Robust design of multimachine power system stabilisers using tabu search algorithm

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  • 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

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  • 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/,, 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 -4.6...

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