CHAPTER 4 EFFECTIVE STABILIZING SYSTEM FOR MULTI-MACHINE...

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CHAPTER 4

EFFECTIVE STABILIZING SYSTEM FOR

MULTI-MACHINE POWER SYSTEM USING

OPTIMAL TUNING OF PSS PARAMETERS

4.1 INTRODUCTION

One of the major issues in the PSS application in the multi-machine

power system for the small signal stability enhancement is tuning of the PSS

parameters. There are so many approaches for finding solution to the problem

of optimal tuning of power system stabilizers in multi-machine power system.

In the past two decades, there has been considerable research in the area of

optimal tuning of power system stabilizers in multi-machine power system. In

multi-machine system with poorly damped modes of oscillations, several

stabilizers have to be used and the problem of tuning of PSS parameters

becomes relatively complicated. This chapter addresses the optimal tuning of

the PSS parameters in the multi-machine power system.

4.2 PROPOSED METHOD FOR THIS WORK

The objective is to find an effective stabilizing system for damping

different critical modes under different operating conditions of a multi-

machine power system using optimal tuning of PSS parameters.

The proposed method has the following steps:

1. Determining the critical modes of power system for different

operating conditions from the corresponding linearized

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models and then selecting a set of dominant critical modes and

ranking the same based on the damping ratio.

2. Identification of the best location of the PSSs for damping the

selected critical modes.

3. Optimal Tuning of the parameters of the PSS for the selected

location using the parameter constrained non-linear

optimization technique.

4. Checking the dynamic responses of the multi-machine power

system after installing the optimally tuned PSSs in the

identified locations for the adequacy of the stabilization.

These steps are explained in detail in the following sections.

4.3 IDENTIFYING AND RANKING THE DOMINANT

CRITICAL MODES OF THE POWER SYSTEM

The multi-machine test system is modeled with each generator has

two axis with four states, '

d∆E , '

q∆E , ∆ω , ∆δ . The IEEE type ST1A model

excitation system has been included for all the generators.

The linearized state equations for the two-axis, fourth order model

generator are as follows:

X AX•

= (4.1)

where A is a state matrix.

As explained in Chapter 2, the linearized state equations for

generator and exciter are given in Equation (4.2) and Equation (4.3)

respectively. The complete state vector of the multi-machine power system

with exciter is given by the Equation (4.4):

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They are as follows:

p∆Ed' i = { - ∆Ed'i - (xqi - x'i) ∆Iqi } / τqo'i ; i = 1,...n

p∆Eq'i = { ∆EFDi - Eq'i + ( xdi - x'i ) ∆Idi } / τdo'i ; i = 1,...n

p∆ωi = { ∆Tmi – (Idi0 ∆Ed'i + Iqi0 ∆Eq'i +

Ed'i0 ∆Idi +Eq'i0 ∆Iqi) - Diωi } / τj ; i = 1,...n

p ∆δi = ∆ωi ; i = 1,….n (4.2)

The state space equation for the exciter,

( )i

Aifdi Ref i fdi

Ai Ai

-K 1p∆E = -∆V +∆V - ∆E ;

T T i=1,…n (4.3)

xT

i = [∆E'di ∆E'qi ∆ωi ∆δi ∆EFDi ] ; i=1,…n (4.4)

From the A matrix, the critical modes are identified for each one of

the critical operating conditions. From all the critical modes corresponding to

different operating conditions, a set of dominant critical modes are identified

and ranked based on the ascending order of the damping ratio.

4.4 IDENTIFICATION OF BEST LOCATION OF THE PSSs IN

A MULTI-MACHINE POWER SYSTEM

For the selected set of dominant critical modes, the best location of

the PSS for each one of the dominant critical modes is identified using RGA

method, In frequency domain, the RGA matrix can be calculated for the

frequency corresponding to the critical mode. Then the rows and columns of

the RGA matrix are rearranged in such a way that the RGA matrix is closer to

identity matrix. This is done by the application of Genetic Algorithms.

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The selection of control loops from the RGA analysis is explained

in such a way that the RGA matrix is used to find out the best input – output

signals which give the good control of PSS which has the speed input signal

and voltage reference signal as output and also that because of the selection of

this input – output pair, the interaction effects by the other PSS control loops

if any over this feedback loop (PSS) should be minimized.

4.5 OPTIMAL TUNING OF THE PARAMETERS OF THE PSS

USING PARAMETER CONSTRAINED NON-LINEAR

OPTIMIZATION TECHNIQUE

In order to determine the tuned parameters of PSS (to damp a dominant

critical mode) connected to the identified machine, the multi-machine power

system is reduced to an equivalent SMIB system retaining that particular machine

alone. This step is repeated for each one of the selected dominant critical modes.

This problem is posed as a non-linear optimization problem.

4.5.1 Problem Formulation for the Optimal Tuning of the PSS

The objective of the optimization problem is to maximize the damping

ratio ζ of the dominant critical mode of that machine which is equivalent to the

minimization of non-linear function (i.e) Damping Index (DI).

DI = (1 )− ζ (4.5)

The optimization programming problem is stated as follows:

To determine KPSS and T1 which will minimize the following function:

K Gc(j ) GEP (j )n nPSSMin: (1- ) = 1 -

2 Mn

ω ω ζ

ω (4.6)

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Subject to KPSS min

≤ KPSS ≤ KPSS max

(4.7)

T1min

≤ T1 ≤ T1max

(4.8)

The PSS parameters used for the optimization are KPSS and T1. Parameter T2 is

fixed.

The expression for damping ratio ‘ζ’ in terms of machine and PSS

parameters is derived and obtained as follows:

Electric torque ∆Te and mechanical torque ∆Tm are assumed to be

on the ∆δ- ∆ω plane .To derive an extra damping ∆TE through the

supplementary excitation, ∆Te must be in phase with the ∆ω. Similarly extra

damping through the governor control must be in phase with the -∆ω. In the

general case of supplementary excitation and governor control of low

frequency oscillations, the extra electric damping ∆TE be included in the ∆Te

and the extra mechanical damping ∆TM be included in ∆Tm (Yu 1983).

The derivation of objective function is done based on the SMIB model of the

system.

Electric torque, ∆Te = K1∆δ + DE∆ω (4.9)

where, DE damping coefficient due to the extra damping

K1 is the Phillips Heffron constant

Assume that the original ∆Tm from the regulator governor control is still

negligible,

So Mechanical torque, ∆Tm = - DM ∆ω (4.10)

where, DM is the mechanical damping coefficient due to extra damping.

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The mechanical damping torque ∆TD synchronous generator model can be

expressed

∆TD = D ∆ω (4.11)

The characteristics equation of the system mechanical loop is given as

(Ms2+ (Dm+DE+D) s+ωb K1) ∆δ = 0 (4.12)

where ∆δ = (ωb ∆ω) / s,

ωb is the system frequency

Normalization yields the equation (4.12) as,

(s2 + 2 ζ ωn s + ωn

2 ) ∆δ = 0 (4.13)

Equating equation (4.12) and equation (4.13) yields

ζ = (Dm+DE+D)/2ωnM (4.14)

where, ωn - undamped mechanical mode oscillating frequency in radian

per second

ζ - damping coefficient in per unit

M = 2H where H is the inertia constant.

So undamped mechanical mode oscillating frequency can be directly

calculated as per the equation, neglecting all damping in the characteristics

equation,

ωn = (ωb K1/ M )1/2

(4.15)

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From the study of SMIB system model, the AVR loop can be

represented as shown in Figure 4.1.

Figure 4.1 AVR loop for Plant Transfer Function (GEP(s))

The transfer function of this loop is given by:

A 2 3s

A 3 do' A 3 6

K K KGEP(s) = Te(s)/ V (s) =

(1+ sT ) (1+sK T ) + K K K∆ ∆ (4.16)

where GEP(s) is the plant tranfer function

K1 to K6 are the Heffron Philips Constants (Padiyar 2002).

The PSS transfer function can be expressed from Figure 2.10 from the

Chapter 2 is as follows:

∆Vs(s)/∆ω(s) = KPSS Gc(s)Gw(s) (4.17)

where, KPSS - PSS gain

Gw(s) - washout transfer function transfer function sTw

(1 sTw)=

+

∆Vt

∆Te(s) A

A

K

1 s T+3

3 do

K

1 sK T '+K2

K6

∆Vs(s)

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Gc(s) - phase compensator transfer function = 1 sT

1

1 sT2

+

+

From equation (4.17):

{∆Vs(s)/∆Te(s)} x {∆Te(s) / ∆ω(s)} = Kpss Gc(s) Gw(s) (4.18)

Since from equation (4.9), ∆Te = K1∆δ+DE∆ω

Te(s)

DE0(s)

∆=

∆δ =∆ω (4.19)

From Equation (4.16),

∆Vs(s)/ ∆Te(s) = 1 / GEP(s) (4.20)

From Equation (4.19),

∆Te(s)/∆ω(s) = DE (4.21)

Using Equation (4.20) and Equation (4.21), the Equation (4.18) can

be written as follows:

[1/GEP(s)] * [DE] = KPSS Gc(s) Gw(s) (4.22)

From the Equation (4.14) neglecting mechanical damping,

DE = 2 ζ ωnM (4.23)

So equation (4.22) by substituting DE,

[1 / GEP(s)]*[2 ζ ωn M] = KPSSGc(s) Gw(s) (4.24)

Assuming, Gw(s) = 1 because Tw >> 1.

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From the equation (4.24), the damping ratio in terms of PSS parameters is

given by

K Gc (s) GEP (s)

PSS = 2 Mn

ζω

(4.25)

where, s = jωn

4.5.2 Parameter Constrained Non-Linear Optimization

The main objective of this method can be very clear with the help

of the Figure 4.2. Among dominant swing modes only those have damping

ratio less than cr

ξ are considered in the optimization. In Figure 4.2, '+' sign

indicates eigen values before optimization and '∗ ' sign indicates eigen values

after optimization. The optimization is done by using the Sequential

Coordinated Programming.

Figure 4.2 Objective of optimization

Cos-1 ζ

+ Inter area modes

+ Local modes *

*

σ

jω

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4.6 PLACEMENT OF OPTIMALLY TUNED PSS IN THE

IDENTIFIED MACHINES

After obtaining the optimized parameters of the PSSs for the

identified individual machines, the PSSs with the optimal parameters are

installed on the identified machines. Then the simulation is carried out in

order to check the adequacy of the damping.

4.7 SYSTEM INVESTIGATED AND SIMULATION RESULTS

In order to verify the performance of the proposed tuning method,

10-machine 39-bus New England test System is used. The system data and

the parameters of all the generating units, transmission lines and loads are

given in Appendix 2.

For the nominal operation condition of the test system, there are

eight critical modes (Table 4.1).

Table 4.1 Critical Modes of New England Test System (Nominal

operating condition)

Mode No. Eigen Values Damping Ratio

1 -1.0361 ± j9.7046 0.1062

2 -0.8359 ± j8.7394 0.0952

3 -1.0388 ± j7.8218 0.1317

4 -0.5390 ± j6.6460 0.0808

5 -0.5013 ± j6.4141 0.0779

6 -1.0606 ± j6.4005 0.1635

7 -0.6701± j5.4632 0.1217

8 -0.7063 ± j3.5853 0.1933

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From the damping factors of the eigen values, it is observed that the

damping of all the critical modes is unsatisfactory. Robustness of the test

system is checked by choosing the following different operating conditions:

a) Line outage (21-22) in the system

b) Line outage (21-22) and 25% load increase in the 16th

and 21st

bus

c) 25% generation increase in generator 7.

The critical swing modes obtained for the above three operating

conditions are given in Table 4.2, Table 4.3 and Table 4.4.

Table 4.2 Critical Modes of Test System (operating condition (a))


1 -0.9652 ± j9.7412 0.0986

2 -0.08668±j8.6973 0.0992

3 -0.9455±j7.9622 0.1179

4 -0.4930±j6.6234 0.0742

5 0.4986±j6.3299 0.0785

6 -0.8394 ±j5.6291 0.1475

7 -0.7620±j4.9294 0.1528

8 -1.1606 ± j8.4005 0.1839

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Table 4.3 Critical modes of Test System (operating condition (b))


1 -0.9568 ± j9.7649 0.0975

2 -0.8588 ± j8.7064 0.0982

3 -0.9319 ± j7.9825 0.1160

4 -0.9607 ± j7.4006 0.1815

5 -0.4976 ± j6.5604 0.0756

6 -0.5281 ± j6.3384 0.0830

7 -0.8243 ± j5.6254 0.1450

8 -0.7550 ± j4.9210 0.1516

Table 4.4 Critical Modes of Test System (operating condition (c))


1 -0.7684 ±j 11.051 0.0694

2 -0.6112 ±j10.1157 0.0603

3 -0.8154 ±j 9.0787 0.0895

4 -0.5308 ±j 8.2943 0.0639

5 -0.3265 ±j 7.5605 0.0431

6 -0.2409 ±j 7.1092 0.0339

7 -0.3061 ±j 6.8213 0.0448

8 -0.2928 ±j 4.3861 0.0666

All the above four different operating conditions (including the

nominal operating condition) are taken and a set of dominant critical modes

(say nine modes) are selected .The damping ratios corresponding to all the

critical modes for the four different critical operating conditions are tabulated

as shown in Table 4.5. Then the dominant critical modes are selected and

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ranked based on ascending order of the damping ratios as shown in the

Table 4.6.

Table 4.5 Selection of Critical Modes

Damping ratio of

the critical modes

of base case

condition

Damping ratio of

the critical modes

of op condition

(a)

Damping ratio of

the critical modes

of op condition

(b)

Damping ratio of

the critical

modes of op

condition(c)

0.1062 0.0986 0.0975 0.0694

0.0952 0.0992 0.0982 0.0603

0.1317 0.1179 0.1160 0.0895

0.0808 0.0742 0.1815 0.0639

0.0779 0.0785 0.0756 0.0431

0.1635 0.1475 0.0830 0.0339

0.1217 0.1528 0.1450 0.0448

0.1933 0.1839 0.1516 0.0666

Table 4.6 Ranking of critical dominant modes

Damping ratio of

dominant critical

modes

0.0339

0.0431

0.0448

0.0603

0.0639

0.0666

0.0694

0.0742

0.0756

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The operating conditions corresponding to all the dominant critical

modes are tabulated in Table 4.7.

Table 4.7 Operating condition corresponding to the selected dominant

critical modes

Critical modes Damping ratio Operating Condition

-0.2409 ±j7.1092 0. 0339 Case (c)

-0.3265±j7.5605 0. 0431. Case (c)

-0.3061±j6.8213 0.0448 Case (c)

-0.6112±j10.1157 0.0603 Case (c)

-0.5308±j8.2943 0.0639 Case (c)

-0.2928± j4.3861 0.0666 Case (c)

-0.7684±j11.0517 0.0694 Case (c)

-0.4930±j6.6234 0.0742 Case (a)

-0.4976 ± j6.5604 0.0756 Case (b)

In Table 4.7, for the first most critical mode (-0.2409 ±7.1092i)

with the damping ratio is 0.0339 which corresponds to the operating

condition- case (c). The optimum locations of PSSs to damp out this dominant

critical mode is found out using RGA method. The RGA matrix is calculated

for the frequency of this critical mode. Then the modified RGA matrix is

found out using Genetic Algorithm (Table 4.8).

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Table 4.8 Modified RGA Matrix for the first mode dominant critical mode

∆Vs

∆ω ∆Vs1 ∆Vs2 ∆Vs3 ∆Vs4 ∆Vs5 ∆Vs6 ∆Vs7 ∆Vs8 ∆Vs9 ∆Vs10

∆ω1 1.0168 -0.001098 0.000687 -0.000765 0.000001 0.0000154 0.0000991 0.0000122 -0.004987 0.000085

∆ω2 0.00089 1.0198 -0.004235 0.000987 0.00067 0.0000234 -0.0000772 -0.000987 0.0002897 0.0000076

∆ω3 -0.00138 2.987E-18 0.98765 1.0987E-18 0.000987 0.000009 3.88E-18 -0.0000002 1.07E-14 0.00001

∆ω4 -0.00548 0.00987 0.000685 1.0190 -0.00097 -0.00000112 5.687E-25 7.87E-18 -1.0497E-15 0.000064

∆ω5 0.00025 -0.00023 -3.82E-18 -1.087E-19 1.0177 -0.000087 -3.87E-18 5.678E-12 0.00006879 7.89E-21

∆ω6 9.04E-17 -0.00045 0.000986 -0.0000099 0.000067 0.93465 0.0000112 0.0000677 0.0002354 0.00009

∆ω71 -0.0009 9.04E-17 -0.004926 -0.0000987 -0.000055 0.0000123 0.93918 -0.000076 -0.0009987 0.00012

∆ω8 -0.00025 0.00067 0.000987 -0.000076 0.000078 0.000998 -1.987E-25 0.92350 -0.00000632 -1.176EE-25

∆ω9 0.00125 0.000099 -1.687E-19 0.000004 -0.000076 -1.468E-18 0.000987 0.000987 1.0021 6.58E-18

∆ω10 -0.00617 0.00187 0.000987 -1.4687E-18 0.000056 3.224E-18 5.687E-14 0.0000076 2.956E-18 1.0035

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From Table 4.8, the machine location order is found out as per

RGA method and shown below.

1. Machine No. 9

2. Machine No. 10

3. Machine No. 1

4. Machine No. 5

5. Machine No. 4

6. Machine No. 2

After identified the best locations of PSS using RGA method, as

per Table 4.8, the optimal parameters are found out using the detail explained

in section 4.4.

Non-linear programming problem for the first machine can be

formulated like this:

n nK GEP (j ) Gc(j )

PSSMin: (1- ) = 1 - 2 Mn

ω ωζ

ω

Subject to 10 ≤ KPSS ≤ 90

0.001 ≤ T1 ≤ 1.6

T2 is fixed as 0.06 seconds. For the above problem, optimized

controller parameters for the first machine are obtained as KPSS = 12.0432 and

T1 = 0.0010 secs.

Like above the optimal PSS parameters are found out for all the

machines to damp out its critical mode and the optimized values are shown in

Table 4.9.

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Table 4.9 Optimized controller parameters

Gen.no KPSS T1 in secs

1 12.0432 0.0010

2 12.099 0.0011

3 11.2791 0.0016

4 11.2792 0.0016

5 12.0056 0.0015

6 11.2792 0.0016

7 10.6075 0.00165

8 12.0432 0.0016

9 11.9235 0.0018

10 12.9567 0.0016

For the first most critical mode (-0.2409 ±7.1092i), the first PSS

with the optimized parameters is connected to the most dominant RGA

element with input signal of ∆ω9 and ∆Vs9 as the output signal to place PSS

on the Machine No.9. The simulation result i.e., dynamic response of the rotor

angle deviation (∆δ13) of the test system, when only one PSS with the

optimized parameters is connected to the 9th

machine is shown in Figure 4.3.

The time axis is taken as time in per unit. The actual time

t actual = time in per unit / ωbase

Figure 4.3 System response of rotor angle deviation ∆δ13 when PSS

connected to the 9th

machine

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From Figure 4.3, it is observed that the damping is not adequate.

The dynamic response is full of oscillations. So one more PSS machine with

its optimized parameters is connected to 10th

which is the machine

corresponding to the next dominant element in the RGA matrix corresponding

to the first critical mode. Simulation result is shown in Figure 4.4.



machine

From Figure 4.4, it is observed that the damping is not adequate.

The dynamic response is full of oscillations. So one more PSS is connected to

first machine which is the machine corresponding to the next dominant

element in the RGA matrix corresponding to the first critical mode.

Simulation result is shown in Figure 4.5.

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connected to the 1st machine

Simulation result of Figure 4.5 reveals that damping of rotor

oscillations is not still adequate. So more PSSs are connected to the system in

machine no.5, machine no.4 and machine no.2 sequentially as per the locating

order of machines. The PSSs are installed in the identified machines with the

optimized controller parameters as shown in Table 4.9.

The system responses when PSSs located in machine no.5, machine

no.4 and machine no.2 sequentially are shown in Figure 4.6 to Figure 4.8.

Finally well damped condition is obtained after connecting six PSS

parameters in the above locating order.

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connected to the 5th machine



machine

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connected to the 2nd machine

Hence finally PSSs are connected to the machines 9, 10, 1, 5, 4 and

2 of the system sequentially. In this case, simulation results demonstrate that

the oscillations are well damped for each mode separately in the system.

Totally six PSSs are connected sequentially in the test system in order to

damp out the first most critical mode. The above steps are repeated for all the

selected modes in the Table 4.7 to get the better damping performance in the

system. After installation, the improvement in damping ratios of all the

selected dominant critical modes is checked and is given in Table 4.10.

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Table 4.10 Comparison between the damping ratios of the dominant

critical modes before and after the placement of PSS

Critical modes Damping ratio

Before PSS

Damping ratio

After PSS

-0.2409 ±j7.1092 0.0339 0.3271

-0.3265±j7.5605 0.0431 0. 4050

-0.3061±j6.8213 0.0448 0.3140

-0.6112±j10.1157 0.0603 0.2981

-0.5308±j8.2943 0.0639 0.3362

-0.2928±j 4.3861 0.0666 0.2980

-0.7684±j11.0517 0.0694 0.3458

-0.4930±j6.6234 0.0742 0.3745

-0.4976 ± j6.5604 0.0756 0.4041

Table 4.10 reveals the enhancement of stability after placing the

PSSs for the selected critical modes. The same procedure is carried out for all

the remaining critical modes to improve the dynamic performance. Figure 4.9

to 4.17 shows the simulation results for the rotor angle deviation for the

operating condition (c) with the step change in 5° rotor angle perturbation.

Figure 4.9 System response of the test system for the rotor angle

deviation (∆δ12) for operating condition (c)

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88





89





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4.8 CONCLUSION

This chapter discussed the design of an effective stabilizing system

for a multi-machine power system. The proposed approach is applied to the

10 machines, 39 bus New England test system. For the test system, the critical

modes are identified for different operating conditions from the linearized

model and a set of dominant critical modes among them are chosen and

ranked based on the damping ratio. For each one of the selected dominant

critical modes, the best location of PSS is identified based on RGA method.

For the identified PSSs as per the RGA method, the parameters are optimally

tuned by non-linear optimization technique using Sequential Quadratic

Programming Method. Then the optimal parameters are installed on the

respective machines. Simulation is carried out by taking the 5% step change

in rotor angle as the disturbance and the results demonstrate the effectiveness

in improving the small signal stability of the system.

CHAPTER 4 EFFECTIVE STABILIZING SYSTEM FOR MULTI-MACHINE...

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