1],[2]: is referred as he characteristic equation of matrix A. ( ) 0det =− AIs ( ) 0det =− AIs
  29. 29. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 29/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 4. Eigenvalues and Eigenvectors
  30. 30. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 30/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 4. Eigenvalues • The eigenvalues of a matrix are given by the values of the scalar parameter which there exist non-trivial solutions to the equation [1]: • For a non-trivial solution [2]: • Expansion of the determinant give the characteristic equation. • The n solution of λ = λ1, λ2, …λn are the eigenvalues of A [2]. ( ) 0det =− AIs ( ) 0=− φλIA
  31. 31. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 31/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 4. Eigenvectors • For every eigenvalue λi, there is an eigenvector ϕi which satisfies: • ϕi is called the right eigenvector of the state matrix A associated with the eigenvalue λi. • Each right eigenvector is a column vector with the length equal to the number of the states. • The right eigenvector is called mode shape. iii φλφ =A
  32. 32. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 32/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 4. Eigenvectors • Left eigenvector associated with the eigenvalue λi is the n-row vector which satisfies: • The right eigenvector describes how each mode of oscillation is distributed among the system states. • It indicates on which system variables the mode is more observable. • The left eigenvector, together with the system’s initial state, determines the amplitude of the mode. • A left eigenvector carries mode controllability information. iii ψλψ =A
  33. 33. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 33/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 4. Eigenvectors The left eigenvector indicates on which system variables the mode is more observable. The right eigenvector is called mode shape.
  34. 34. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 34/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and stability
  35. 35. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 35/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and Stability • The stability of the system is determined by the eigenvalues λi. • Real eigenvalues: Non oscillatory mode. – Negative real eigenvalue represent a decaying mode. – Magnitude define the decay. – Positive real eigenvalues represent aperiodic instability. • Complex eigenvalues: Occurs in conjugate pair, and each pairs correspond a oscillatory mode.
  36. 36. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 36/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and Stability • For a complex pairs of eigenvalues [1]: • The frequency of oscillation (f) in Hz is given by [1]: • This represents the actual or damped frequency (f). • T he damping ratio (ζ) is given by: • The damping ratio ζ determines the rate of decay of the amplitude of the oscillation [1]-[2]. ωσλ j±= π ω 2 =f 22 ωσ σ ς + − =
  37. 37. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 37/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and Stability σ ωj X X z1 z2 (a) Stable focus Eigenvalues ωσλ j±= Trajectory Type of singularity (b) Unstable focus σ ωj X X z1 z2
  38. 38. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 38/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and Stability (f) Vortex Eigenvalues ωσλ j±= Trajectory Type of singularity (g) Saddle σ ωj X X z1 z2 σ ωj XX z1 z2
  39. 39. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 39/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 5. Eigenvalues and Stability (c) Stable focus Eigenvalues ωσλ j±= Trajectory Type of singularity (d) Unstable focus σ ωj XX z1 z2 σ ωj XX z1 z2
  40. 40. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 40/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6. Indexes
  41. 41. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 41/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6. Indexes • Numerous indices, can be calculated from eigenvectors such as [4]: – Participation factors, – Transfer function residues and – Mode sensitivities. • Those are very useful in system analysis and controller design.
  42. 42. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 42/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.1. Participation factors • The Participation matrix (P), combine the right and left eigenvectors [1]: where where ψki is the kth element in the ith row of the the left eigenvector ψi, and φki is the kth element in the ith column of the right eigenvector φi [4]. [ ]npppP 21=             =             = nini ii ii ni i i p p p ψφ ψφ ψφ  22 11 2 1 1p
  43. 43. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 43/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.1. Participation factors • The participation factor pki is a measure of the relative participation of the kth state variable in the ith mode, and vice versa. • The sensitivity of a particular eigenvalue λi to the changes in the diagonal elements of the state matrix A. kikikip ψφ= kk i ki a p ∂ ∂ = λ kikikip ψφ=
  44. 44. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 44/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.2. Controllability and observability • The system response in presence of input is given as: • Expressing in terms of the transform ed variables Z: where Φ is the modal matrix of A. • Then yield to: uDxCy uBxAx ΔΔΔ ΔΔΔ += += zx Φ=Δ uDzCy uBxAz ΔΔ ΔΔ +Φ= +=Φ
  45. 45. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 45/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.2. Controllability and observability • The state equations in the normal form (decoupled) may therefore [1], [4]: where: • They are the modal controllability (B’) and modal observability matrices (C’). uDzCy uBΛzzΦ ΔΔ Δ += += ' ' CΦC BΦB = = − ' ' 1
  46. 46. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 46/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.2. Controllability and observability • If the ith row of matrix B’ is zero, the inputs have not effect on the ith mode. • ith mode is said to be uncontrollable [1] • If the ith row of matrix C’ determines whether or not variable zi contribute to the formation of outputs. • If the ith Coolum of matrix C’ is zero, then the corresponding mode is unobservable [1]. BΦB 1 ' − = Mode controllability matrix CΦC =' Mode observability matrix
  47. 47. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 47/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.3. Residues • For small-signal stability analysis of power systems, this primarily is related on the eigenvalue of the state matrix. • For control design the open-loop transfer function between specific variables is useful [4]. • Consider transfer function between the variables y and u: • Let asume y is not direct function of u (D = 0) xy uxAx cΔΔ bΔΔΔ = += ( ) ( ) ( )su sy sG ∆ ∆ = ( ) ( ) bAIc 1− −= ssG
  48. 48. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 48/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.3. Residues • G(s) can be factorized [1], [2], [4]: • Using partial fractions: where and Ri is known as the residue of G(s) function at pole pi. ( ) ( )( ) ( ) ( )( ) ( )n l pspsps zszszs KsG −−− −−− =   21 21 ( ) n n ps R ps R ps R sG − ++ − + − =  2 2 1 1 ( ) ( ) ( )su sy sG ∆ ∆ = ( ) [ ] ΨbΛIcΦG 1− −= ss
  49. 49. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 49/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 6.3. Residues • Since Λ is a diagonal matrix [4]: • This equation gives the residues in terms of eigenvalues. ( ) ∑= − = n j j j s R sG 1 λ bΨcΦ iiiR =
  50. 50. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 50/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 7. References
  51. 51. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 51/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve 7. References [1] P. Kundur, Power System Stability and Control. New York: McGraw- Hill, 1994. [2] J. Arrillaga and C.P. Arnold, Computer Modelling of Electrical Power Systems, John Wiley & Sons, 1983. [3] P.M. Anderson and A.A. Fouad, Power System Control and Stability, The Iowa State University Press, 1977. [4] R.Sadikovi’c, Use of FACTS Devices for Power Flow Control and Damping of Oscillations in Power Systems. PhD Thesis in Swiss federal Institute of Technology, Zurich, 2006.
  52. 52. Dr. Francisco M. Gonzalez-Longatt, fglongatt@ieee.org .Copyright © 2009 52/63 Allrightsreserved.Nopartofthispublicationmaybereproducedordistributedinanyformwithoutpermissionoftheauthor. Copyright©2009.http:www.fglongatt.org.ve Please visit: http://www.fglongatt.org.ve Comments and suggestion are welcome: fglongatt@fglongatt.org.ve

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