DC power system using Prony signal analysis

Electric Power Systems Research, 26 (1993) 31 39

31

Evaluation of nonconventional HVDC converter controls in an AC/DC power system using Prony signal analysis P. A. E m m a n u e l Energy Systems Computer Application, Belleview, WA (USA)

M. H. Nehrir* and D. A. Pierre Department of Electrical Engineering, Montana State University, Bozeman, MT 59717-0378 (USA)

R. A d a p a Electric Power Research Institute, Palo Alto, CA (USA) (Received June 25, 1992)

Abstract This paper presents an e v a l u a t i o n of various n o n c o n v e n t i o n a l high voltage DC (HVDC) c o n v e r t e r controls in an AC/DC power system. A method based on Prony signal analysis is used to obtain reduced-order

models of the system with different nonconventional HVDC converter control schemes. The linear models provide information on local and interarea modes of the system and the degree of damping that each control scheme provides for the system. A detailed swing program is used to simulate the system. The evaluation procedure is explained and results are presented. The system studied is a 42-bus, 17-machine test system which exhibits some steady-state and dynamic characteristics of the western North American power system.

1. I n t r o d u c t i o n

The c o n v e n t i o n a l method of operating HVDC systems is with the rectifier controlling the current and the inverter operating in the c o n s t a n t extinction angle (CEA) control mode. An advantage of this c o n v e n t i o n a l method, which keeps the inverter at minimum extinction angle during normal operation, is minimum reactive power c o n s u m p t i o n of the DC line. However, the inverter in this mode exhibits a negative impedance c h a r a c t e r i s t i c t h a t can cause voltage and p o w e r instability [1]. Also, during AC system faults at the inverter end, there is a risk of c o m m u t a t i o n failure, and if the fault is electrically close to the inverter, the inverter m a y not be able to recover. A n o t h e r problem with the c o n v e n t i o n a l mode of o p e r a t i o n arises when the rectifier voltage ceiling exceeds that of the inverter. In this case the rectifier and inverter c h a r a c t e r i s t i c s intersect in the region of c u r r e n t margin which can lead to three-point instability.

*To whom correspondence should be addressed.

0378-7796/93/$6.00

One w a y to improve the voltage and p o w e r stability of the c o n v e n t i o n a l control scheme is to provide c o m p e n s a t i o n equipment, for example, a static VAR compensator, at the inverter terminal. Using this approach, the principle of minimizing the reactive power c o n s u m p t i o n by operating the inverter at minimum extinction angle becomes partially ineffective owing to the need for the reactive support equipment. The other economical solution is to use the high speed control capability of the HVDC converter. It has been shown t h a t by changing the inverter mode of o p e r a t i o n it is possible to improve system stability. Various n o n c o n v e n t i o n a l modes of operation have been suggested in the literature for this purpose [1-10]; most of the studies, however, have focused on voltage stability i m p r o v e m e n t in power systems with a w e a k AC side. Also, those studies focusing on system stability have not studied controller effects on system oscillatory modes. In this paper, a c o m p a r a t i v e e v a l u a t i o n of different c o n v e r t e r control performances is made in a 42-bus, 17-machine AC/DC system which has some of the steady-state and dynamic characteristics of the w e s t e r n N o r t h A m e r i c a n p o w e r system. © 1993 - - Elsevier Sequoia. All rights reserved

32

The comparison is made using time-domain simulation and Prony signal analysis [11-14]. P r o n y analysis of the power system provides reduced-order linear transfer function models for the system with different HVDC n o n c o n v e n t i o n a l c o n v e r t e r control schemes. These models are then used in root-locus studies to investigate the effect of each n o n c o n v e n t i o n a l control scheme on the stability of the system. It is found that the nonconventional control schemes do not have significant effect on the damping of the oscillatory modes of the system studied.

2. N o n c o n v e n t i o n a l configurations

HVDC

system

control

In this section the n o n c o n v e n t i o n a l HVDC control schemes that have been used in this study are listed. For detailed information regarding the application of the n o n c o n v e n t i o n a l control schemes the reader is referred to the literature (e.g. refs. 1 10). A review of these control schemes is given in ref. 15. The v o l t a g e - c u r r e n t (VI) or o p e r a t i o n a l characteristics of the c o n v e n t i o n a l and nonconventional HVDC control schemes are given in Fig. 1. U n d e r c o n v e n t i o n a l control, the rectifier operates under c o n s t a n t c u r r e n t control and the inverter operates under minimum extinction angle

INPUTS , G N A L SET POINT

NONCOnVENTIONaL CONTROI.LER FIRING PULSE

(MEASURED) EXT ANGLE {SET)

EXT. ANGLE REGULATOR

i ~

THYa STCA FIRING

j

Fig. 2. N o n c o n v e n t i o n a l c o n t r o l l e r scheme.

control or the rectifier operates under minimum ignition angle control with the inverter operating under c o n s t a n t c u r r e n t control (curves 1--4 in Fig. 1). The n o n c o n v e n t i o n a l control schemes are applied at the inverter to maintain a certain control action. The control schemes studied, whose characteristics are shown in Fig. 1, are c o n s t a n t DC voltage control (curve 5), c o n s t a n t reactive c u r r e n t control (curve 6), c o n s t a n t active power control (curve 7), and c o n s t a n t reactive power control (curve 8). There are two other important control schemes applied at the inverter terminal, namely, c o n s t a n t AC voltage control and c o n s t a n t power factor control. A functional block diagram for implementation of the n o n c o n v e n t i o n a l control schemes at the inverter terminal of the DC line is given in Fig. 2. The inverter firing pulses are determined based on the set and m e a s u r e d values of the inverter extinction angle signal or the measured and the set value of a signal corresponding to a p a r t i c u l a r n o n c o n v e n t i o n a l control scheme, for example, the inverter-end AC voltage signal. A brief description of the test power system on which the DC line has been modeled follows.

8 ... ..........

..,.o,--"" .,...,"

3. T h e t e s t s y s t e m

.°.,"

Idl DE: Currlmt

Fig. 1. N o n c o n v e n t i o n a l c o n t r o l l e r c h a r a c t e r i s t i c s : c u r v e 1, m i n i m u m i g n i t i o n a n g l e c o n t r o l at rectifier; 2, c o n s t a n t c u r r e n t c o n t r o l at rectifier; 3, c o n s t a n t c u r r e n t c o n t r o l at inverter; 4, m i n i m u m e x t i n c t i o n a n g l e c o n t r o l at inverter; 5, c o n s t a n t DC v o l t a g e c o n t r o l at inverter; 6, c o n s t a n t r e a c t i v e c u r r e n t control at inverter; 7, c o n s t a n t active power c o n t r o l at inverter; 8, c o n s t a n t r e a c t i v e power c o n t r o l at inverter.

The n o n c o n v e n t i o n a l HVDC control schemes were e v a l u a t e d using the 42-bus, 17-machine test power system shown in Fig. 3. A t h o r o u g h description of this system is given in ref. 17. The intent is to have a relatively small system which can effectively represent some of the important dynamic characteristics of the western N o r t h American power system. This system is b r o k e n into nine main areas designated A, C, M, N W l , NW2, P, SC1, SC2, and U. Each area consists of a main bus having the total load in t h a t area lumped directly to it, and a number of smaller buses each c o n n e c t e d to a g e n e r a t o r t h r o u g h a transformer. The transmission lines connecting the areas have voltage ratings of 500 kV except for the lines connecting area C to M, M to V and V to A, which are r a t e d at 230 kV. The general p a t t e r n of power flow in this system is from north

33 TABLE 2. G e n e r a t i o n and inertia p a r a m e t e r s for each generator 12

MV

345 MW 122

2767

NW1

•

Generation

Energy a

(MW)

(MW s)

12500 4500 2000 1600 10300 1250 6700 1200 2500 14000 700 10600 2469 12500 1000 700 3200

57000 18000 7680 10000 13000 6OOO 8000 5478 7856 17000 2206 13000 10000 63000 4576 2539 14000

MV

2788 MW

3£88

Generator

7 MW

MV

NW2

MW

|/

PACIFIC INTER/ MDUNTAIN HVDC HVDC ~/" 2700/3100 I 1600 / - ~" MW MW

U

III~

P

_

MW

9855 MW

SC1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Fig. 3. The 42-bus, 17-machine test system. a Total kinetic energy of g e n e r a t o r s in m e g a w a t t seconds.

to s o u t h (top to bottom) in r o u g h l y the same proportions as might o c c u r in the western power system. Total load and generation of each area is also r o u g h l y comparable. The e l e c t r o m e c h a n i c a l dynamics of this model are established by including two types of g e n e r a t o r in each area. G e n e r a t o r s shown bold are large inertia machines; they have large power generation capability and are represented by a c o n s t a n t voltage behind their react a n c e and classical swing equations. The other type comprises smaller machines with a c t u a l ratings which are modeled using the d q-axis generator model and the I E E E type AC4 exciter model [ 18]. Both types of g e n e r a t o r have governor models included. G e n e r a t i o n and load for each area are given in Table 1, and generation and kinetic energy data associated with each machine are listed in Table 2. All system c o m p o n e n t values and a detailed description of the system appear in ref. 17. TABLE 1. G e n e r a t i o n and load data for each area Area

Generation

Load

(MW) A C M NWl NW2 P SC1 SC2 U

19000 11900 1250 9169 17700 11300 14200 0 3200

15600 11750 900 11100 9000 12600 9800 11700 3800

The system in Fig. 3 also contains two +500 kV HVDC transmission lines ( b r o k e n lines in the Figure). One, b e t w e e n areas NW2 and SC1, represents the Pacific HVDC intertie and is modeled as a two-terminal system carrying 2700 M W for e v a l u a t i o n of n o n c o n v e n t i o n a l control schemes. The second, b e t w e e n areas U and SC1, represents the I n t e r m o u n t a i n tie and is modeled by equivalent real and reactive c o n s t a n t c u r r e n t loads at the respective terminals. The a c t u a l Pacific HVDC intertie is, however, a multiterminal _+500 kV line with a p o w e r c a p a c i t y of 3100 M W [19], and the I n t e r m o u n t a i n intertie is a two-terminal _+500kV DC line r a t e d at 1600 M W [201. The 42-bus system is strong on the AC side of the inverter end of the Pacific HVDC line. The strength or w e a k n e s s of an AC/DC system is defined by its effective short-circuit ratio (ESCR). The ESCR for an AC/DC system, as defined in ref. 1, is the ratio of the MVA shortcircuit c a p a c i t y of the AC system at the converter bus to the DC t r a n s m i t t e d power. An AC/DC system is said to be strong in AC for an ESCR of 2 or higher. In the case of the 42-bus system studied here, the ESCR is a b o u t 4 at the inverter end (bottom end of the DC line), thus making the AC system strong relative to the DC side. The n o n c o n v e n t i o n a l control schemes will be e v a l u a t e d for the inverter end of the DC line in the system of Fig. 3.

34

4. H V D C m o d e l i n g

i ......................

The block diagram for the DC c o n v e r t e r control model used in the simulation of the HVDC system b e t w e e n areas NW2 and SC1 in Fig. 3 is shown in Fig. 4. During normal operation the converter at the rectifier end (area NW2) and t h a t at the inverter (area SC1) are c o o r d i n a t e d to give c o n s t a n t c u r r e n t or power control at the rectifier and c o n s t a n t extinction angle control at the inverter. When the inverter t a k e s control of the current, the c u r r e n t order (desired current) is set to a value 10% to 15% less t h a n the c u r r e n t order at the rectifier. This difference is referred to as Imargm in Fig. 4. In addition to the above normal controls, voltage d e p e n d e n t c u r r e n t order limit (VDCOL) and margin switching unit (MSU) blocks are shown in the c o n v e r t e r model. These features help maintain c o n s t a n t power on the DC line under a b n o r m a l conditions and enhance its overall performance. The VDCOL is introduced in the c u r r e n t order d e t e r m i n a t i o n logic and its function is to reduce the c u r r e n t order linearly from its normal value to some specified minimum when the DC bus voltage drops below some specified percentage of its r a t e d value. It is used to maximize the power transfer capability of the DC line under adverse conditions and to reduce the DC tie power and consequently the VAR demand from the AC system during periods of depressed AC system voltage [21]. The M S U may be used to p r e v e n t any sustained decrease in the scheduled power of the DC line when the inverter t a k e s control of the c u r r e n t at a value which is less t h a n the c u r r e n t order at the rectifier by an a m o u n t equal to the c u r r e n t margin. More e x p l a n a t i o n on modeling of the M S U is given in ref. 21. However, b e c a u s e this s t u d y is focused on P r o n y analysis

cooo

"

!

PI

alRcOdfier

CongoUer

Unf~mcd

ACIDC ConvmlJc~nal SystemSignals z Non-

Power

Inverter

FirinA g~gl¢

~

Fig. 4. DC c o n v e r t e r c o n t r o l model. * i n d i c a t e s b l o c k s for inv e r t e r only.

Filtered la

Signal

Low.Pss Fill~

wfiere A=Z,ff=2n(3)=20.0

Fig. 5. I m p l e m e n t a t i o n of a g e n e r a l n o n c o n v e n t i o n a l s c h e m e for P r o n y b a s e d a n a l y s i s .

control

of the system, the M S U was not implemented in the simulation. I m p l e m e n t a t i o n of the Pacific HVDC intertie in the 42-bus power system of Fig. 3 is given in ref. 22. In this study the rectifier is provided with a proportional-plus-integral (PI) c o n s t a n t c u r r e n t controller along with minimum firing angle control and VDCOL. The inverter is provided with a n o n c o n v e n t i o n a l controller along with c o n s t a n t extinction angle control. The nominal gain values used are Kp = 0.15 for the proportional controller and K , - - 5 . 8 for the integral controller. The functional block diagram for implementation of the n o n c o n v e n t i o n a l control schemes at the inverter terminal is s h o w n in Fig. 5.

5. Prony based analysis of the 42-bus system P r o n y analysis is an off-line algorithm for analyzing signals to determine magnitude, phase, modal, and damping information c o n t a i n e d in the signals. The algorithm gives an optimal fit (in the least squared error sense) to a signal y(t) in the form

y(t)-- ~ Bi exp(~it)

VDCOL

AC:~nverte~rk ~ C~*-

......

(1)

and identifies the signal residues (the Bi) and eigenvalues (the 22). The s t a n d a r d P r o n y algorithm does not use knowledge of the system input and therefore is not a system identification routine to directly identify system transfer functions. However, system transfer function residues and eigenvalues can be o b t a i n e d for the optimal fit of the o u t p u t signal y(t) to the a c t u a l response of the system. A s t a n d a r d P r o n y signal analysis method which results in residue and eigenvalue decomposition of an o u t p u t signal is described in ref. 11. P r o n y based analysis has r e c e n t l y been used

35

to analyze p o w e r system oscillatory modes [12, 13] and in p o w e r system stabilizer design for m u l t i m a c h i n e p o w e r systems [14]. A t h o r o u g h derivation of system transfer function residues and eigenvalues from P r o n y analysis is given in ref. 16. The P r o n y algorithm is used to obtain the residues and eigenvalues of a reduced-order transfer function for the 42-bus, 17-machine AC/ DC power system under study for each of the n o n c o n v e n t i o n a l c o n v e r t e r control schemes described in §2. The root-locus technique is used to analyze the transfer functions to provide information a b o u t the a m o u n t of damping that each controller can c o n t r i b u t e to the oscillatory modes present in the system. References 14 and 16 describe the P r o n y method t h a t results in a transfer function model for the system in the form

Y(s) ~ R i a(s) - I(s) - i:0 s _ ~

(2)

It is shown in ref. 14 t h a t the residues and eigenvalues of (2) can be o b t a i n e d from P r o n y analysis of the o u t p u t for a class of inputs I(s). This input can be w r i t t e n as a finite s u m m a t i o n of delay signals as follows:

I(s)

1

[Co + c, exp(

- s D 1 ) -~ c 2

exp( - sD2)

S -- ~n+l

+ ' • " + ct~ exp( -sDj~)]

(3)

where Di < D/+I, the c~ are a r b i t r a r y real constants, and n is the order of the model. The above method was used to obtain transfer functions for the AC/DC test system, each time with an o u t p u t signal corresponding to a n o n c o n v e n t i o n a l control mode at the inverter. The o u t p u t signals used are AC voltage, DC voltage, reactive current, p o w e r factor, active and reactive power, with the input signal having the general form given by (3). The input signal given as a disturbance to the inverter firing angle was a low amplitude r e c t a n g u l a r pulse of width 1 s. The p a r a m e t e r s used in (3) to yield the r e c t a n g u l a r

pulse are k = 1, co = 3, cl = - 3, D1 = 1, and A n + l : 0. The steps in obtaining the system oscillatory modes from the P r o n y analysis are as follows. Step 1. Apply the r e c t a n g u l a r pulse as a disturbance to the inverter firing angle in the 42-bus test system and obtain the signal corresponding to the n o n c o n v e n t i o n a l control scheme of interest in each case as output. A block diagram showing the AC/DC system with the input pulse at the inverter firing angle and the o u t p u t signal observed after going t h r o u g h a low pass filter is given in Fig. 5. Step 2. Perform P r o n y analysis on the o u t p u t signals obtained in step I using the r e c t a n g u l a r pulse input defined by (3). A reduced-order linear model for the system is then obtained based on the optimal fit of the response of the linear systern with the actual one described in step 1. Results from this P r o n y based analysis and the time-domain simulation of the system studied are given in the next section. 6. R e s u l t s

The residues and eigenvalues of the linear P r o n y based models of the 42-bus system yield information a b o u t the local and i n t e r a r e a oscillatory modes of the system. These modes, along with the system residues and eigenvalues, are given in Tables 3 8 for signals corresponding to different n o n c o n v e n t i o n a l HVDC control schemes used as o u t p u t of the model. Note t h a t the order of the linear model that provided an optimal fit to the response of the actual (simulated) nonlinear system is 9 or 10 for the cases studied. In order to validate the low-order transfer function models, their responses were compared with that of the actual system. For each nonconventional HVDC control configuration the r e c t a n g u l a r pulse was applied to the model and the a c t u a l system, and the o u t p u t s were observed. In every case the fit b e t w e e n the a c t u a l system response and the response of the P r o n y model was very good after the pulse d i s t u r b a n c e

T A B L E 3. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n m o d e l f o r A C v o l t a g e as o u t p u t s i g n a l Mode

Freq. (Hz)

Eigenvalue

1 2 3 4 5

1.71 0.91 0.71 2.08 4.70

-0.1710E + 0.2988E - 0.5575E -0.6141E -0.8541E

+ + +

Residue 02 01 01 00 00

_+ j 0 . 1 0 7 4 E _+ j 0 . 5 7 4 2 E _+ j 0 . 4 4 8 4 E _+ j 0 . 1 3 0 7 E ± j0.2948E

+ + + + +

02 01 01 02 02

+0.1773E + 0.6172E + 0.2541E +0.2766E - 0.3078E

+ -

01 01 01 01 01

± ± ± ± ±

j0.1974E j0.1110E j0.4492E j0.1306E j0.2269E

+ + -

02 00 01 01 01

36 T A B L E 4. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n mode l for DC v o l t a g e as o u t p u t s i g n a l Mode

Freq. (Hz)

Eigenvalue

1 2 3 4

5.00 1.68 0.69 0.91

-0.3184E -0.1319E -0.1632E + 0.3192E - 0.2501E

Residue

4. 02 + 02 + 00 - 01 + 01

+ + + + +

j0.3142E j0.1058E j0.4388E j0.5710E j0.0000E

+ 02 + 02 4. 01 + 01 + 00

+0.1237E 0.8809E + 0.3053E +0.2563E - 0.9447E

+ + +

03 01 02 01 00

+ + + + -

j0.1221E j0.1218E j0.1798E j0.1287E j0.1080E

+ + + + +

03 02 00 00 00

+ + + + +

04 01 01 00 01

+ j0.6975E + j0.3996E + j0.3442E _+ j0.1480E + j0.1637E

+ + + + -

04 01 01 01 01

+ -

01 ± j0.2453E 03 _+ j0.3617E 03 ± j0.4867E 03 ± j0.7346E 04 _+ j0.1597E

+ -

01 03 03 04 03

+ + + + + +

00 00 00 03 01 00

_+ j0.1167E _+j0.2714E _+ j0.2822E + j0.5213E + j0.5088E + j0.3494E

+ + + -

01 01 01 14 15 15

+ + + + +

03 00 00 00 01

_+ j0.5514E _+ j0.9385E _+j0.8197E + j0.7546E + j0.4083E

4. 04 + 00 4- 00 - 01 - 14

T A B L E 5. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n model for r e a c t i v e c u r r e n t as o u t p u t s i g n a l M ode

Freq. (Hz)

Eigenvalue

1 2 3 4 5

2.42 0.36 0.92 2.12 4.57

-0.2278E - 0.3175E - 0.8378E -0.8531E - 0.6664E

Residue

+ 02 + 00 - 01 + 00 4. 00

+ j0.1523E _+ j0.2271E + j0.5750E + j0.1333E + j0.2871E

+ + + + +

02 01 01 02 02

+0.1352E 4. 0.2278E 0.3003E - 0.9537E -0.1195E

T A B L E 6. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n mode l for p o w e r f a c t o r as o u t p u t s i g n a l Mode

Freq. (Hz)

Eigenvalue

1 2 3 4 5

1.66 0.37 0.91 3.39 1.59

-0.3257E - 0.3314E -0.2354E - 0.3231E -0.4991E

+ + + +

Residue 02 00 01 01 00

+ + + + +

j0.1044E j0.2337E j0.5695E j0.2133E j0.9962E

+ + + + -

02 01 01 02 01

-0.1010E -0.3377E 4. 0.1564E + 0.3744E + 0.9770E

T A B L E 7. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n m o d e l for a c t i v e p o w e r as o u t p u t s i g n a l M ode

Freq. (Hz)

Eigenvalue

1 2 3

0.69 0.36 0.91

-0.1142E -0.3522E - 0.3597E -0.3046E 0.2210E - 0.4918E

+ + + + +

Residue 00 00 02 02 01 00

+ j0.4383E ± j0.2240E ___j0.5703E ± j0.0000E + j0.0000E + j0.0000E

+ + + + + +

01 01 01 00 00 00

-0.1751E +0.8635E -0.5387E -0.9009E - 0.3926E -0.5556E

T A B L E 8. I d e n t i f i e d r e d u c e d - o r d e r t r a n s f e r f u n c t i o n mode l for r e a c t i v e p o w e r as o u t p u t s i g n a l M ode

Freq. (Hz)

Eigenvalue

1 2 3 4

1.45 0.36 0.71 O.93

-0.3261E -0.3462E -0.1913E -0.3105E - 0.1635E

+ 02 + 00 + 00 4. 00 + 01

Residue + j0.9092E ± j0.2236E +_ j0.4446E ± j0.5864E + j0.0000E

was removed. B e c a u s e of space limitation, only comparison of responses for AC voltage control and DC voltage control of the inverter are given here (Figs. 6 and 7). For comparison of model and actual responses for other inverter control schemes, see ref. 15. It is clear from Figs. 6 and 7

Jr 01 + 01 + 01 + 01 + 00

-0.8153E - 0.4055E -0.1910E +0.5671E + 0.2928E

that the model response ( b r o k e n line) fits well with the response of the a c t u a l system (full curve) except during the one-second period that the r e c t a n g u l a r pulse was applied. A method has also been proposed to include a t h r o u g h p u t term in the P r o n y signal analysis algorithm to make

37 3.5 3"

30

2.5" :

1

.ft

I 20

10 1,5" 0 -10

0.5O"

-20

-0.5

1

2

3

4

5 TLme ( 5 ¢ . )

6

7

8

9

10

-30

-0,9

Fig. 6. T r a n s f e r f u n c t i o n model o u t p u t data v e r s u s actual data for AC voltage signal: - - , actual output; . . . . . . , estimated output.

.O.S

.0,7

-0.6

portion

.0.5

of

-0.4

interest

-0.3

-0,2

.0,|

0

Ol

expanded

Fig. 8. Root-locus plot ~ r AC voltage signal at the inverter.

25-

203O 20 |0

-

-10

O-20 -5 0

1

2

3

4

T~e

5 (S~.)

6

7

8

9

10

Fig. 7. T r a n s f e r f u n c t i o n model o u t p u t data versus actual data for DC voltage signal: , actual output; . . . . . . . , estimated output.

-30

-2.5

-2

portion

-1.5

of

-I

interest

-0.5

0

0,5

expanded

Fig. 9. Root-locus plot for DC voltage signal at the inverter.

the P r o n y based model response fit the a c t u a l system response b e t t e r during the d i s t u r b a n c e [15]. The P r o n y based transfer functions were used to obtain a root-locus plot for each control scheme used. Information o b t a i n e d from these plots is helpful in determining w h e t h e r a particular n o n c o n v e n t i o n a l control scheme is a good c a n d i d a t e for damping certain oscillatory modes of the system. Root-locus plots are given for AC voltage control and DC voltage control in Figs. 8 and 9. These plots are o b t a i n e d for transfer function gain variations from 0 to 10. Similar root-locus plots for o t h e r n o n c o n v e n t i o n a l control schemes are given in ref. 15. In all cases the poles in the frequency range of interest covering the local and i n t e r a r e a oscillatory modes (below 3 Hz) were very close to zeros, forming pole-zero pairs, and indicating lack of controllability. Hence, little damping of such modes is possible t h r o u g h feedback. This can also be seen from the fact that the residues associated with b o t h system i n t e r a r e a modes and local modes for the n o n c o n v e n t i o n a l DC control schemes studied are relatively small (see Tables 3-8). The magnitudes of these residues give a good m e a s u r e of

the a m o u n t of damping t h a t can be achieved for each mode (eigenvalue) t h r o u g h feedback. From the system eigenvalues for the AC voltage control and DC voltage control schemes shown it is noticed t h a t one pair of eigenvalues corresponding to 0.91 (or 0.93 Hz) is in the right half plane (see Tables 3 and 4). This indicates t h a t these two control schemes are u n s t a b l e for that mode. These u n s t a b l e modes can also be observed from the pulse response of the P r o n y based model and t h a t of the a c t u a l system (Figs. 6 and 7). Time-domain simulation of the 42-bus system was also carried o u t with different control schemes applied at the inverter, and the system response was obtained for different faults. Gain values used for both the rectifier and inverter are K p = 0 . 1 5 for the proportional controller and K~--5.8 for the integral controller. The system response to a 1400 M W resistive b r a k e pulse applied at bus NW1 in the test system for 0.5 s is reported here. Bus NW1 represents the Chief J o s e p h area in the Pacific N o r t h w e s t where the Bonneville P o w e r Administration applies its 1400 M W resistive brake. The time variations of

38

5i 10-

+51 +°i,

......

-5:

j *15

V' ~;i V ",' i l 'h' v v ~ , '

',j ',

1'0

-

1'5

2"0

+

2'5

30

Fig. 10. Relative g e n e r a t o r a n g l e s for f a u l t 1 ( m a c h i n e s 4, 8, 11 a n d 15) with DC v o l t a g e control: - - , g e n e r a t o r a n g l e for m a c h i n e no. 11; . . . . . . . , g e n e r a t o r a n g l e for m a c h i n e no. 15.

15" 10" 5" o-5" -10"

V -15"

: tj ! :

[i

v

t ,' '

i L"

¢

5

10

'~' ',,"

J 15

20

25

3O

Fig. 11. R e l a t i v e g e n e r a t o r a n g l e s for f a u l t 1 ( m a c h i n e s 4, 8, 11 a n d 15) w i t h AC v o l t a g e control: - - , g e n e r a t o r a n g l e for m a c h i n e no. 11; . . . . . . . , g e n e r a t o r a n g l e for m a c h i n e no. 15.

sample generator angles (for machines 11 and 15 in Fig. 3) were Obtained relative to generator 1. This generator is considered to be a large machine at bus A. The swing curves are given in Figs. 10 and 11 for DC and AC voltage control schemes, respectively, at the inverter. It is clear from these Figures that the system responses for both control schemes are very similar. Similar responses were also obtained for other control schemes and are reported in ref. 15. The similarity of responses can also be observed from the system eigenvalues predicted from Prony based analysis of the system (Tables 3-8). The eigenvalues for different control schemes show similar low frequency modes, and the dampings corresponding to these modes are also similar. The system response was also obtained for other types of fault with different inverter control schemes [15]. Similar conclusions can be drawn for the cases for which the system remained stable.

7. C o n c l u s i o n s This paper presents a method based on the Prony algorithm for analysis of an AC/DC power

system. It provides information regarding the low frequency oscillatory modes of the system and the degree of damping of these modes with different nonconventional HVDC control scherues used at the inverter. The Prony algorithm is used to obtain a reduced-order transfer function model of the system, and the classical root-locus technique as a tool to analyze the model in order to obtain information on the amount of damping available to each of the oscillatory modes present in the system. Information from the Prony analysis regarding the low frequency oscillatory modes of the system has been validated by carrying out time-domain simulation. From the time-domain simulation results it is noticed that the damping effect of the HVDC control schemes is not significant in the interconnected system studied. This system has a high effective short-circuit ratio at the inverter. In systems with high ESCR (greater than 2) the AC system is strong and cannot be effectively controlled by the DC line converter controllers using local signals only. For the system studied the ESCR is greater than 4 at the inverter bus.

Acknowledgements The authors gratefully acknowledge the assistance of Dr. James Smith of Montana State University, Dr. Dan Trudnowski of Battelle Pacific NW Labs., and Dr. John Hauer of the Bonneville Power Administration. This work was supported by the Electric Power Research Institute (contract RP2665-1), by the Montana Electric Power Research Affiliates Program, and by Montana State University.

References 1 M. S z e c h t m a n , W. W. Ping, E. S a l g a d o a n d J. P. Bowles, U n c o n v e n t i o n a l H V D C c o n t r o l t e c h n i q u e for s t a b i l i z a t i o n of w e a k p o w e r s y s t e m , I E E E Trans., PAS-103 (1984) 2244-2248. 2 A. E. H a m m a d , D. A. W o o d f o r d a n d R. M. M a t h u r , AC v o l t a g e c o n t r o l at a n HVDC t e r m i n a l , Proc. Can. Communications and Power Conf., Montreal, Canada, 1978, pp. 341 349. 3 A. E. H a m m a d , K. Sadek, J. K o e l s c h a n d G. G u e t h , A d v a n c e d s c h e m e for AC v o l t a g e c o n t r o l at H V D C c o n v e r t e r t e r m i n a l s , I E E E Trans., PAS-104 (1985) 697-703. 4 O. P. Bowles, A l t e r n a t i v e t e c h n i q u e s a n d o p t i m i z a t i o n of v o l t a g e a n d r e a c t i v e p o w e r c o n t r o l at H V D C c o n v e r t e r stations, Proc. I E E E Conf. Overvoltages and Compensation on

Integrated A C / D C Systems, Winnipeg, Canada, 1980. 5 Y. Yoshida, D e v e l o p m e n t of a c a l c u l a t i o n m e t h o d of AC v o l t a g e s t a b i l i t y in H V D C t r a n s m i s s i o n s y s t e m s , Electr. Eng. Jpn., 94 (1974) 77-85.

39 6 F. Nishimura, A. Watanabe, N. Fujii and F. Ogata, Constant power factor control system for HVDC transmission, I E E E Trans., PAS-95 (1976) 1845 1853. 7 M. Yuki, T. Nambu, N. Nagai, K. Katsuki and J. Tsukamoto, Development of digitalized control equipment for HVDC transmission, IEEE Trans., PAS-103 (1984) 190 196. 8 E. Rumph and S. Ranade, Comparison of suitable control systems for HVDC stations connected to weak AC systems. Part I: New control system; Part II: Operational behavior of the HVDC transmission, IEEE Trans., PAS-91 (1972) 549 564. 9 R. Jotten et al., Control in HVDC systems the state of the art. Part I: Two-terminal systems; Part II: Multiterminal systems, Proc. CIGRE, Paris, France, 1978, Paper No. 14-10. 10 M. Z. Tarnawecky, HVDC transmission control schemes,

Proc. Manitoba Power Conf. E H V DC, Winnipeg, Canada, 1971. 11 F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956. 12 J. F. Hauer, C. J. Demeure and L. L. Scharf, Initial results in Prony analysis of power system response signals, IEEE P E S Summer Meeting, Long Beach, CA, USA, 1989, Paper No. 89 SM 702-2-PWRS. 13 J. F. Hauer, The use of Prony analysis to determine modal content and equivalent models for measured power system response, IEEE P E S Summer Meeting, Long Beach, CA, USA, 1989, Panel Session on Eigenvalue and Frequency Domain Methods for System Dynamic Performance.

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