EMTP simul(10)

7/27/2019 EMTP simul(10)

1/14

100 Power systems electromagnetic transients simulation

Equation 5.1 can be expressed as:

ik = etR/Lik1 +

1 etR/L

vk (5.2)

Although the exponential form of the difference equation can be deduced from the

difference equation developed by the numerical integrator substitution method, this

approach is unsuitable for most transfer functions or electrical circuits, due to the

difficulty in identifying the form of the exponential that has been truncated. The

root-matching technique provides a rigorous method.

Numerical integrator substitution provides a mapping from continuous to discrete

time, or equivalently from the s to the z-domain. The integration rule used will influ-

ence the mapping and hence the error. Table 5.1 shows the characteristics of forward

rectangular, backward rectangular (implicit or backward Euler) and trapezoidal inte-

grators, including the mapping of poles in the left-hand halfs-plane into the z-plane.If the continuous system is stable (has all its poles in the left-hand half s-plane) then

under forward Euler the poles in the z-plane can lie outside the unit circle and hence

an unstable discrete system can result. Both backward Euler and the trapezoidal rule

give stable discrete systems, however stability gives no indication of the accuracy of

the representation.

The use of the trapezoidal integrator is equivalent to the bilinear transform (or

Tustin method) for transforming from a continuous to a discrete system, the former

being the time representation of the latter. To illustrate this point the bilinear transform

will be next derived from the trapezoidal rule.In the s-plane the expression for integration is:

Y(s)

X(s)= 1

s(5.3)

In discrete time the trapezoidal rule is expressed as:

yn = yn1 +t

2(xn + xn1) (5.4)

Transforming equation 5.4 to the z-plane gives:

Y(z) = z1Y(z) + t2

(X(z) + X(z)z1) (5.5)

Rearranging gives for integration in the z-domain:

Y(z)

X(z)= t

2

(1 + z1)(1 z1) (5.6)

Equating the two integration expressions (i.e. equations 5.3 and 5.6) gives the wellknown bilinear transform equation:

s 2t

(1 z1)(1 + z1) (5.7)

7/27/2019 EMTP simul(10)

2/14

The root-matching method 101

Table5.1

Integratorcharacteristics

Name

Forw

ardrectangular

B

ackwardrectangular

Trapezoidal

(forw

ardEuler)

(implicit/backwardEuler)

Waveform

u

t

t

u

u

t

Integrator

yk=

yk1+

tfk1

yk=

yk1+

tfk

yk=

yk1+

t

2(fk+

fk1)

Differentiator

yk=

yk+1

yk

t

yk=

yk

yk1

t

yk=

2 t

(yk

yk1)

yk1

Approximationtos

s

z

1t

s

z

1

tz

s

2 t

(z

1)

(z+

1)

stoz-plane

j

Forward

rec

tangular

1

j

B

ackward

re

ctangular

1

1

j

Trapezoidal

rule

1

1

7/27/2019 EMTP simul(10)

3/14


Hence the trapezoidal rule and the bilinear transform give the same mapping between

the s and z-planes and are therefore identical.

Equation 5.7 can also be derived from an approximation of an exponential. The

actual relationship between s and z is:

z = est (5.8)

Hence

z1 = est (5.9)Expressing est as two exponential functions and then using the series approxima-tion gives:

z1

=est

=

est/2

est/2

(1 st/2)

(1 + st/2)(5.10)

Rearranging for s gives:

s 2t

(1 z1)

(1 + z1) (5.11)

which is identical to equation 5.7. Hence the trapezoidal rule (and many other inte-

gration rules for that matter) can be considered as a truncated series approximation

of the exact relationship between s and z.

5.3 z-domain representation of difference equations

Digital simulation requires the use of the z-domain, either in the form of a transfer

function or as an equivalent difference equation.

In the transfer function approach:

H(z) = a0 + a1 z1 + a2 z2 + + am zm

1 + b1 z1 + b2 z2 + + bm zm= Y(z)

U(z)(5.12)

or expressed as a two-sided recursion [2]a0 + a1 z1 + a2 z2 + + am zm

U(z)

=

1 + b1 z1 + b2 z2 + + bm zm

Y(z) (5.13)

Equation 5.13 can be implemented directly and without any approximation as a Norton

equivalent.

Rearranging equation 5.13 gives:

Y(z) =

a0 + a1 z1 + a2 z2 + + am zm

U(z)

b1 z1 + b2 z2 + + bm zm

Y(z) (5.14)

7/27/2019 EMTP simul(10)

4/14


The corresponding difference equation is:

y(kt) = (a0 u + a1 u1 + a2 u2 + + am um)

(b

1 y

1 +b

2 y

2 + +b

m y

m)

(5.15)The first term on the right side of equation 5.15 is the instantaneous term between

input and output, while the other terms are history terms. Hence the conductance is

a0 and the history term is:

a1u1 + a2u2 + + amum b1y1 + b2y2 + + bmym (5.16)Whereas in the s-domain stability is ensured if poles are in the left-hand half-plane,

the equivalent criterion in the z-plane is that the poles must reside inside the unit

circle.

In the transformation from the s to z-plane, as required by digital simulation, the

poles and zeros must be transformed correctly and this is the purpose of the root-

matching technique. In other words, to ensure that a difference equation is suitable

to simulate a continuous process the poles, zeros and final value of the difference

equation should match those of the actual system. If these conditions are met the

difference equations are intrinsically stable, provided the actual system is stable,

regardless of the step size. The difference equations generated by this method involve

exponential functions, as the transform equation z1 = est is used rather thansome approximation to it.

When integrator substitution is used to derive a difference equation, the polesand zeros usually are not inspected, and these can therefore be poorly positioned or

there can even be extra poles and zeros. Because the poles and zeros of the difference

equation do not match well those of the continuous system, there are situations when

the difference equation is a poor representation of the continuous system.

The steps followed in the application of the root-matching technique are:

1. Determine the transfer function in the s-plane, H(s) and the position of its poles

and zeros.

2. Write the transfer function H(s) in the z-plane using the mapping z=

est, thus

ensuring the poles and zeros are in the correct place. Also add a constant to allow

adjustment of the final value.

3. Use the final value theorem to compute the final value ofH(s) for a unit step input.

4. Determine the final value ofH(z) for unit step input and adjust the constant to be

the correct value.

5. Add extra zeros depending on the assumed input variation between solution points.

6. Write the resulting z-domain equation in the form of a difference equation.

The final value ofH(s) must not be zero to allow the final value matching constant

in H(z) to be determined. When that happens the final value is matched for a different

input. For example some systems respond to the derivative of the input and in such

cases the final value for a unit ramp input is used.

Appendix E (sections E.1 and E.2) illustrate the use of the above procedure with

a single order lag function and a first order differential pole, respectively. Table 5.2

7/27/2019 EMTP simul(10)

5/14


Table 5.2 Exponential form of difference equation

Transfer function Expression for Norton

H(s) = G1 + s

R = 1/kIHistory = et/ Itt

k = G (1 et/)

H(s) = G (1 + s )

R = 1/kIHistory = k et/ Vtt

k = G(1 et/)

H(s) = G s1 + s

R = 1/kIHistory = et/ Itt k Vtt

k = G (1 et/)

t

H(s) = G (1 + s1)(1 + s2)

R = 1/kIHistory = et/2 Itt k Vtt et/1

k

=

G (1 et/1 )

(1 et/2

)

H(s) = G 2n

s2 + 2 ns + 2n

R = 1/kIHistory = A Itt B It2t

k = G (1 etp1) (1 etp2)= G (1 A + B)

H(s) = G s2

ns2 + 2 ns + 2n

R = 1/kIHistory = k Vtt + A Itt B It2t

k = G (1 et

p1

) (1 et

p2

)t

= G (1 A + B)t

H(s) = G (s2 + 2 n + 2n)

sn

R = kIHistory = Itt

A

k Vtt +

B

k It2t

k = G (1 etp1) (1 etp2)

t

= G (1 A + B)t

7/27/2019 EMTP simul(10)

6/14


gives expressions of the exponential form of difference equation for various s-domain

transfer functions.

In Table 5.2, A and B are as follows:

If two real roots ( > 1):

A = 2e nt

et n

21 + et n

21

B = e2 nt

If two repeated roots ( = 1):

A = 2ent

B=

e2nt

If complex roots ( < 1):

A = 2e nt cos

nt

1 2

B = e2 nt

By using the input form shown in Figure 5.13(a) on page 113, the homogeneous

solution of the difference equation matches the homogeneous solution of the dif-

ferential equation exactly. It also generates a solution of the differential equationsresponse that is exact for the step function and a good approximation for an arbitrary

forcing function.

5.4 Implementation in EMTP algorithm

The exponential form of the difference equation can be viewed as a Norton equivalent

in just the same way as the difference equation developed by Dommels method, the

only difference being the formula used for the derivation of the terms. Figure 5.1illustrates this by showing the Norton equivalents of a series RL branch devel-

oped using Dommels method and the exponential form respectively. Until recently

it has not been appreciated that the exponential form of the difference equation

can be applied to the main electrical components as well as control equations, in

time domain simulation. Both can be formed into Norton equivalents, entered in

the conductance matrix and solved simultaneously with no time step delay in the

implementation.

To remove all the numerical oscillations when the time step is large compared to

the time constant, the difference equations developed by root-matching techniquesmust be implemented for all series and parallel RL, RC , LC and RLC combinations.

The network solution of Dommels method is:

[G]v(t) = i(t) + IHistory (5.17)

7/27/2019 EMTP simul(10)

7/14


2LR+

t

IHistory =

IHistory IHistory

ikm (t) ikm (t)

vkm (t)

i(tt) + v (tt) IHistory= e tR / L

i (tt)

mm

k k

Dommels method Exponential form

R

(1e tR / L)

(1tR/(2L))

(1+tR/(2L)) (1+tR/(2L))

(t/(2L))

Figure 5.1 Norton equivalent forRL branch

Structurally the root-matching algorithm is the same as Dommels, the only differ-

ence being in the formula used for the derivation of the conductance and past history

terms. Moreover, although the root-matching technique can also be applied to single

L or C elements, there is no need for that, as in such cases the response is no longer

of an exponential form. Hence Dommels algorithm is still used for converting indi-vidual L and C elements to a Norton equivalent. This allows difference equations,

hence Norton equivalents, based on root-matching methods to be used in existing

electromagnetic transient programs easily, yet giving unparalleled improvement in

accuracy, particularly for large time steps.

In the new algorithm, IHistory includes the history terms of both Dommels and the

root-matching method. Similarly the conductance matrix, which contains the conduc-

tance terms of the Norton equivalents, includes some terms from Dommels technique

and others of the exponential form developed from the root-matching technique.

The main characteristics of the exponential form that permit an efficientimplementation are:

The exponential term is calculated and stored prior to entering the time step loop. During the time step loop only two multiplications and one addition are required

to calculate the IHistory term. It is thus more efficient than NIS using the

trapezoidal rule.

Fewer previous time step variables are required. Only the previous time step currentis needed for an RL circuit, while Dommels method requires both current and

voltage at the previous time-step.

Three simple test cases are used to illustrate the algorithms capability [3]. The

first case shown in Figure 5.2 relates to the switching of a series RL branch. Using

a t = time step ( being the time constant of the circuit), Figure 5.3 showsthe current response derived from Dommels method, the exponential method and

7/27/2019 EMTP simul(10)

8/14


Vdc =100V

t=0.01s

R =1.0

L=0.05mH

Figure 5.2 Switching test system

Current(amps)

0.0008 0.0011 0.0014 0.0017 0.002

Time (s)

Exponential form Dommels method Theoretical curve

40.0

60.0

80.0

100.0

0.0

20.0

Figure 5.3 Step response of switching test system fort =

continuous analysis (theoretical result). At this time step, Dommels method does not

show numerical oscillations, but introduces considerable error. The results shown in

Figure 5.4 correspond to a time step of t = 5 ( = 50s). Dommels methodnow exhibits numerical oscillations due to truncation errors, whereas the exponential

form gives the correct answer at each solution point. Increasing the time step to

7/27/2019 EMTP simul(10)

9/14


0.0

20.0

40.0

60.0

80.0

100.0

120.0

Current(amps)

Time (s)


0.0005 0.001375 0.00225 0.003125 0.004

Figure 5.4 Step response of switching test system fort = 5

Current(amps)

Time (s)


0.0

20.0

40.0

60.0

80.0

100.0

120.0

0 0.001 0.002 0.003 0.004

Figure 5.5 Step response of switching test system fort = 10

t = 10 results in much greater numerical oscillation for Dommels method, whilethe exponential form continues to give the exact answer (Figure 5.5).

The second test circuit, shown in Figure 5.6, consists of a RLC circuit with a

resonant frequency of 10 kHz, excited by a 5 kHz current source. Figures 5.7 and 5.8

7/27/2019 EMTP simul(10)

10/14


Sine-wave excitation

5 kHz or 10 kHzLC

f=2

1

L= 0.2533mH

R = 1

C= 1F

=10kHz

Figure 5.6 Resonance test system

Voltage(volts)

0.0001 0.0004 0.0007 0.001

Time (s)

Exponential form Dommels method

0

10

20

30

40

50

60

Figure 5.7 Comparison between exponential form and Dommels method to a 5 kHzexcitation for resonance test system. t = 25 s

show the voltage response using 25s and 10s time steps, respectively. Consid-erable deviation from the expected sinusoidal waveform is evident for Dommels

method. Figure 5.9 shows the comparison when the excitation contains a 10 kHz

component of 1 A peak for a time-step of 10s. At that frequency the inductance and

capacitance cancel out and the exponential form gives the correct response, i.e. a 2 V

peak-to-peak 10 kHz sinusoid on top of the d.c. component (shown in Figure 5.10),

whereas Dommels method oscillates. The inductor current leads the capacitor voltage

by 90 degrees. Therefore, when initialising the current to zero the capacitor voltage

should be at its maximum negative value. If the capacitor voltage is also initialised to

zero a d.c. component of voltage (|V

| =I/C) is effectively added, which is equiv-

alent to an additional charge on the capacitor to change its voltage from maximum

negative to zero.

A third test circuit is used to demonstrate the numerical problem of current chop-

ping in inductive circuits. A common example is the modelling of power electronic

7/27/2019 EMTP simul(10)

11/14


Voltage(volts)

0.0001 0.0004 0.0007 0.001

Time (s)


0

14

28

42

56

70

Figure 5.8 Comparison between exponential form and Dommels method to a 5 kHzexcitation for resonance test system. t = 10 s

V

oltage(volts)

Time (s)


5

5

15

25

35

0.0001 0.0004 0.0007 0.001

Figure 5.9 Comparison between exponential form and Dommels method to 10 kHz

excitation for resonance test system

7/27/2019 EMTP simul(10)

12/14


Voltage(volts)

0.0001 0.0004 0.0007 0.00114.4

14.8

15.2

15.6

16

16.4

Time (s)

Exponential form

Figure 5.10 Response of resonance test system to 10 kHz excitation, blow-up ofexponential forms response

100V

(RMS)VLOAD

= 50s

L=0.05mH

R = 1

Figure 5.11 Diode test system

devices such as diodes and thyristors. Although the changes of state are constrained to

occur at multiples of the step length, the current falls to zero between these points [4];

thus the change occurs at the time point immediately after and hence effectively turn-

ing the device off with a slight negative current. To demonstrate this effect Figure 5.11

uses a simple system where an a.c. voltage source supplies power to an RL load via a

diode. Figure 5.12(a) shows the load voltage for the exponential form and Dommels

method using a time-step of 500s. This clearly shows the superiority of the expo-

nential form of difference equation. The numerical oscillation at switch-off depends

on how close to a time point the current drops to zero, and hence the size of negative

current at the switching point. The negative current at switching is clearly evident in

the load current waveform shown in Figure 5.12(b).

7/27/2019 EMTP simul(10)

13/14


Voltage(volts)

Time (s)

Current(amps)

Time (s)

40

10

20

50

80

110

140(a)

(b)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Exponential Form Dommels method


40

0

40

80

120

160

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 5.12 Response to diode test system (a) Voltage (b) Current

These three test circuits clearly demonstrate the accuracy and stability of the

exponential form of the difference equation regardless of the time step.

5.5 Family of exponential forms of the difference equation

In the root-matching technique used to derive the exponential form of a difference

equation the poles and zeros of the s-domain function are matched in the z-domain

7/27/2019 EMTP simul(10)

14/14


u (t) u (t) u (t) u (t)

t t t t

(a) (b) (c) (d)

Figure 5.13 Input as function of time

u 1y

R

L

G

R

R

1+ sL

1

Gs

G

1 +s

1 +s

H(s) =I(s)

V(s)H(s) =

==

Figure 5.14 Control or electrical system as first order lag

function. Extra zeros are added based on the assumed discretisation on the input,

which is continuous [5]. Figure 5.13 shows some of the possible discretisations

and these result in a family of exponential forms of the difference equation. The

root-matching technique is equally applicable to equations representing control or

electrical systems [6]. For each of the discretisation types, with reference to the first

order lag function shown in Figure 5.14, the use of the root-matching technique

expressed as a rational function in z1 produces the following exponential formdifference equations.

Input type (a):

y(z)

u(z)= b/a(1 e

at)(1 z1eat)

Input type (b):

y(z)

u(z) =b/a(1

eat)z1

(1 z1eat)Input type (c):

y(z)

u(z)= b/(2a)(1 e

at)(1 + z1)(1 z1eat)

EMTP simul(10)

Documents

Transcript of EMTP simul(10)