EMTP simul(7)

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58 Power systems electromagnetic transients simulation

NPLO

VALFIR

EXTNCT

Th ree p h aseinput v o ltages

Feed b ack co ntr o lsystem

o rder

, measured

V dc , I dc , etc.

T o n,

T o ff

P 6 o r P 12

P 1 o r P 7Co nverter

firingco ntr o l system

Valve states

Figure 3.17 Firing control mechanism based on the phase-locked oscillator

o rder o rder

actual

T 0

B2

c (1) c (1) o r c (7)

TIME (rad)

Figure 3.18 Synchronising error in ring pulse

The time between two consecutive zero crossings, of the positive to negative (ornegative to positive) going waveforms of the same phase, is dened here as the half-period time, T / 2. The measured periods are smoothed through a rst order real-polelag function with a user-specied time constant. From these half-period times thea.c. system frequency is estimated every 60 degrees (30 degrees) for a six (12) pulsebridge.

Normally the ramp for the ring of a particular valve (c (1) , . . . , c(6)) starts fromthe zero-crossing points of the voltage waveforms across the valve. After T / 6 time(T / 12 for twelve pulse), the next ramp starts for the ring of the following valve insequence.

It is possible that during a fault or due to the presence of harmonics in the voltagewaveform, the ring does not start from the zero-crossover point, resulting in asynchronisation error, B2, as shown in Figure 3.18. This error is used to update thephase-locked oscillator which, in turn, reduces the synchronising error, approaching

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State variable analysis 59

zero at the steady state condition. The synchronisation error is recalculated every60 deg for the six-pulse bridge.

The ring angle order ( order ) is converted to a level to detect the ring instant asa function of the measured a.c. frequency by

T 0 = order (rad .)

f ac (p.u.)(3.46)

As soon as the ramp c(n) reaches the set level specied by T 0 , as shown inFigure 3.18, valve n is red and the ring pulse is maintained for 120 degrees. Uponhaving sufcient forward voltage with the ring-pulse enabled, the valve is switchedon and the ring angle recorded as the time interval from the last voltage zero crossingdetected for this valve.

At the beginning of each time-step, the valves are checked for possible extinc-tions. Upon detecting a current reversal, a valve is extinguished and its extinctionangle counter is reset. Subsequently, from the corresponding zero-crossing instant,its extinction angle is measured, e.g. at valve 1 zero crossing, 2 is measured, andso on. (Usually, the lowest gamma angle measured for the converter is fed back tothe extinction angle controller.) If the voltage zero-crossover points do not fall on thetime step boundaries, a linear interpolation is used to derive them. As illustrated inFigure 3.17, the NPLO block coordinates the valve-ring mechanism, and VALFIRreceives the ring pulses from NPLO and checks the conditions for ring the valves.If the conditions are met, VALFIR switches on the next incoming valve and measuresthe ring angle, otherwise it calculates the earliest time for next ring to adjust thestep length. Valve currents are checked for extinction in EXTNCT and interpolationof all state variables is carried out. The valves turn-on time is used to calculate thering angle and the off time is used for the extinction angle.

By way of example, Figure 3.19 shows the response to a step change of d.c.current in the test system used earlier in this section.

3.6 Example

To illustrate the use of state variable analysis the simple RLC circuit of Figure 3.20is used ( R = 20 .0 , L = 6.95 mH and C = 1.0 F), where the switch is closed at0.1 ms. Choosing x1 = vC and x2 = iL then the state variable equation is:

x1x2

=0

1C

1L

RL

x1x2

+01L

E S (3.47)

The FORTRAN code for this example is given in Appendix G.1. Figure 3.21 displaysthe response from straight application of the state variable analysis using a 0.05 mstime step. The rst plot compares the response with the analytic answer. The resonantfrequency for thiscircuit is1909.1 Hz(or a periodof0.5238 ms), hence having approx-imately 10 points per cycle. The second plot shows that the step length remained at

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Time (s)

d.c. currentRectified d.c. v o ltage (pu)

Firing angle (rad)Extincti o n angle (rad)

0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650

Figure 3.19 Constant order (15 ) operation with a step change in the d.c. current

i L

E S v C

L

R

C

Figure 3.20 RLC test circuit

0.05 ms throughout the simulation and the third graph shows that 2024 iterationswere required to reach convergence. This is the worse case as increasing the nominalstep length to 0.06 or 0.075 msreduces the error as the algorithm is forced tostep-halve(see Table 3.1). Figure 3.22 shows the resultant voltages and current in the circuit.

Adding a check on the state variable derivative substantially improves the agree-ment between the analytic and calculated responses so that there is no noticeabledifference. Figure 3.23 also shows that the algorithm required the step length to be0.025 in order to reach convergence of state variables and their derivatives.

Adding step length optimisation to the basic algorithm also improves the accu-racy, as shown in Figure 3.24. Before the switch is closed the algorithm convergeswithin one iteration and hence the optimisation routine increases the step length. Asa result the rst step after the switch closes requires more than 20 iterations and theoptimisation routine starts reducing the step length until it reaches 0.0263 ms whereit stays for the remainder of the simulation.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2S tate varia b le analysisAnalytic

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

Time (ms)

C a p a c

i t o r v o

l t a g e

S t e p

l e n g

t h

I t e r a t

i o n c o u n

t

Figure 3.21 State variable analysis with 50 s step length

Table 3.1 State variable analysis error

Condition Maximumerror (Volts)

Time (ms)

Base case 0.0911 0.750xcheck 0.0229 0.750Optimised t 0.0499 0.470

Both Opt. t and xcheck 0.0114 0.110t = 0.01 0.0037 0.740t = 0.025 0.0229 0.750t = 0.06 0.0589 0.073t = 0.075 0.0512 0.740t = 0.1 0.0911 0.750

Combining both derivative of state variable checking and step length optimisationgives even better accuracy. Figure 3.25 shows that initially step-halving occurs whenthe switching occurs and then the optimisation routine takes over until the best steplength is found.

A comparison of the error is displayed in Figure 3.26. Due to the uneven distrib-ution of state variable time points, resampling was used to generate this comparison,

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1

0

1

2v Cv Lv R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (ms)

i Li C

0.01

0.005

0

0.005

0.01

V o l

t a g e

C u r r e n

t

Figure 3.22 State variable analysis with 50 s step length

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

Time (ms)

C a p a c

i t o r v o

l t a g e

S t e p

l e n g

t h

I t e r a t

i o n c o u n

t

Figure 3.23 State variable analysis with 50 s step length and x check

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

S tate varia b le analysisAnalytic

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (ms)

0

1

2

0

0.02

0.04

0.06

0

10

20

C a p a c

i t o r v o

l t a g e

S t e p

l e n g

t h

I t e r a t

i o n c o u n t

Figure 3.24 State variable with 50 s step length and step length optimisation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

Time (ms)

C a p a c

i t o r v o

l t a g e

S t e p l e n g

t h

I t e r a t

i o n c o u n

t

Figure 3.25 Both x check and step length optimisation

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E r r o r

i n c a p a c

i t o r v o

l t a g e

( v o

l t s )

Time (ms)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Base caseS V derivative c h eck Optimised step lengt hBo th

0.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 3.26 Error comparison

that is, the analytic solutions at 0.01 ms intervals were calculated and the state variableanalysis results were interpolated on to this time grid, and the difference taken.

3.7 Summary

In the state variable solution it is the set of rst order differential equations, ratherthan the system of individual elements, that is solved by numerical integration. Themost popular numerical technique in current use is implicit trapezoidal integration,due to its simplicity, accuracy and stability. Solution accuracy is enhanced by the use

of iterative methods to calculate the state variables.State variable is an ideal method for the solution of system components withtime-varying non-linearities, and particularly for power electronic devices involv-ing frequent switching. This has been demonstrated with reference to the statica.c.d.c. converter by an algorithm referred to as TCS (Transient Converter Simu-lation). Frequent switching, in the state variable approach, imposes no overhead onthesolution. Moreover, theuseof automatic step length adjustment permits optimisingthe integration step throughout the solution.

The main limitation is the need to recognise dependability between systemvariables. This process substantially reduces the effectiveness of the state variablealgorithms, and makes them unsuited to very large systems. However, in a hybridcombination with the numerical integration substitution method, the state variablemodel can provide very accurate and efcient solutions. This subject is discussed ingreater detail in Chapter 9.

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Chapter 4

Numerical integrator substitution

4.1 Introduction

A continuous function can be simulated by substituting a numerical integrationformula into the differential equation and rearranging the function into an appropriate

form. Among the factors to be taken into account in the selection of the numericalintegrator are the error due to truncated terms, its properties as a differentiator, errorpropagation and frequency response.

Numerical integration substitution (NIS)constitutes the basis of Dommels EMTP[1][3], which, as explained in the introductory chapter, is now the most generallyaccepted method for the solution of electromagnetic transients. The EMTP methodis an integrated approach to the problems of:

forming the network differential equations collecting the equations into a coherent system to be solved numerical solution of the equations.

The trapezoidal integrator (described in Appendix C) is used for the numericalintegrator substitution, due to its simplicity, stability and reasonable accuracy in mostcircumstances. However, being based on a truncated Taylors series, the trapezoidalrule can cause numerical oscillations under certain conditions due to the neglectedterms [4]. This problem will be discussed further in Chapters 5 and 9.

The other basic characteristic of Dommels method is the discretisation of thesystem components, given a predetermined time step, which are then combined in asolution for the nodal voltages. Branch elements are represented by the relationshipwhich they maintain between branch current and nodal voltage.

This chapter describes the basic formulation and solution of the numericalintegrator substitution method as implemented in the electromagnetic transientprograms.

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4.2 Discretisation of R , L , C elements

4.2.1 Resistance

The simplest circuit element is a resistor connected between nodes k and m , as shownin Figure 4.1, and is represented by the equation:

ikm (t ) =1R

(v k (t ) vm (t)) (4.1)

Resistors are accurately represented in the EMTP formulation provided R is not toosmall. If the value of R is too small its inverse in the system matrix will be large,resulting in poor conditioning of the solution at every step. This gives inaccurateresults due to the nite precision of numerical calculations. On the other hand, verylarge values of R do not degrade the overall solution. In EMTDC version 3 if R isbelow a threshold (the default threshold value is 0.0005) then R is automatically setto zero and a modied solution method used.

4.2.2 Inductance

The differential equation for the inductor shown in Figure 4.2 is:

vL = vk vm = L

di kmdt (4.2)

v k

ikm

R

v m

Figure 4.1 Resistor

v k

ikm

v m

Figure 4.2 Inductor

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v k

ikm

v m

Figure 4.4 Capacitor

Rearranging gives the following transfer between current and voltage in the z-domain:

I km (z)(V k (z) V m (z))

= t 2L

(1 + z 1)(1 z 1)

(4.7)

4.2.3 Capacitance

With reference to Figure 4.4 the differential equation for the capacitor is:

ikm (t ) = Cd(v k (t ) vm (t))

dt (4.8)

Integrating and rearranging gives:

vkm(t) = (v k(t) vm(t) ) = (v k(t t ) vm(t t ) ) +1C

t

t t ikm dt (4.9)

and applying the trapezoidal rule:

vkm (t ) = (v k (t ) vm (t)) = (v k (t t ) vm (t t)) +t

2C(i km (t ) + ikm (t t))

(4.10)Hence the current in the capacitor is given by:

ikm (t ) =2Ct

(v k (t ) vm (t)) ikm (t t ) 2Ct

(v k (t t ) vm (t t))

=1

R eff [vk (t ) vm (t ) ] + I History (t t ) (4.11)

which is again a Norton equivalent as depicted in Figure 4.5. The instantaneous termin equation 4.11 is:

R eff =t

2C(4.12)

Thus very large values of C , although they are unlikely to be used, can cause illconditioning of the conductance matrix.

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