Transients on Transmission Lines Represented by State Equations ... When a lumped parameters line model is adopted, it is very common to use state space ...
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An Efficacious Computational Procedure to Solve Electromagnetic Transients on Transmission Lines Represented by State Equations E. C. M. Costa1 , S. Kurokawa2 , A. J. Prado2 , and J. Pissolato1 1
School of Electric Engineering and Computation, State University of Campinas, Brazil 2 School of Engineering of Ilha Solteira, S˜ao Paulo State University, Brazil
Abstract— This work presents an analytic solution to evaluate electromagnetic transients on transmission lines. Several papers apply a line modeling by lumped parameters and state space representation due the relative facility implementation and accuracy over considerable frequency rage. Although the efficacy and precision of transients simulations is intrinsically attached to the methodology applied for resolution of state equations which represent these phenomena. Therefore the current work describes a procedure to solve the state equations based on matrix diagonalization and Eigensystem-Based Solution for state equations. This procedure is compared to results obtained by well known numerical solution of Trapezoidal Rule and to results obtained by EMTP. The transients are evaluated considering a variable integration step ∆t and from this procedure, the proposed solution proves to be greatly robust, numerically stable and more efficient than the widely applied Trapezoidal Rule. The transmission line modeling applied in the proposed analyses takes into account the variation of longitudinal parameters in function of frequency and then it is inserted in a cascade of π circuits through the synthesis of a rational function F (ω), this method is entitled as Vector Fitting and also described by this paper.
1. INTRODUCTION
When a lumped parameters line model is adopted, it is very common to use state space techniques to evaluate the currents and voltages along the line. In this way, it is possible for the model to carry out simulations directly in time domain without the explicit use of inverse transforms and it can be easily implemented. These characteristics of the lumped parameters model are the same as those used to simulate electromagnetic transients on lines with non-linear components, such as corona effects and fault arcs [1–2], or when a detailed voltage and current profile is needed. The line distributed parameters behavior is approximated by a rational function F (ω) and, from this function, it is possible to synthesize an equivalent electric circuit by lumped resistances and inductances that describes the line longitudinal impedance Z(ω). The use of the state space techniques presents several advantages as the possibility of the methodology to be extended to time-variable and non-linear systems. Another advantage is that if state space techniques are used, the transmission line is represented by a system of first order differential equations obtained from this approach and this system can be easily evaluated by numerical integration methods [3]. The integration method known as Trapezoidal Rule for lumped parameters is widely used in computational algorithms for transients simulation and commercial softwares based on Electromagnetic Transients Program (EMTP). The simulation precision is intrinsically dependent of the procedure applied to solve the state equations and emphasizing that Trapezoidal Rule as well as other numerical methods are iterative procedures and largely dependent of integration step ∆t, thus the simulation results are also directly dependent of ∆t. However, numerical solutions have an ideal performance when ∆t is significantly small otherwise the simulation presents numerical oscillations hence several inaccuracies. Another limitation is associated to the great number of lumped elements used to represent adequately long transmission lines, thus this fact associated to a very small ∆t results in a great computational processing and sometimes it results in numerical errors and computational faults. Based on this assertion, this paper proposes an analytical procedure to solve the state equations, less dependent of ∆t and then less computational cost. The results are obtained by numerical solution based on trapezoidal integration method, by analytical solution and from MICROTRAN (EMTP) and then all results are compared and analyzed in function of integration step ∆t.
2 2. FITTING LINE LONGITUDINAL PARAMETERS
To include the frequency dependence of the longitudinal parameters in the state matrices, initially it is necessary to approximate it by a rational function which can be associated with an equivalent electric circuit representation. The per unit length (p.u.l.) longitudinal impedance Z(ω) of a transmission line, tabulated taking into account soil and skin effects, is an improper function. In this way, for fitting Z(ω) it is necessary initially to obtain a modified function F (ω) [4]. Many fitting procedures are available to obtain an approximated rational function for F (ω), starting from tabulated values from Z(ω) and in this paper the Vector Fitting algorithm will be used [4]. In a general way, the Vector Fitting algorithm is accurate, robust and can be applied to both smooth and resonant responses with high orders and wide frequency bands. Once the functions Z(ω)fit has been fitted, it is possible to associate it with the equivalent circuit shown in Figure 1. R' 1
R' 0
R' 2
R' m
L' 0 L' 1
L' 2
L' m
Figure 1: Equivalent circuit used for fitting the p.u.l. parameters of a transmission line. 3. FREQUENCY DEPENDENCE OF LONGITUDINAL PARAMETERS IN LUMPED MODEL
It is known that if the frequency effect in the longitudinal parameters is disregarded, a single phase line can be approximated by a cascade of nπ circuits [1–5]. If soil and skin effects are taken into account, each π circuit of the cascade will be the aspect of the network shown in Figure 2. In Figure 2, R0 , R1 , R2 , . . . , Rm are resistors and L0 , L1 , L2 , . . . , Lm are inductors. These elements are used to represent the frequency dependence of the longitudinal parameters of the line. The terms G and C are, respectively, the shunt conductance and shunt capacitance. The cascade shown in Figure 2 describes a non-homogeneous system with n(m + 2) state equations that represents the currents and voltages along the line. The parameters of the circuit shown in Figure 2 are calculated as: d n d Lk = L0 k n 0 d Ck = C k n d Lk = L0 k n
Rk = R 0 k
(1) (2) (3) (4)
In (1)–(4) d is the length of the line and n is the quantity of π circuits used to represent the line. The terms Rk0 and L0k are resistances and inductances used for fitting p.u.l. longitudinal parameters of the line whereas Ck0 and G0k are, respectively, p.u.l. shunt parameters of the line.
A
R0
G 2
L0
C 2
R1
R2
Rm
L1
L2
Lm
B
G 2
C 2
Figure 2: A generic π circuit that represents a frequency dependent single phase line.
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Considering the circuit described in Figure 1, presented in Figure 2, where the terms Ik0 , Ik1 , Ik2 , . . . , Ikm are currents in the inductors L0 , L1 , L2 , . . . , Lm , respectively. Considering that currents in inductances L0 to Lm and voltage between terminals A and B are written in time domain, it is possible to write the differential equations of the line: m m X X dIk0 Ik0 1 1 1 = − Rj + − Rj Ikj + Vk−1 − Vk (5) dt L0 L0 L0 L0 j=1
j=1
dIk1 R1 R1 = Ik0 − Ik1 (6) dt L1 L1 dIkm Rm Rm = Ik0 − Ikm (7) dt Lm Lm 1 1 G dVk = Ik0 − I(k+1)0 − Vk (8) dt C C C The equations presented in (5)–(8) are applied in each π circuit of the cascade shown in Figure 2, thus it describes a non-homogeneous system with n(m + 2) state equations that represents the currents and voltages along the line. These equations are written in state space as: •
[ X ] = [A][X] + [B]u(t)
(9)
The vector [Xn ] has dimension (m + 2) and u(t) is a voltage source. A generic vector [Xk ] is written as being: [X k ]T = [ Ik0 Ik1 Ik2 · · · Ikm Vk ] (10) where [Xk ]T is the transposed vector of [Xk ]. Vector [X] has dimension n(m + 2). Matrix [A], written in (9), is a square matrix with dimension n(m + 2) and it is a tridiagonal matrix that elements are also square submatrices with dimension (m + 2). A generic submatrix [Akk ] in main diagonal of matrix [A], is written as: j=m P Rj j=0 R1 R2 Rm 1 − ··· − L L L L L 0 0 0 0 0 R1 R1 − 0 ··· 0 0 L1 L1 R2 R2 0 − · · · 0 0 [Akk ] = (11) L2 L2 .. .. .. .. .. .. . . . . . . Rm Rm 0 0 · · · − 0 Lm Lm 1 G 0 0 ··· 0 − C C It is observed in (11) that a generic submatrix [Akk ] has only non-null elements in main diagonal, in first row and in first column. The elements of the first diagonal above, in matrix [A], are matrices with only one non-null element. This non-null element is the first element of the last row and it is expressed as −1/C. In the matrices situated in the first diagonal below, the non null element is the last element of the first row and it is written as 1/L0 . The vector [B] in (9) has n(m + 2) elements and it is written as: ¸T · 1 0 ... 0 (12) [B] = L0 Taking into account (11), it is observed that matrix [A] is highly sparse. Therefore, to save space in computer memory and to reduce processing time, sparsety techniques can be used to store only non-null elements.
4 4. ANALITICAL SOLUTION FOR STATE EQUATIONS
There are several analytic methods that can be used to solve differential equations as Eigensystembased solution, Vandermond Matrix and Lagrange Interpolation Formula [3]. In this paper the analytic solution of the state equations was obtained by a procedure based on Eigensystem-based solution and matrices diagonalization. The state equations describe a non-homogeneous or coupled system, because [A] is not a diagonal matrix. Although matrix [A] can be converted in a diagonal matrix by a similarity transformation. Therefore this coupled system is decoupled hence it is possible to obtain n(m + 2) decoupled equations. Considering a full square matrix [T] with dimension n(m+2) whose the columns are eigenvectors of matrix [A], thus it is possible to define a vector [Y] written in function of [X]. [X] = [T][Y] In state space form:
(13)
•
[ Y ] = [T]−1 + [A][T][Y] + [T]−1 [B]u(t) From similarity transformation described in [3]:
(14)
•
[ Y ] = [λ][Y] + [G]
(15)
[G] = [T]−1 [B]u(t)
(16)
where −1
[λ] = [T]
[A][T]
(17)
Matrix [λ] is a diagonal matrix whose non-null elements are the eigenvalues of matrix [A]. This way, the column Tk of matrix [T] is an eigenvector associated with eigenvalue λk . It is important to observe that any eigenvalue of matrix [A] is repeated and they are real eigenvalues or pairs of complex conjugate eigenvalues. Therefore an element λk is written as: λk = αk + jβk
(18)
Considering that [λ] is a diagonal matrix, the system shown in (16) is an uncoupled system. Taken into account that the initial conditions are considered null, the integration of (16) results in a vector [Y] described as: [Y]T = [y1 y2 y3 · · · yn(m+2) ] (19) where [Y]T is the transposed of [Y]. The k-th element of vector [Y] is written as: gk u(t) λk In (20) gk is the k-th element of the vector [G] and ck is the integration constant. Then, it is possible to obtain the vector [X]: yk = ck eλk t −
[X] = [T] [y1
y2
y3
···
yn(m+2) ]T
(20)
(21)
If the initial conditions of the line are null, a generic element xk of vector [X] in (21) will be written as: ϕ ϕ X X xk (t) = eαi t (aki − bki )sin β i t + eαi t (aki + bki )cos β i t (22) i=1
i=1
where ϕ = (m + 2)n µ ¶ gi aki = Re Tki λi
µ and
bki = Im Tki
gi λi
¶
(23) (24)
In (26) aki and bki are, respectively, the real and imaginary parts of the term Tki λgii . The term Tki is the element T (k, i) of matrix [T], gi is i-th element of vector [G] and λi is i-th eigenvalue of the matrix [A]. By using (22) it is possible to obtain the state variables xk (t) that are the currents and voltages along the line.
5 5. ANALISYS OF METHODOLOGIES
To analyze the efficacy of analytical methodology for state equations solution and numerical solution based on the classical trapezoidal integration, it describes a single-phase transmission line with length 100 km, illustrated by Figure 3. In Figure 3, the commuter S and the voltage source u(t) simulate the transients occasioned from line energization. For simulations of electromagnetic transients from transmission line energization, a transformer can be represented by a capacitance connected between line terminal and ground [5]. Thus, the transformer illustrated in Figure 4 is modeled by a shunt capacitance CT equal 6ηF [5]. To represent this line is applied a cascade with 110π circuits. From Figure 4, it verifies the voltage transients calculated by trapezoidal integration and by analytical procedure, considering ∆t equal 0.05 µs. It is possible to verify that the results obtained from both solution methods, when a small integration step ∆t is determined, and from Microtran (EMTP) are similar. Then, from this procedure, it possible to assert that analytical solution method is according to the simulation evaluated by EMTP. To demonstrate the influence of ∆t over the iterative solution method using trapezoidal rule, it S
u ( t)
Figure 3: Single-phase transmission line.
Figure 4: Transient voltage with ∆t = 0.05 µs: analytical solution (1); trapezoidal integration solution (2) and EMTP (3).
Figure 5: Transient voltage calculated using trapezoidal integration: ∆t = 2 µs (1) and ∆t = 0.05 µs (2).
Figure 6: Transient voltage calculated using trapezoidal integration: ∆t = 8 µs (1) and ∆t = 0.05 µs (2).
Figure 7: Transient voltage calculated using analytical solution: ∆t = 8 µs (1) and ∆t = 0.05 µs (2).
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is evaluated the transients voltages on transformer terminal, considering ∆t 0.05 and 8 µs. From Figures 5 and 6, it is possible to observe the sensibility of trapezoidal rule solution in function of ∆t. Observing that increasing ∆t, numerical oscillations increases as well, as verified in Figures 5 and 6. From Figures 5 and 6, it is possible to conclude that trapezoidal rule is greatly dependent of variation of ∆t, describing numerical instabilities in function of ∆t increment. The same procedure applied to analyze the trapezoidal rule in function of ∆t is used for analytical solution as well. Thus, Figure 7 shows the analytical solution considering ∆t equal 0.05 and 8 µs. It verifies, from Figure 7, that the analytical solution is practically invariable with variation of ∆t and without numerical oscillations. Thus, considering the time interval adopted between 0.05 and 8 µs, it is possible to assert that the presented analytical methodology is robust in function of integration step ∆t. 6. CONCLUSION
It verifies, from Figure 7, that the analytical solution is practically invariable with variation of ∆t and without numerical oscillations. Thus, considering the time interval adopted between 0.05 and 8 µs, it is possible to assert that the presented analytical methodology is robust in function of integration step ∆t. Firstly, considering the results described in Figure 4, it concludes that both solution methodologies applied for lumped parameters modeling are according with the results obtained from EMTP. Thus, it is possible to conclude that when a small integration step ∆t is adopted, both methodologies presents adequate results. Afterward, the behavior of trapezoidal rule solution was analyzed in function of a variable ∆t. Concluding that with increment of ∆t, trapezoidal integration solution presents numerical oscillations, decreasing the quality of the results, as described in Figures 5 and 6. After that, a similar procedure was done to analytical method and from this procedure the analytical solution shown to be invariable in function of ∆t, as described in Figure 7. Thus, it is possible to assert that the analytical solution is robust, at lest, in analyzed interval for ∆t. Therefore, from this work, it is possible to conclude that the analytical solution is less dependent of ∆t and, from this assertion, this methodology shows to be an efficient computational tool to evaluate electromagnetic transients on transmission lines, once that using a large integration step the computational processing is considerably reduced. ACKNOWLEDGMENT
This research was supported by Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) and National Counsel of Technological and Scientific Development (CNPq). REFERENCES
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