Modeling and Simulation of RLC Networks & Modal Equivalents for - PowerPoint PPT Presentation

Model Based Systems Engineering (MBSE) Lecture Series N. Martins Talk - Part 2 Modeling and Simulation of RLC Networks & Modal Equivalents for Transmission Networks Containing Distributed Parameter Lines Sergio L. Varricchio, CEPEL and multiple co-authors* (*) Sergio Gomes Jr. (CEPEL), Nelson Martins (CEPEL), Francisco D. Freitas (University of Brasilia), Carlos M. Portela (COPPE), Leonardo Lima (Kestrel Power), Franklin C. Veliz (CEPEL) 1

Outline of Part II • Introduction; • Modeling electrical network components in the formulations Descriptor System (DS) and Y ( s ) matrix; • Distributed parameter transmission line model for Y(s) matrix • The Sequential MIMO Dominant Pole Algorithm (SMDPA) for computing the dominant poles and residue matrices associated with MIMO TFs of infinite systems; • Performance of a multi-bus equivalent (MIMO ROM) for a transmission network with distributed parameter lines (poles computed by SMDPA); • Modeling infinite systems by Linear Matrix Approximations

Introduction to Part II (1/2) • Modal Analysis – Involves the calculation of the system matrix, its poles & zeros and their sensitivities to system parameters; – Provides system structural information: mode shapes, participation factors, TF dominant poles, reduced order models; – Matrix models are used for the study of different power system phenomena: • Eletromechanical transients (Algebraic network modeling, R+jX); • Subsynchronous resonance (Lumped R-L-C dynamic network modeling); (High-frequency network modeling, • Harmonic performance; all transmission lines having • Electromagnetic Transients. distributed parameters)

Introduction to Part II (2/2) • High Frequency Modeling of Electrical Networks – 3 formulations: State Space (SS), Descriptor Systems (DS) and Y ( s ) matrix; – The distributed parameter nature of transmission lines (TL) can be modeled by transcendental functions having infinite poles – Infinite systems; – Infinite systems are neatly modeled by the Y ( s ) matrix formulation; – Finite approximations of infinite systems can be modeled in the SS and DS formulations, where TLs are represented by cascaded RLC circuits; – Various NLA methods exist to efficiently compute ROMs for large scale DS models; – A main disadvantage of the Y ( s ) matrix formulation is the inexistance of robust and efficient algorithms for the computation of the poles and residue matrices of multivariable TFs.

Descriptor Systems (1/5) • Basic equations:          T x t A x t B u t         T y t C x t D u t The components of the system are described by first-order ordinary differential • equations and algebraic equations as well; The Kirchhoff Law of Currents for each individual node of the network is then added to • these equations, to define the connection among the various existing system components; The DS model is a generalization of the SS model and leads to a simpler and more • efficient computer implementation.

Descriptor Systems (2/5)  Transfer Functions      1    TF MIMO  T H s C s T A B D     y s    b    1  TF SISO  T H s c s T A d   u s  Frequency Response      b     1  T H j c j T A d  Time Response (trapezoidal rule of integration)     2 2                        T A x t t T A x t B u t u t t       t t               T y t t C x t t D u t t

Descriptor Systems (3/5) RLC Series RLC Parallel Voltage Source v k v k v j i kj v j i f v f R v k k j v j j k i kj i L R f L f C R L k j v C L di f      L R i v v v f f f k j f dt di      kj L v R i v v C kj k j dt C dv v C  C C i Kirchhoff Current Law kj dt 1 dv     C C i v i L C kj dt R  →  0 Node k i mk di     L   0 L v v v v m C k j C dt  → Nodes connected to k

Descriptor Systems (4/5) Matlab script vs PSCAD Validation Parameters for 3-bus system C 1 L 1 v 1 Res. (  ) Cap. (  F) Ind. (mH) 8.0 R 2 80.0 23.9 L 1 C 1 barra 1 L 2 424.0 R 3 133.0 C 2 8.0 L 3 531.0 R 12 0.46 C 3 11.9 R 13 R 12 9.7 R 13 0.55 L 12 L 13 11.9 L 13 L 12 barra 2 barra 3 Input → i 2 = 1 pu Output → v 1 (pu) i 2 Nominal Frequency: 50 Hz i v 2 R 2 L 2 C 3 R 3 L 3 C 2 L 2 Nominal Voltage: 20 kV MVA base: 10 MVA

Descriptor Systems (5/5) Matlab script vs PSCAD Validation Voltage at Bus # 1 following a step in the current injected in Bus 2 0.4 MATLAB 0.3 PSCAD 0.2 Voltage (pu) Tensão (pu) 0.1 0.0 -0.1 -0.2 -0.3 0 2 4 6 8 10 12 14 16 18 20 Tempo (ms) Time (ms)

    C v 1 C dv    1  1   i C         10 3 7 C i v i     3 30 L C L i 3 3 2 L dt R     2 3  C   v  2 C 2     i    20      L i 3 L 3       (7)      C v d     3  0 13 C i i i 3     20 12 2 i dt 30     i L     12 12     L i 13 13         L i f f     v     1     (13)  v 2         v 3       1 v C     1    1 1 i       10       1 i L       2   1 1 1 R  v      2 C 2        1 1 i  20       1      i L   3 v       (7)    f 1 1 1   R v       3  C 3 i    2       1 1 i   30         i   3 1 1 R i       12 12         1 1 R i 13 13         1 1 R i       f f          1 1 1 1 v 1       (13)   1 1   1   v   2         1 1     1  v 3

Descriptor System Matrices (2/2) • Let us consider the nodal voltages as the output variables:   v C 1   i   10   i  L  2  v  C 2   i   20   i         L 3 1 v   v 1 f         v     C 3 1 v i           2 2 i         30   1  v       i  3 3 i   12   i  13    i f   v   1   v 2     v 3

Y(s) Matrix Formulation • Basic equations:        Y s x s B u s         T y C x D u s s s • Elements • Diagonal y ii : Summation of the all elementary admittances connected to node i ; • Off-diagonal y ij : negative value of summation of all elementary admittances connected between nodes i and j ; • SS and DS formulations are particular cases of Y(s): Y(s) = (sT – A) • Voltage sources are modeled by additional equations; • The derivative of Y ( s ) with respect to s , for the computation of the system poles, is automatically built by coding simple rules that are similar to those used for building Y ( s ) .

Y(s) Matrix Formulation  Transfer Function      1 MIMO TF    T H s C Y s B D     y s    b SISO TF    1  T H s c Y s d   u s  Frequency Response      b    1  T H j c Y j d

Y(s) Matrix Formulation – Basic Elements Series RLC Parallel RLC Voltage Source R v i j j k k k f f k k C R L L C R L f f k k 1  n y series    1 0 1 1 y v i      kj j f R s L y parallel sC k  s C 1 j R sL   v z i v 1   dy parallel 1 k f f f L k k k   C dy series 2 s C  2 ds s L where: 2   ds 1       R s L   z R s L f f f   s C k k k dz f  k L f k ds