Robust design of multimachine power system stabilisers using tabu search algorithm

M.A.Abido and Y.L.Abdel-Magid

Abstract: Kobust design of mullimacliine power system stabilisers (PSSs) using the tabu search (TS) optimisation technique is presented. The proposed approach employs TS for optimal parameter settings of a widely used convcntional fixed-structurc lead-lag PSS (CPSS). The parameters of the proposed stabilisers arc sclcctcd using TS in order to shift the system poorly damped clcctromcclianical modes at sevcral loading conditions and system configurations simultaneously to a prescribed ~ o n c in the left hand side of tlie .r-plane. Incorporation oT TS as a derivative-free optimisation technique in PSS design significantly reduces tlic comptdational burden. I n addition, tlie quality of the optimal solution does not rely on the initial guess. The pcrforniancc of the proposed PSSs tinder different disturbances and loading coliditions is investigated for niultiniacliine power system. The eigenvalue aiialysis and tlie nonlinear simulation results show the effectiveness of tlic proposed PSSs in damping out the local, as well a s the intct'arca, modes and cnhance greatly the system stability over a wide range of loading conditions and system configurations.

1 Introduction

In the past two decades, the utilisation of supplementary excitation control signals for improving the dynamic stahil- ity of power systems has rcccivcd much allclition [ILIX]. Nowadays, the convcntional power system stabiliser (CPSS) is widely used by power system utilities. llcccntly, scvcral approaches based on modern control theory havc been applied to tlie PSS design problem. These includc optimal, adaptive, variable structure, and intelligent control [2-S]. Despite the potential of modcrii control Icchniqucs with dif- fcrciit S ~ ~ L K ~ L I ~ C S , powcr system utilitics still p r c h the CPSS structure [6]. The reasons behind that niiglit he the eiasc of on-line tuning and the lick of assurance of Ihc slability related to some adaptive or variable s~rticture Lechniques.

Different techniques of scqucntial design of PSSs are prc- sciited to damp out one of the electromeclianical modes iat a time [7]. Howcvcr, this approach inay not liiially lead to an overall optimal choice of PSS parameters. Moreover, the stabilisers designed to damp one mode can producc adverse cffccts in other modes. Also, the optimal sequence of design is a very involved question. The scqucntial design of PSSs is avoided in [X, 91. Unfortunately, tlie proposed techniques are iterative and require a heavy computation burden due to the system reduction proccdure. In addition, tlic initialisation step of Ilicse algorithms is crucial and affects the final dyilamic rcsponsc of tlic controlled system. Therefore, a final selection criterion is required lo avoid long rutis of validation tests on the nonlinear model.

Generally, it is important to recognise that macliinc parameters change with loading, making thc macliinc

behaviour quite different for different operating conditions. Since these parameters change in a rather complex matiner, a set of CPSS parameters which stabilises the system tinder a certain operating condition tnay no longcr yield satisfac- tory rcst11ts whcn thcrc is ia drastic change in power system operating conditions and conligurations. I-lence, PSSs should provide some dcgrcc of robustness to the variations in system parameters, loading conditions and contigura- tions.

H , oplimisation tccliniqucs [IO, 1 I ] have been applied to thc robust PSS design problem. However, the importance and diflicultics in the selection of weighting fiinctioiis of the I / , optimisation problem havc bccii reported. In addition, the additive and/or multiplicative uncertainty rcprcscnta- tion cannot t r a i t situations whcrc i a nominal stable system becomes unstable after being perturbed [12]. Moreover, the pole-zero cwnccllation phenomenon associated with this approach produces closed loop poles whose damping is directly dependent on the open loop system (nominal sys- tem) [13]. On the other hand, tlic order of the Hm-based stabiliser is 11s high as that of the plant. This gives rise to the complex structure of such stabilisers and reduces their applicability. Although the scqucntial loop closure method [14] is well suited for on-line tuning, there is no analytical tool to dccidc the optinial scqucncc of the loop closurc.

On the other hand, Knndur r t d. [IS] havc prcscntcd a comprehensivc analysis of the cffccts of the different CPSS parameters on the overall dynamic pcrforniancc of the powcr system. It is shown that the appropriate selection of CPSS paramctcrs rcs~ilts in satisfactory pcrformancc during systcm t~pscls. I n addilion, Gibhard [ I61 dcmonstratcd that the CPSS provides satisfactory damping perlbrmance over ii wick range of systcm loading conditions. The robust nature of the CPSS is due to the fact that tlic torque refer- ence voltage transfer function remains approximately invar- iant over a wide range of operaling conditions.

For the robust design of llic CPSS, scvcral operating conditions and system contigul-ations are simultaneously considered in Ihc CPSS design process [16, 171. A genetic

3x7

algorithm-based approach to robust CPSS design is prc- sented in [17]. It is shown that the optimal sclcction of PSS parameters results in a robust pcrhrinaiice of the CPSS. However, there exist some structural problciiis in the con- ventional genetic algorithm such as preinaturc convergence and duplicatious among strings as evolution is processing. A gradient proceclurc for optimisation of PSS parameters at different operating conditions is prcsented in [ I 81. Unfor- tunately, the optimisatioii proccss requires the compulation of sensitivity factors and cigeiivcclors at each itcration. This gives risc to a heavy computatioiial burden and slow con- vergence. hi addition; tlie search proccss is susceptible to becoming trapped in local minima and the solution obtained will not be optimal. Thererore, a TS-based approach to robust PSS design is proposcd in this paper.

In the last few years, the tabu search algorithm [IO-231 appeared as another promising heuristic algorithm for han- dling combinatorial optiniisation problems. The tabu search algorithm uses a flexiblc incmory of scarch history to prevent cycling and to avoid ciilrapnient in local optiiiia. It has been shown that, under certain conditions, the tabu search algorithm can yicld a global optimal soltition with probability 1 1221.

In this paper, the problem of robust PSS design is formu- lated as an optimisatioii problem and the TS algorithm is employed to solve this problem Tlic proposed design approach has been applied to diffcrcnt examples of multi- machine power systems. The eigenvalue analysis and the nonlincar simulation results have been carried otic to assess the effectiveness of t l ie proposed PSSs under dilrerent dis- turbances, loading conditions and system configurations.

2 Problem statement

2. I Power system model A power system can be modelled by a set of nonlinear dif- rerential equations as:

k =: f(X, U ) (1)

where X is the vector ol' the state variables and U is the vector of input variables. In this study X = [S, U, E;,, E/<dT and U is the PSS output signals. Here, S and (11 arc tlie rotor angle and spccd, respectively. Also, K;, and I$(, arc tlie iiitcmal and lield voltages, rcspcctively.

I n the design of PSSs, the liiicarised incremental models around an cquilibl-iuni point are usually employed [24]. Therefore, the state equation of a power system with 17

machines and 171 stabilisers can bc written as:

~ = A X + B U (2)

where A is a 4n x 411 matrix and equals a/ldX while B is a 4n x 171 inatrix and cquals df7dU. Both A and B are evalu- ated at a certain operating point. X is a 4n x 1 slate vector and U is an I N x 1 input vector.

2.2 PSS structure A widely used conventional lead-lag PSS is considered in this study. It can be described a s

where T,, is the washout time conslant, (i, is the PSS out- put signal at the ith machine, and Awl is tlie speed devia- tion of this machine. The time conslants Ti,., T2 and T, are usually prcspecified. The stabiliser gain Ki and time con- slants TI i and T,, are tlie parameters to be dctcrniined.

188

2.3 Objective function To iiicrcasc the system damping, two eigenvalue-based objective functions are considered:

1 L j )

.TI = c (0" -.?,,I2 (4)

.I2 = (C" - Ct,,I2 (3

j = l n,

r , i'

. 7 = ~ 1 <,,,<<,, whcrc t ip is the nuiiibcr or operating points considered in lhc design proccss. qj and i;,j arc the rcal part and the damping ratio of the ith eigeavaluc of the ,jlh operating point. respectively. Also, q, and b arc chosen thresholds. Here, JI and .J2 rcllcct the system response settling time and overshoot, respectively. Tlic value of IT" represents tlie desirable level of system damping. This level can be achieved by shifting the domilmiit eigenvalues to the left of the s = (J" line in the s-plane. This insures some dcgrcc of relative stability. Also, the value of 5;) represents the desira- ble clamping ratio which ~ t i i i be achieved by shifting the dominant cigciivalues to the left ol' the = I;" linc in the .Y-

plaiic. This ensures a good lime-domain response in terms of overshoots and settling time. The conditions q,j z CJO and I;i,i 5 CO arc imposed to consider only thc tinslablc or poorly damped modes which mainly belong to the elcctromechaii- ical ones. The problem constraints are the CPSS parameter bounds. It is worth noting that it is also possible to use a combination of cqns. 4 arid 5 in the foriii ol' .Il + I I J ~ [17]. LoadiIig-depeiident q, arid & can also be tiscd. The design problem can be forinulatcd as the following optimisation problem:

Sul?jer.t to (6)

( 7 )

T;:"" 5 5 T;l'"" ( 8 )

T;:"' 5 r?, 5 TA:'"" ('3)

j;;nii> < < I<;CLX - , -

The proposed approach cinploys tlie TS algorithm to solve this optimisation problem and search for an optimal or ncar optimal set of PSS parameters, {Ki, T I , T2i, i = 1, 2, ... ( m}.

3 Tabu search algorithm

3.1 Overview Thc tabu search is a higher level heuristic algorithm for solving combinatorial optimisation problems. It is an itera- tivc improvement proccdure that starts ftoiii any initial solution and attempts to dctcrminc a better solution. TS was proposed in its present form a few years ago by Glover [20- 231. It has now become an established optimisation approach that is rapidly spreading to many new fields. Together with other hetirktic scarcli algorithms such as CA, TS has been singled out a s 'extremely promising' for tlie fultirc treatment of practical applications [20]. Gcncr- ally, TS is characterised by its ability to avoid cntrapmeiil in a local optimal solution and prevent cycling by using flexible memory 01' search history.

3.2 TS algorithm The basic elcmcnts of TS are briefly stated and dcliued as follows:

Current solu/ion, x ~ , ~ ~ ~ ~ , , ~ ~ ; A set of the optimiscd paramctcr values at any iteration. It plays a central role in generating the neighbour trial solutions.

//I/; / 'ioc.-Ue,,cr. ' f r m w ? ~ . Lli,\iril>,, Vol. 147. N o . 6 . Noiwiiihci. 20110

- Muvex: They characterise the process of gcncrating trial solutions that are related to x,,,,,,,~.

Ser of ccozdihte I I U J D ~ . S , N(.u,.,,,,,,,); Thc set of all possiblc movcs or trial solutions, .ulri~ll, i n the ncighbourliood or xcurrcnl. 111 case of continuous variable optiniisation problems, this set is too large or even an infinite set. Tlicrcforc, one could opcratc with ii subset, S(s,,,,.,,,,,) with a liinitcd number of trial solutions 121, of this set, i.e. S C N and xlrial E S(.Y,,,,.,,,,,). . T& iestriction.~; These are certain conditions imposed on moves that make some of them forbidden. These forbidden movcs arc listed to a ccrvain sizc and known as tabu. This list is called tlie tabu list. The reason bcliind classirying a certain inovc as forbiddcn is basically to prevcnt cycling and avoid returning to the local optimum just visited. The labu list size plays a great role in the search for high-quality solutioiis. The way to identify a good tabu list size is simply watch for tlie occurrence of cycling when tlie sizc is too small, and the deterioration in solution quality when the size is too large caused by [orbidding too many moves. In some applications a simple choice of the tabu list size in a range centred at 7 seems to be quite effective [21]. Gener- ally, the tabu list size should grow with the sizc of the givcn problem. In our implcincntation, tlic sizc 7 is found to be quite satisfxtory.

Aspirution criterion (Level): A rule that overrides labu restrictions, i.e. if a certain move is forbidden by tabu restriction, the aspiration criterion, when satisfied, can make this move allowablc. Diffcrcnt rorms or aspiration criterion arc uscd in the literatim [19-23]. The one consid- ered here is to override the tabu status of a move if this move yields a solution which has better objective hnction, J, than the one obtained earlier with the same move. Thc importance of using an aspiration criterion is to add somc flexibility to the tabu scarcli by dirccting it towards the attractive movcs.

Stopping criteriu; These are tlie conditions under which thc search process will tcrminatc. In this study, the search will terminate if one of the following criteria is satisfied: (a) tlie number of iterations sincc the last clmngc of the bcst solution is greater than a prespccificd number: (h) the number of itcrations rcachcs the maximum allowable numbcr; or (c) the valuc or the objective function reaches zero. The general algorithm of TS can be describcd in stcps as follows: Step 1; Sct tlic iteration counter I< = 0 and I-andomly geiier- ate an initial solution qnili;,l. Set this solution as tlie current solution as well as the best solution, x,,~~,,,,, i.e. xillilllll =

StcJp 2; Randomly generate a set of trial solutions xliz,l in tlie neighbourhood of the current solution, i.e. crcatc S(x,,,,,,,,,). Sort tlie elements of S bascd on their objective function values in ascending order as the problem is a min- imisation one. Lct LIS dcfinc xYlil( as thc it11 trial solution in the sortcd set, I 5 i 5 nt. Here, ~~~~~~l represents the best trial solution in S i n terms of the objcctivc function valuc associ- ated with it. Step 3: Set i = I . If J(,Y,,~,~~) > J(.Y,,~,~) go to step 4, else set xl,cql = x,,.~~~( and go to stcp 4. Step 4: Check tlie tabu status of xlrial'. If it is not in the tabu list then put it in the tabu list, set ,U,,,,.,,,,, = s,i>,{, and go to step 7. If it is in the tabu list go to stcp 5. Step 5; Check tlie aspiration criterion of then override tlie tabu restrictions, update the aspiration level, set x,,,,,,,,~ = ,qriali, and go to step 7. i f not, set i = i + I and go to step 6.

- in1 - 'hcsl.

//(/i l + o r , - ( ; w w T'ro,>vm / l i \ r r i /> . , I ' d 147. N o 6 , N o i r w h c r 2lli)ll

2 7 8

1 Load A

-

9 3

6

Load B

4

1 3 Table 1: Generator operating conditions of example 1

Generator Base case Case 1 Case 2 Case 3

P Q P Q P Q P Q

G7 0.72 0.27 2.21 1.09 0.36 0.16 0.33 1.12

G2 1.63 0.07 1.92 0.56 0.80 -0.11 2.00 0.57

G3 0.85 -0.11 1.28 0.36 0.45 -0.20 1.50 0.38

Table 2 Loads of example 1

Load Base case Case 1 Case 2 Case 3

P Q P Q P Q P Q

A 1.25 0.50 2.00 0.80 0.65 0.55 1.50 0.90

B 0.90 0.30 1.80 0.60 0.45 0.35 1.20 0.80

C 1.00 0.35 1.50 0.60 0.50 0.25 1.00 0.50

4

4. I Test system In this example, the 3-machine 9-bus system shown in Fig. 1 is considered. Details oftlie system data arc givcn in

Example 1: Three machine power system

38')

Table 3: Eigenvalues and damping ratios of example 1 without PSSs

Base case Case 1 Case 2 Case 3

-0.01 i j9.07. 0.001 -0.021 2 j8.91, 0.002 -0.30e j8.95, 0.034 0.38 * j8.87,-0.034 -0.78i i13.86.0.056 -0.52i- i13.83.0.038 -0.84~i13.72.0.061 -0.342 i13.69.0.025

[24]. The participation factor nicthod [25] and the sensitiv- ity of PSS effect nicthod [26] were iiscd to identify the opti- mum locations of PSSs. The results of both methods indicate that Gz and C, are thc optimum locations Tor installing PSSs.

4.2 PSS design To design the proposed PSSs, four operating GISCS arc coil- sidered. The generator operating conditions and the loads at these cases are given in Tables 1 and 2, respectively. The electromechanical modes eigenvalues and their damping ratios without PSSs are given in Table 3. It is clear that the electronicchanical modes are poorly damped and sonic of them are unstable. In this example, tlie optiiiiiscd pa"- tcrs arc Kl, Tli and ?;, i = 2, 3. 7>,,,, 7; and T, arc set to be 5s, 0.05s and 0.05s, respectively [24].

Table 4: Optimal values of proposed PSS parameters for example 1

Generator

G2 11.833 0.140 0.133 5.821 0.118 0.300 G3 0.438 0.238 0.150 0.138 0.340 0.374

Objective function .I1 Objective function .I2

k Tl J3 k Jl J3

800 7

In the casc of J , , q, is chosen to be 3.0, while is cho- sen to be 0.25 in the case of J2. With each c a ~ c , the TS

algorithm has been applied to search for the optimised parameter settings so as to shift simultancously the poorly damped eigenvalues of the four cases to the left of the s- plane. The final values of thc optiinised parameters in each case are given in Tablc 4. The convergence rates or tlic objective functions are shown in Fig. 2. With the optiinal values of the proposed PSSs, thc system eigenvalues with J , and ,I2 settings arc given in Tables 5 and 6, respectively. lt is quite clear that tlie system damping with the proposed PSSs is greatly enhanced.

4.3 Nonlinear time-domain simulation To dcinonstrate the effectiveness of the proposed PSSs over a wide range of loading conditions, two different distur- bances are considered as rollows:

(61) A 6-cycle fault disturbance at bus 7 at the end of line 5- 7 with case 3. The h d t has been cleared without tripping.

( / I ) A 6-cyclc f:,tult disturbance at bus 7 at the end of line 5- 7 with casc 1. The fault is cleared by tripping the line 5-7 with successful rcclosure after I .Os.

The system responses to the considered faults with and without the proposed PSSs are shown in Figs. 3-8. It is clear that the proposed PSSs provide good damping char- acteristics to low-frequency oscillations and greatly enhance the dynamic stability of power systems.

0.04 1

Table 5 Eigenvalues and damping ratios of example 1 with proposed PSSs U, settings)

Base case Case 1 Case 2 Case 3

-3.OOij18.40.0.161 -3.392j18.47,0.181 -3.632j16.71,0.212 -3.16~j18.15.0.172 -4.47 i j8.27, 0.457 -3.06 +j7.60,0.373 -3.01 2 j8.65, 0.329 -3.57 2 j8.32, 0.394

Table 6 Eigenvalues and damping ratios of example 1 with proposed PSSs (J2 settings)

Base case Case 1 Case 2 Case 3

-4.13i j18.09,0.223 -4.64+ j18.32.0.246 -5.242 j16.34, 0.305 -4.41 2 j17.88, 0.239 -2.63+17.86,0.317 -1.98+17.56,0.253 -2.06+j8.06,0.250 -1.96+17.96,0.239

0.02 1 a

3 0.02 -

6 Q

0 c ._ .: 0.00

I 7J n

-0.02 -

3 0.02 Q

. . . . . . : : : : :: . . . . ; : 1

0.04 7

3 0.02 -

m- Q

0

m

V V

Q

r .-

‘Z 0.00

% -0.02 -

-0.04 ~1 0.00 2.00 4.00 6.00

time, 5

Sy,,ion r q i i ) i i . s i ~ i f~~.wniple I rsitli i b l r a l x i t u ~ (1)): ivi/hoii/ 1’SS.s Fig. 6 Am1

. . . . . . . . A(% ~ ho,,

5

5.7 Jest system In this examplc, the IO-machine 39-bus New England powcr system shown in Fig. 9 is considered. Generator GI is an equivalcnt power source reprcscnting parts or the US- Canadian interconnection systcm. Details ol‘ the systcm data arc given in [27]. Although the numbcr and location of PSSs required can hc investigated [25, 261, it is assumed hcrc that all generators except GI arc eqiiippcd with PSSs for illustration and comparison purposes.

~ ~~

Example 2: New England power system

IE1: Proc.-Gerr~i. T r ~ m w ~ , l>i.uril,., I ’ d 147. No. 6. N o w r i h r w 211011

3 Q 0.02 0’04 i

15--

1 3 3 14

23 1 4- I 34

13

35 36

’ k 2 11

20 38 30 I

5.2 PSS design To design the proposed TSPSS, three different operating conditions that represent thc system under severe loading conditions and critical line outaecs in addition to the base case are considered. These conditions are extremely harsh

3‘1 I

Table 7: Eigenvalues and damping ratios of example 2 without PSSs

Base case

0.191 + j5.808, -0.033

0.088 L j4.002, -0.022

-0.028 + j9.649, 0.003

-0.034 t j6.415, 0.005

-0.056 + ]7.135,0.008

-0.093 k j8.117,0.011

-0.172 + j9.692,0.018

-0.220 t j8.013, 0.027

-0.270 + j9.341, 0.029

Case 1 Case 2

0.195 + j5.716, -0.034 0.189 f j5.811, -0.033

0.121 *j3.798,-0.032 0.006~j3.113,-0.002

0.097 cj6.006, -0.016 0.001 + j6.180, -0.0002

-0.032 i j9.694, 0.003 -0.028 2 j9.650, 0.003

-0.104+ j8.015.0.013 -0.032+j1.105,0.005

-0.109t j6.515, 0.017 -0.091 5 j8.115, 0.011

-0.168+j9.715.0.017 -0.172tj9.693,0.018

-0.204 t j8.058, 0.025 -0.218 2 j8.024, 0.027

-0.250 + j9.268, 0.027 -0.269 + j9.342,0.029

from the stability point of view [28]. They can be described

(i) Base case; (ii) Case I ; outage of line 21-22; (iii) Case 2; outage of line 1-38. (iv) Casc 3; outage of line 21-22, 25'% incrcasc in loads at buses 16 and 21, and 25% increase in generation of G7. The electromechanical modes and damping ratios without PSSs for these conditions are given in Table 7. It is clear that these modes are poorly damped and some of them are unstable. In this example, the optimised parameters arc Kj, TI, and T, , i = 2, 3, ..., 10, i.e. tlie number of optiniised paranieters is 27. T,,,, T2 and T4 are set to be 5s [IX], 0.05s and 0.05s, respectively.

In this case, uo and are chosen to be -1.0 and 0.2, respectively. The final values of the optimised parameters with both objective functions are given in Table 8. The convergence rate of the objective functions are shown in Fig. IO. With tlie optimal values ol' the proposed PSSs, the system eigenvalues with J , and J2 settings are given in Tables 9 and IO, respectively. It is quite clear that the sys- tem eigenvalues associated with the electromechanical modes have been shifted to the left of the +plane with the proposed PSSs. This demonstrates that the system damping with the proposed PSSs is greatly improved.

Table 8 Optimal values of proposed PSS parameters for example 2

as:

Obiective function J, Obiective function J, Generator

k TI 73 k TI J3

Gz 26.963 0.399 0.880 25.465 0.865 0.993

G3 15.733 0.650 0.826 49.336 0.848 0.986

G4 26.842 0.425 0.966 42.820 0.632 0.924

G5 43.727 0.102 0.427 29.322 0.202 0.347

G6 18.260 0.974 0.393 49.384 0.409 0.642

G7 2.737 0.460 0.202 10.328 0.236 0.151

G8 0.278 0.734 0.743 26.630 0.948 0.998

G9 18.732 0.171 0.337 31.461 0.291 0.176

GI 0 26.598 0.945 0.871 48.940 1.000 0.962

5.3 Nonlinear time-domain simulation To demonstrate the effectiveness of the proposed PSSs over a widc range of operating conditions, the following distur- bances are considered for nonlinear time simulations: (U) A 6-cycle fault disturbancc at bus 29 at the end of line 26-29. The fault is cleared by tripping the line 26-29 with succcssfiil reclosure after I .Os.

392

1

Case 3

0.205 2 j5.638, -0.036

0.152e j3.714.-0.041

0.126 f j5.964, -0.021

0.051 + j9.648, -0.005

-0.098k j8.013, 0.012

-0.101 t j6.512.0.016

-0.167 t j9.727, 0.017

-0.202 + j8.079, 0.025

-0.238 j9.296, 0.026

1 o'60

iterations Fig. 10 ?(,jct.iilr,/iiizciio,, l r w i n i m q / ~ , w l p / c , 2

~ alycctwc liinclion .I , ~ objective Cunclion .I2

(b) A 6-cycle fault disturbance at bus 14 at the end of line 14-15. The fault is cleared by tripping the line 14-15 with successful rcclosurc after I .Os. The perrormance of the proposed PSSs is comparcd to that of PSSs with the settings given in [ 181, where the stabilisers were designed using gradicnt mcthods with the Same condi- lions considered in this study. For disturbance ((I), the speed deviation of G,, as the nearest generator to the fault location, is shown in Fig. 11. It is clear that the system response with the proposed PSSs is stable, while with PSSs of [18] the system is unstable. Additionally, PSSs of [I81 h i 1 to stabilise the system with disturbance ( / I ) , the proposed PSSs provide good damping characteristics and tlm system is stablc under this severe disturbance as shown in Fig. 12. In addition, the proposed PSSs arc quite efficient in damp-

0'04 1 0.03 4

Table 9: Eigenvalues and damping ratios of example 2 with proposed PSSs (J, settings)

Base case

-1.236 i j14.83, 0.083 -1.148+j11.13,0.103

-2.064 t j10.94, 0.185 -1.160i jlO.99,0.105

-1.787 t j9.507, 0.185

-1.101 t j9.159, 0.119 -1.730 ~ ~ 7 . 9 2 5 , 0.213 -1.096 i j5.569, 0.193 -1.008 + j3.864, 0.252

Case 1

-1.321 kj14.54, 0.090

-1.129 * j11.12.0.101

-2.004t j10.87, 0.181 -1.029 t j10.75, 0.095

-1.082 f j8.927, 0.120

-1.785 t j8.603, 0.203 -1.974 + ~7.223, 0.264 -1.047 t j5.479, 0.188 -1.035f j3.614, 0.275

Case 2

-1.239 t j14.83, 0.083

-1.145 1 j11.13, 0.102 -2.040 A j10.91, 0.184

-1.1591 jlO.99,0.105

-1.777 + j9.466, 0.185 -1.056i j9.137.0.115

-1.803 1 17.526.0.233 -1.000 t j5.584, 0.176 -1.000 t j2.333. 0.394

Case 3

-1.234t jl4.60, 0.084

-1.1161 jl l .11, 0.100 -1.982 i j10.83, 0.180

-1.000 1 j10.73,0.093

-1.082 1 j8.903, 0.121 - 1 . 7 2 8 ~ j8.556, 0.198 -1.971 117.036. 0.270

-1.028 t j5.429, 0.186 -1.069 + j3.443, 0.297

Table IO: Eigenvalues and damping ratios of example 2 with the proposed PSSs (J2 settings)

Base case

-3.035 t j14.72, 0.202

-2.482 + j13.16, 0.185

-3.330 f j12.74, 0.253 -2.317 tj12.74, 0.179 -2.216+ j11.95.0.182

-3.351 t j11.48, 0.280

-2.335 t jl1.06, 0.207 -1.915+ j9.553, 0.197 -0.764f j2.912, 0.254

Case 1

-3.076i j14.61, 0.206

-2.444 1 j12.98, 0.185 -3.342 1 j12.74. 0.254

-2.206 1 j12.64, 0.172 -2.363 t j l l . 6 7 , 0.194

-2.962i j l l .12, 0.257 -2.149tj11.17,0.189 -1.954 i j5.105, 0.357

-0.660 t j2.927, 0.220

Case 2

-3.040 t j14.72, 0.202

-2.490t j13.15, 0.186

-3.339i j12.73, 0.254 -2.300 f j12.74, 0.178 -2.1 48 t j11.86, 0.178

-3.351 t j11.46, 0.281 -2.439 1 j10.99, 0,217 -1.923 t j9.520, 0.198 -0.471 t j2.195, 0.210

Case 3

-3.100 t j14.55, 0.208

-2.417 i j12.91, 0.184

-3.394 i j12.73, 0.258 -2.189 + j12.62, 0.171 -2.1851 j11.73.0.183

-2.923t j11.01, 0.257 -2.242 i j l l .09, 0.198 -1.825 + j4.847, 0.352 -0.605 i j2.936, 0.202

ing out the local modes as well as the inter-arm modes or oscillations. This illustrates the superiority of the proposed TSPSS design approach to get optimal or near optiiiial PSS paramctcrs.

0’04 1 0.03 1 0.02 -

3 a 5 0.01 - .z

0.00 -

-0.01 -

-0.02 ~

0.00 2.00 4.00 6.00 8.00 10.0

time, s

Due to space limitations and to give a clear perceptive of t l ie system responses, two pcdorniance indiccs that reflect the settling time and ovcrshoots arc introduced and evalu- ated. Thcsc indices are defined as:

where 17 is the number or nlachincs and tsi,,, is the imula- lion time. The values of these indices with the distiirb~mces (U) and (/I) arc given in Table 1 I , I t is clcar that the values of these indices with thc proposed PSSs arc much smaller. This demonstrates that the settling time and tlie spccd dcvi- ations or all tinits are much reduced by applying the pro- posed PSSs.

Table 11: Values of performance indices for example 2

P/l p12

[I81 [I81 Fault PSSs Proposed PSSs PSSs Proposed PSSs

J1 settings J2 settings J, settings J2 settings

a 4964 74.382 78.592 191.8 8.233 9.034 b 4944 78.828 49.307 163.6 8.488 7.543

6 Discussion

Some comnicnts on the proposcd approach arc now in order:

Unlike tlic methods of [7, XI, the proposcd TS based approach docs not rely on the initial solution. Starting aiiy- where in the scarch space, the TS algorithm ensures the convcrgcnce to the optimal solution. Example I is rcconsid- ered to ctemonstratc this point. In this case, the main target is to shift the dominant eigenvalues as Car a s possible to the left of the .qdanc. Diflfcrent initial solutions are considered by changing tlic seed of the random number generator that generates tlic initial solution. Tlie convcrgeiice or the objc- tivc fimctions with din‘crcnt initial solutions is shown in Fig. 13. Tlie results emphasise that the proposcd TS-based approach finally leads to the optimal solution rcgardlcss of the initial otic.

Bascd on the above conclusion, tlic proposed approach can bc used to improve the solution quality of other meth- ods described in [5-8].

looool

1 ; I I I I I I I I 0 100 200 300 400

iterations Fig. 13 Ohjer/iwfiinctiiin J , ofe,~uinpk I ivirli i([fereii/ /n/tiu//s~u/iiri,s

7 Conclusions

In this study, the tabu search algorithm is proposcd for the robust PSS design problem. Thc proposed design approach cmploys TS to search for optimal settings of CPSS param- eters. The proposed objective function shifts simultaneously the electromechanical mode eigenvalues of different operat- ing conditions to the left in the s-plane. The proposed approach has been applied to two different multimachine power systems with different loading conditions and system configurations. The main featurcs of thc proposed approach can be summarised as: (i) The solution quality of the proposed approach is inde- pendent of the initial guess. Hence, the proposed approach can be used to improvc the quality of the sohitions of other classical optimisation methods. (ii) Since eigenvector calculations and sensitivity analysis are not required to evaluate the proposed objective func- tions, heavy computations of the design process are avoided. (iii) The eigenvalue analysis reveals thc effectiveness of the proposed PSSs to damp out local as well as inter-area modes of oscillations. (iv) The nonlinear time-domain simulation results show that thc proposed PSSs work effectively over a wide rangc of loading conditions and system configurations.

8 Acknowledgment

This projcct has becn Uunded by King Fahd University of Petroleum and Mincrals uiidcr project no. EEiPOWER SYSTEM SI2 12.

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