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6.3.6 Example with Modified Euler’s method

As an illustration of the Modified Euler’s method, let us consider a three machine, 9 bus system.

The schematic diagram of this system is shown in Fig. 6.4. The bus data of this system is given

in Table A.7 while the line data are given in Table A.8. Further, the values of x∕

di, di (damping

constant) and H i (inertia constant) for all the three generators are given in Table 6.1.With the data given in Tables A.7 and Table A.8, the load flow solution of this system has been

carried out and the load flow results are given in Table 6.2. With the help of load flow results and

values of x∕

di, the magnitudes and angles (E and δ ) of the internal voltages of all the generators

have been calculated by utilising equations (6.16)-(6.18) and are also shown in Table 6.2. It is to

be noted that, throughout the transient stability simulation, the values of E i (i = 1, 2, 3) are to

be kept constant at the values given in Table 6.2. Also, at steady state, the speed of all generators

are assumed to be equal to ωs (i.e. ωio = ωs; i = 1, 2, 3). Now, following the arguments given in

step 7 of sub-section 6.3.3, under steady state, the values of δ i (i = 1, 2, 3) will remain constant

at those values given in Table 6.2. Similarly, under steady state, the values of ωi (i = 1, 2, 3) will

all be equal to ωs (= 376.9911184307752 rad./sec for a 60 Hz. system). Moreover, for transient

stability simulation, a time step of 0.001 sec. (∆t = 0.001) has been taken. Further, to start with,

the damping of the generators have been neglected.

Figure 6.4: 3 machine, 9 bus system

Now, let us assume that a three phase to ground fault takes place at bus 7 at t = 0.5 sec. Tosimulate this fault, the element Y 77 is increased 1000 times to represent very high admittance to

ground. With this modification in Y 77, the equation set (6.23) are solved to calculate the faulted

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Table 6.1: Machine data for 9 bus system

Gen. x∕

di di H ino. (p.u)1 0.0608 0.0254 23.64

2 0.1198 0.0066 6.403 0.1813 0.0026 3.01

Table 6.2: Load flow result for 9 bus system

Gen. P G QG V θ E δ no. (MW) (MVAR) (p.u) (deg.) (p.u) (deg.)

1 71.64102147 27.0459235334 1.04 0.0 1.0566418430 2.27164584042 163.0 6.6536603184 1.025 9.2800054816 1.0502010147 19.7315857693

3 85.0 -10.8597090709 1.025 4.6647513331 1.0169664112 13.1664110346

values of V i (i = 1, 2, 3). These values are shown in second column of Table 6.3. With these

calculated values of the generator terminal voltages, the values of P (o)ei (i = 1, 2, 3) have been

calculated using equations (6.24)-(6.25) and are shown in third column of Table 6.3. Now, from Fig.

6.4, bus 7 is the terminal bus of generator 2 (just after the transformer) and therefore, for any short

circuit fault at bus 7, the real power output of generator 2 is expected to fall drastically. Indeed,

from Table 6.3, after the fault, P (o)e2 is indeed very low (0.0003691143 p.u.). Also, note that the

values of δ io and ωio (i = 1, 2, 3), which are to be used for calculation at this time instant, are all

equal to the corresponding steady state values as there was no fault prior to this time instant.

Table 6.3: Calculations with Euler’s method for predictor stage at t = 0.5 sec. (damping = 0)

Gen.V (p.u)

P (o)e

no. (p.u)1 0.8515307492 - 0.0053320553i 0.67917489392 0.3391142785 + 0.1215867116i 0.0003691143

3 0.6169489129 + 0.0743553183i 0.3821503052

With these values of P (o)ei and ωio, the initial estimates of

dδ i

dtand

dωi

dt(i = 1, 2, 3) are

calculated from equations (6.27) and (6.26) respectively and are shown in columns 2 and 3 of Table

6.4 respectively. Finally, the values of δ (1)i and ω

(1)i (i = 1, 2, 3) are calculated by using equations

(6.32) - (6.33) and are shown in columns 4 and 5 of Table 6.4 respectively. From Tables 6.1 - 6.4 it

is observed that the steady state values of δ i (for i = 1, 2, 3; shown in the last column of Table 6.1)

are equal to the corresponding values of δ (1)

i(for i = 1, 2, 3; shown in the fourth column of Table

6.4). This is due to the fact that at this predictor stage,dδ i

dt(i = 1, 2, 3) are all equal to zero.

However, due to fall in P ei (i = 1, 2, 3), from equation (6.33), the values of ω(1)i (i = 1, 2, 3) are all

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greater than ωs.


Gen. dδ

dt

dω

dt

δ (1) ω(1)

no. (deg.) (rad/sec.)1 0 0.2968990107 2.2716458404 376.99141532972 0 47.9965914231 19.7315857693 377.03911502213 0 29.2982026019 13.1664110346 377.0204166333

Once the calculations pertaining to the predictor stage at t = 0.5 sec. are over, we move on

to the calculations pertaining to the corrector stage. Towards this goal, initially the values of P ei

(i = 1, 2, 3) are updated using the values of δ (1)i (i = 1, 2, 3) in equations (6.23) - (6.25). As

discussed earlier, the updated value of P ei is denoted as P (1)

ei

. Now, as the values of δ (1)

i

(i = 1, 2, 3)

are equal to the corresponding steady state values, the values of P (1)ei (i = 1, 2, 3) are also equal

to the values given in Table 6.3. Subsequently, with the values of P (1)ei and ω

(1)i (i = 1, 2, 3), the

values of dδ i

dtand

dωi

dt(i = 1, 2, 3) at the end of the present time step are calculated from equations

(6.34) - (6.35) and are shown in columns 2 and 3 of Table 6.5 respectively. Lastly, the final values

of δ i and ωi (i = 1, 2, 3) at t = 0.5 sec. are calculated by using equations (6.36) - (6.37), which are

shown in columns 4 and 5 of Table 6.5 respectively.

Table 6.5: Calculations with Euler’s method for corrector stage at t = 0.5 sec. (damping = 0)

Gen. dδ

dt

dω

dt

δ ωno. (deg.) (rad/sec.)

1 0.0002968990 0.2968990107 2.2716543459 376.99141532972 0.0479965914 47.9965914231 19.7329607703 377.03911502213 0.0292982026 29.2982026019 13.1672503663 377.0204166333

With these final values of δ i and ωi (i = 1, 2, 3) corresponding to t = 0.5 sec. at hand, we now

increment the time by ∆t (= 0.001 sec.) and repeat the calculations for predictor and correctorstages for t = 0.501 sec. The detailed calculations are shown in Tables 6.6 - 6.9. Initially, with the

values of δ i and ωi (i = 1, 2, 3) just obtained, the generator terminal voltages and the generator

output electrical powers are calculated and are shown in Table 6.6. With these newly calculated

values of P ei (i = 1, 2, 3), the values of dδ i

dt,

dωi

dt, δ

(1)i and ω

(1)i (i = 1, 2, 3) corresponding to the

predictor stage are calculated and are shown in Table 6.7. With these updated values of δ (1)i and ω

(1)i

(i = 1, 2, 3), the values of the generator terminal voltages and P ei (i = 1, 2, 3) are re-calculated

corresponding to the corrector stage and are shown in Table 6.8. Lastly, using these updated values

of P ei (i = 1, 2, 3), the values of dδ i

dt anddωi

dt corresponding to the corrector stage are calculated andare shown in Table 6.9. Finally, the values of δ i and ωi (i = 1, 2, 3) at t = 0.501 sec. are calculated

by using equations (6.36)-(6.37), which are shown in columns 4 and 5 of Table 6.9 respectively.

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Table 6.6: Caculations with Euler’s method for predictor stage at t = 0.501 sec. (damping = 0)

Gen.V (p.u)

P eno. (p.u)

1 0.8515306823 - 0.0053313633i 0.6791650241

2 0.3391113605 + 0.1215948479i 0.00036912883 0.6169476981 + 0.0743625981i 0.3821597463


Gen. dδ

dt

dω

dt

δ (1) ω(1)

no. (deg.) (rad/sec.)1 0.0002968990 0.2969777087 2.2716713570 376.99171230742 0.0479965914 47.9965909948 19.7357107724 377.08711161313 0.0292982026 29.2976113769 13.1689290296 377.0497142447

Table 6.8: Caculations with Euler’s method for corrector stage at t = 0.501 sec. (damping = 0)

Gen.V (p.u)

P eno. (p.u)

1 0.8515305484 - 0.0053299791i 0.67914528442 0.3391055238 + 0.1216111204i 0.0003691579

3 0.6169452683 + 0.0743771578i 0.3821786281

Table 6.9: Calculations with Euler’s method for corrector stage at t = 0.501 sec. (damping = 0)

Gen. dδ

dt

dω

dt

δ ωno. (deg.) (rad/sec.)

1 0.0005938767 0.2971351045 2.2716798648 376.99171238612 0.0959931824 47.9965901381 19.7370857735 377.0871116127

3 0.0585958139 29.2964289348 13.1697683443 377.0497136535

For subsequent time instants, the calculations proceed exactly in the same way as described

above. The fault is assumed to be cleared at t = 0.6 sec. To simulate this event (clearing of fault),

the value of Y 77 is restored to its pre-fault value and subsequently, the values of P ei (i = 1, 2, 3)

are calculated from equations (6.23)-(6.25). Please note that, while doing so, the latest values of δ i

(i = 1, 2, 3) obatained at t = 0.599 sec. are used in equation set (6.23). For subsequent instances

(beyond t = 0.6 sec.), the calculations proceed in the identical manner and finally, the simulationstudy is stopped at t = 5.0 sec. The variations of δ i (i = 1, 2, 3) with respect to the center of inertia

(COI) are shown in Fig. 6.5. For calculating the value of δ i with respect to the COI (denoted as

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δ COI i ) at each time step, the following expression has been used:

δ COI i = δ i − δ COI ; for i = 1, 2, 3; (6.60)

where,

δ COI =

m

i=1

H iδ i

m

j=1

H j

(6.61)

In equation (6.61), ‘m’ denotes the number of generators in the system (in our present case, m

= 3). At each time step, with the final calculated values of δ i (i = 1, 2, 3) calculated at the end

of corrector step, the value of δ COI is computed and thereafter, each value of δ COI i (i = 1, 2, 3)

is computed with the help of equation (6.60). With the values of δ COI i thus obtained for all time

steps, the plots shown in Fig. 6.5 are obtained. These plots show that the generators experiencesustained oscillations. This is due to the fact that in this study, the damping of the generators have

been neglected.

Figure 6.5: Variation of δ COI 1

(with no damping) obtained with Euler’s method

Let us now consider the damping of the generators. As discussed earlier (in the context of

equations (6.14) and (6.15)), when damping is considered, an extra term (representing damping)

is introduced in the differential equation corresponding to rate of change of speed. However, there

would be no change in the differential equation representing the rate of change of generator angle.

Therefore, the set of differential equations for ith generator is given by;

dδ i

dt= ωi −ωs (6.62)

2H i

ωs

dωi

dt

= P mi − P ei − di(ωi − ωs) (6.63)

In equation (6.63), the extra term di(ωi − ωs) represnts the damping of the generator. There

would be, of course, no change in the algebraic equations and the set of algebraic equations would

270



still be represented by equation set (6.23). With these sets of differential and algebraic equations,

the calculations proceed in identically the same way as described above and the variations of δ COI i

(i = 1, 2, 3) for the same fault considered above are shown in Fig. 6.6. Comparison of these

three plots with those shown in Fig. 6.5 shows that when damping of the generators are taken into

consideration, the oscillations in all the three generators reduce gradually with time, which, indeed

should be the case. As the generator oscillations are decreasing with time, the generators will remain

in synchronism and therefore, the system is stable.

Figure 6.6: Variation of δ COI 1

(with damping) obtained with Euler’s method

We will now discuss the application of Runga Kutta (RK) 4th order method of integration for

solving the transient stability problem in the next lecture.

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