artigo em analise

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011 2371

An Eigenstructure-Based Performance Index andIts Application to Control Design for Damping

Inter-Area Oscillations in Power SystemsD. P. Ke, C. Y. Chung, Senior Member, IEEE, and Yusheng Xue, Member, IEEE

Abstract—An eigenstructure-based performance index is pro-posed in this paper to measure the dynamic performance of thesystem as well as control efforts. Calculation of this index isbased on eigenstructure of the closed loop system and the designparameters; it does not rely on control structures. Therefore, thisindex can be applied for solving structurally constrained controlproblems. A tuning scheme based on this index is proposed forcoordinating power system stabilizers (PSSs) and supplementarydamping controllers (SDCs) for flexible AC transmission systems(FACTS) devices to damp inter-area oscillations of systems and tooptimize their control efforts under multiple operating conditions.Both PSSs and SDCs utilize control structures as a low ordersingle-input-single-output phase lead-lag compensator. Wide-areasignals are employed to upgrade their effectiveness in dampinginter-area oscillations. Time delays caused by usage of wide-areasignals are also considered in the tuning scheme. Results of simula-tion on a four-machine two-area system and the New England andNew York interconnected system show that the proposed index iseffective in measuring dynamic performance of the system andthe coordinatedly tuned PSSs and SDCs based on this index canrobustly damp inter-area oscillations of systems with optimizedcontrol efforts.

Index Terms—Coordination, damping control, eigenstructure,inter-area oscillation, structural constraint.

I. INTRODUCTION

P OWER system stabilizers (PSSs) have been widelyemployed to provide additional damping for inter-area

oscillations in power systems [1]–[3]. It is also recognized thatsupplementary damping controllers (SDCs) for flexible ACtransmission systems (FACTS) devices have great potential indamping inter-area oscillations [4]. Generally it is expectedthat as these damping controllers (PSSs and SDCs) work in acoordinated manner to provide adequate additional damping to

Manuscript received September 16, 2010; revised January 16, 2011; acceptedFebruary 22, 2011. Date of publication March 28, 2011; date of current ver-sion October 21, 2011. This work was supported in part by Research GrantsCouncil of Hong Kong (PolyU 5154/08E) and in part by the Department ofElectrical Engineering of The Hong Kong Polytechnic University. Paper no.TPWRS-00748-2010.

D. P. Ke and C. Y. Chung are with the Computational IntelligenceApplications Research Laboratory (CIARLab), Department of ElectricalEngineering, The Hong Kong Polytechnic University, Hong Kong (e-mail:[email protected]; [email protected]).

Y. Xue is with State Grid Electric Power Research Institute, State Grid Cor-poration of China, Nanjing 210003, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2011.2123119

inter-area oscillations, their control efforts should be optimal[5], [6]. Moreover these controllers should be robust for dif-ferent operating conditions or network configurations [7], [8].Besides, structural constraints are usually imposed on dampingcontrollers: they must use dynamic output feedback controlschemes because it is difficult to implement state feedbackcontrol in power systems; they should be low order and with asimple structure familiar to engineers so that implementationand the subsequent tuning are easy [7]–[9]. Moreover, althougha decentralized implementation scheme using only local signalsas inputs is more practical for damping controllers, they canbe configured for quasi-decentralized implementation whenwide-area signals are employed as inputs to enhance theireffectiveness in damping inter-area oscillations [9]. Howevertime delays that occur because of use of wide-area signalsshould be considered in the design.

Several methods have been proposed for the design ofdamping controllers. Approaches based on robust controltheories and linear matrix inequalities (LMI) have been appliedfor damping controller design to deal with uncertainties ofoperating conditions [5], [8], [10]–[12]. However they cannotbe applied for design of structurally constrained controllers.In [8], for designing structurally constrained SDCs for coor-dinated functioning, the LMI-based regional pole placementproblem is converted into the iteratively solved bilinear matrixinequalities (BMI) problem. Nevertheless, this method assumesthat input matrices of state space equations of the controllersare known. Methods that directly optimize eigenvalues ofthe closed loop system can readily be applied to structurallyconstrained controllers while considering multiple operatingconditions [7], [13]–[16]. However, system dynamics in timedomain are not only related to eigenvalues, but also associatedwith eigenvectors [17]. Moreover, control effort cannot beexplicitly optimized in these methods. Therefore, one salientmerit of optimal control [6], [9] is that the control process,together with the system dynamics objective, can be explicitlyconsidered in the cost function. Unfortunately standard optimalcontrol cannot be applied to structurally constrained controllersalthough it can perform well for state feedback controllers [12].The method applied in [9], [18], and [19] tries to solve thisproblem by assuming that poles of synthesized controllers areknown. However this assumption is only applicable for somecertain control structures and the method cannot take multipleoperating conditions into consideration.

In this paper, an eigenstructure-based performance index isproposed to measure the system dynamic performance as wellas the control efforts. Computation of this index does not rely on

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2372 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011

the control structures because it is only related to the eigenstruc-ture of the closed loop system and the design parameters. Min-imizing this index can improve system dynamics and optimizethe control efforts. Therefore, by utilizing this index to constructobjective functions, structurally constrained controllers can bedesigned for optimization of control. Based on this idea, a tuningmethod is proposed for coordination of structurally constrainedPSSs and SDCs as stated above, to damp inter-area oscillationsas well as to optimize the control efforts under multiple oper-ating conditions.

This paper is organized as follows. In Section II, the proposedindex is introduced. The closed loop power system model usedto calculate the index is synthesized in Section III. In Section IV,the proposed tuning scheme for coordination of PSSs and SDCsis introduced. Simulation results are reported in Section V. Con-clusions are presented in Section VI.

II. PROPOSED EIGENSTRUCTURE-BASED PERFORMANCE INDEX

Irrespective of the structure adopted by damping controllers,the synthesized linear model of a closed loop power systemaround an operating point can generally be described as follows:

(1)

(2)

(3)

where and are the state vector and statematrix, respectively, for the th operating condition taken intoconsideration; is the output vector representing system dy-namics objective; is the output vector of damping controllers;and and are output matrices.

For the th operating condition, a quadratic performancemeasurement (cost function) is defined as

(4)

where and are diagonal matriceswith positive entries on their respective diagonals; H is the con-jugate transpose operator; and is the initial value of . Asvariables denoting relative power angles of generators are oftenselected as components of , minimization of (4) will suppresspower angle oscillations of the system and also optimize controlefforts of damping controllers. Substituting (2) and (3) into (4),

can be rewritten as

(5)

where is a Hermite matrix defined as

(6)

In power system dynamic analysis, state matrix can besimilarly diagonalized [20], [21]. Thus the time domain solutionof (1) can be derived as

(7)

where and are right and left eigenvector matrices, re-spectively, of is a diagonal matrix defined as follows:

(8)

where are eigenvalues of .If the system is stable, i.e., all eigenvalues of are with

negative real parts, then (5) can be calculated based on (7), asfollows:

(9)

where is termed as cost matrix for the th operating condi-tion and is defined as follows:

(10)

Here denotes dot production and is a Hermite matrix withthe following definition:

......

......

(11)

where is the conjugate operator.It is seen from (4) that is positive for any given .

Therefore, from (9) and (10), it is inferred that is a positivedefinite matrix. Consequently (9) can be further decomposed asfollows:

(12)

(13)

where are eigenvalues of and are realpositive numbers; are the correspondingcoordinates’ values when projecting onto the orthogonalbasis formed by the right eigenvectors of . It is noted that

are only related to the eigenstructure of thesystem ( , and ) and design parameters ( ,and ) for the th operating condition.

Actually, denotes the initial disturbed deviation from theoperating point and it cannot be determined in the control de-sign process [5], which means that are also un-determined. Therefore, though is a direct time domain in-dicative of the dynamic performance of the system and has beenutilized in the state feedback optimal control [6], it is generallyineffective to use directly for designing the structurallyconstrained controllers because the undetermined cannotbe dealt with in such cases. Nevertheless, according to (12),a way to reduce for the undetermined is to reduce

, and vice versa. Therefore, a new performanceindex is proposed in this paper as follows:

(14)

Obviously this index is independent of , and it is not equiv-alent to but it too can measure the performance of system

KE et al.: EIGENSTRUCTURE-BASED PERFORMANCE INDEX AND ITS APPLICATION TO CONTROL DESIGN 2373

Fig. 1. Overall system structure.

dynamics as well as the control efforts. Furthermore, irrespec-tive of control structures, derivation of this index can be justbased on the synthesized closed loop system model.

It is clear that is the function of parameters of dampingcontrollers. The system dynamics as well as control efforts canbe optimized by adjusting these parameters to minimize .However in order to calculate , the linear model of the closedloop system has to be constructed; this is introduced in the nextsection.

III. MODELING OF THE CLOSED LOOP SYSTEM

A. System Structure

The structure of the overall system in which PSSs and SDCswork in a coordinated manner to damp inter-area oscillationsis illustrated in Fig. 1. Both PSSs and SDCs are assumed to bea classical phase lead-lag compensator. They are implementedin a quasi-decentralized manner, and wide-area signals are em-ployed to enhance their effectiveness in damping inter-area os-cillations. Possible time delays are approximately considered inthe design. The modeling of each part in Fig. 1 and synthesisof the closed loop system model are presented in the followingsubsections.

B. Reduced-Order Open Loop Power System Model

The order of a practical power system model is usually high,but most internal dynamics of the model can be neglectedwithout causing considerable inaccuracy in describing theinput-output relationship since frequencies of the concernedsystem dynamics are between 0.2 and 0.8 Hz (inter-area oscilla-tions) [12]. Thus, the low-order approximated models obtainedvia model reduction can be employed to facilitate the subse-quent design process [6], [9], [11]. There are a variety of modelreduction techniques suitable for reducing the order of practicalpower system models [22], [23]. Therefore, to reduce the timerequired for computation of and to accelerate the tuningprocess, the Schur balanced model truncation algorithm [24]is applied in this paper to obtain the following reduced-orderopen loop power system model:

(15)

(16)

where and are the state vector andstate matrix, respectively, of the reduced-order system for theth operating condition; is the output vector for inputs of

damping controllers; is the input matrix; and areoutput matrices.

C. Time Delay Approximation

Time delays are approximated by using the second order Padeformula [9]:

(17)

where is the time delay vector. The state space equations de-scribing the dynamics of time delays are obtained as follows:

(18)

(19)

where is the state vector of time delays; is the input vectorof damping controllers; , and are state matrix,input matrix, output matrix, and feed-forward matrix, respec-tively.

By incorporating (18) and (19) into (15) and (16), the fol-lowing linear model is obtained:

(20)

(21)

with the following matrix definition:

D. Modeling of Controllers

Suppose there are damping controllers (specificallyin Fig. 1, ) and the th controller has parameters

and . These controllers can bemodeled as

(22)

(23)

where and are output and input, respectively, of the thcontroller; is the state vector; and the following matrices aredefined:

Then state space equations of the synthesized controller are ex-pressed as

(24)

(25)


where

The linear model (1)–(3) can then be constructed by incor-porating (24) and (25) into (20) and (21). The correspondingmatrix relationships are obtained as follows:

Consequently the proposed index can be calculated from thesynthesized linear model of the closed loop system. A tuningscheme based on this index is proposed in the next section tosimultaneously adjust parameters of PSSs and SDCs to dampinter-area oscillations and to optimize their control efforts undermultiple operating conditions.

IV. PROCEDURE OF CONTROLLER TUNING

A. Tuning Problem Formulation

Choosing variables to form should take into account twopoints: one is that inter-area modes should be sufficiently ob-served in so that minimizing will result in suppressionof inter-area oscillations; and another is that if the damping ofa mode (i.e., a local mode) deteriorates dramatically after con-troller tuning due to its poor visibility in , variables that sig-nificantly participate in this mode should be included in recon-struction of so that it can be considered in controller tuningand thus appropriately damped.

To ensure robustness of the controllers, typical multiple op-erating conditions are considered in the design. In this paper,parameters , and of PSSs and SDCs are as-sumed to be adjustable, while filter constant and washouttime constant are preset and remain fixed during the tuningprocess. Therefore, an optimization-based tuning scheme forcoordination of PSSs and SDCs is proposed as follows:

(26)

(27)

(28)

where is the number of operating conditions; is the weightof the th operating condition; is the damping ratio ofis a real positive number (2% in this paper) to ensure some smallsignal stability margin while the objective of system dampingcontrol is achieved by minimization of (26); and is the pa-rameter vector with the following definition:

and are lower and upper limits, respectively, of ;calculation of these limits is presented in the following.

Suppose there is a phase lead-lag block shown as follows:

(29)

If the phase of this block reaches maximum (or minimum) ofat frequency , then time constants and can be deter-

mined by

(30)

(31)

When the maximum and minimum compensated phases pro-vided by the block and frequencies of inter-area modes aregiven, upper and lower limits of and can be calculateddepending on (30) and (31), respectively. Upper and lowerlimits of the gain of the phase compensator can be calculatedby residue analysis [8].

B. Solving the Optimization Problem

The optimization problem (26)–(28) is a standard constrainednonlinear programming problem (NLP) solved in this paper bysequential quadratic programming (SQP), a highly effective andmatured method for the NLP [25].

Initial values of controller parameters used as a starting pointfor the SQP are given by the conventional sequential tuningmethod, which is also employed for comparison with the pro-posed tuning scheme [16]. Firstly, the compensated phase sup-plied by a controller to an inter-area mode is derived by residueanalysis in the nominal operating condition [12]. The gain ofthis controller is then increased gradually to enhance dampingof the mode while considering the control effort of the controlleras well as side effects on other modes. Each controller is tunedsequentially while the other already tuned controllers are online.

To depict the solving process more conveniently, (26)–(28)are expressed in a more general and compact form as follows:

(32)

(33)

(32)–(33) are solved by an iteration process based on the La-grangian function, constructed as follows:

(34)

where is the Lagrangian multiplier vector for . At thebeginning of the th iteration, the controller parameter vectorand the positive definite Hessian matrix (which is initiallyan identity matrix and is updated iteratively to finally convergeto the real Hessian matrix of the Lagrangian function) are avail-able. Then, the following steps are executed [25]:

S.1) Formulate and solve the following convex quadraticprogramming (QP) subproblem:

(35)

(36)


where is the search direction vector of parameters at theth iteration; and T are Hamilton and transpose opera-

tors, respectively.S.2) Based on , the controller parameter vector at the nextiteration is calculated as follows:

(37)

where is the optimal step length along the search di-rection and it can be determined by minimizing the fol-lowing merit function:

(38)

where is the penalty parameter vector.S.3) If the stopping criteria are satisfied, terminate the iter-ation process; otherwise, go to the next step.S.4) The new Hessian matrix is obtained by using thequasi-Newton method, as follows:

(39)

(40)

(41)

(42)

(43)

S.5) Set and go to the next iteration.

C. Selection of Design Parameters , and

Firstly, the appropriate initial guesses for , and aredetermined. From (6), (10), and (14), can be rewritten in analternative expression, as follows:

(44)

where and are th and th diagonalentries of and , respectively; is the -dimensionalvector and its th component is computed fromwhen setting , the rest of entries of arezeros and is the -dimensional vector andits th component is computed from whensetting , the rest of entries of are zerosand . Depending on the initial values of controllerparameters, and can be calculated. It is expectedthat all additive terms on the right-hand side of (44) can beweighted in the same order of magnitude in so that allcomponents of and can be equivalently considered inoptimization. Hence, the initial guesses for and aredetermined by simply setting these additive terms equal. Withthese initial guesses, can be calculated. Thus, the initialguesses for are derived by assuming that all additive termsin objective function (26) are equal as well.

Fig. 2. Control structure of Statcom.

Based on the initial guesses, a trial process is then performedto adjust , and until acceptable controller tuning re-sults for all operating conditions are derived [9]. The adjustmentof design parameters is according to the following heuristic prin-ciples: enhancing corresponding entries of may increasedamping of the system, while controller outputs would be signif-icantly constrained by larger entries of ; increasing couldgive more control priority to the th operating condition.

V. EXAMPLES OF APPLICATIONS OF THE PROPOSED

INDEX AND THE TUNING METHOD

A. Four-Machine Two-Area System

The classic four-machine two-area system is employed todemonstrate the proposed performance index and its applica-tion in simultaneous tuning of PSSs and SDCs. The diagram andparameters of this system can be found in [21]. A static com-pensator (Statcom) is installed at Bus 8 to maintain its voltage.The steady voltage at Bus 8 will rise to nearly 1.0 p.u. with re-active power support from the Statcom. The Statcom is mod-eled as a current injection always kept in perpendicular withbus voltage so that there is only reactive power exchange be-tween the grid and the Statcom (Fig. 2) [26]. The voltage reg-ulator of the Statcom is an inert block, with = 100 and= 0.005. The unit of time constants used in this paper is sec-onds, unless otherwise specified. The loads are modeled as acombination of constant impedances (50%) and induction mo-tors (50%). The 47-order open loop system, therefore, has 24state variables of generators (power angle, angular speed, d-axisand q-axis transient voltages, and d-axis and q-axis sub-transientvoltages), 16 state variables of excitation systems, one state vari-able of the Statcom and six state variables of induction motors(angular speed, and d-axis and q-axis voltages behind transientreactance).

Operating conditions considered for this system are as inTable I, where the first five are used for design while the lastone is applied for validation of robustness of the controllers.The loads remain fixed for all operating condition. Two localmodes in this system are well damped. However, an inter-areamode with frequencies at about 0.65 Hz residing between Area1 and Area 2 is poorly damped. Hence, an SDC is equipped inthe Statcom, together with a PSS installed in Generator 4, toprovide additional damping for inter-area oscillation. PSS isinstalled in Generator 4 on the basis of analysis of participationfactors.

Selection of wide-area feedback signals is based on residueanalysis. Large residues of a system’s input-output pair, with re-spect to a mode, indicate that this mode can be effectively con-trolled by the input-output pair using closed loop feedback con-trol [12]. Accordingly, it is found that active power in line 10-9 isthe most effective input signal for both PSS and SDC in damping


TABLE IOPERATING CONDITIONS FOR THE FOUR-MACHINE TWO-AREA SYSTEM

Fig. 3. Open loop frequency response of output of SDC to input of SDC.

inter-area oscillation and, therefore, it is chosen as the controlinput for these two controllers. The communication latency willbe around 20 ms for sending this signal to the remote SDC sitethrough a dedicated fiber-optic communication channel [27],[28]. Moreover, since the time required for phasor measurement(about three 60-Hz cycles or 50 ms [27]) and signal processingis also considered, the total delay of 80 ms in feedback signalfor the SDC is used in this design. Furthermore, the time delayin feedback signal for the PSS is assumed to be zero. The filtertime constant and the washout time constant are set to 0.01 and10, respectively, for both controllers.

Since the inter-area mode is mainly dominated by the rela-tive motions of generators between the two areas, output vectorsused to form the cost function are defined as follows:

(45)

(46)

where , and are power angles of Generators 1, 2, 3,and 4, respectively; and and are outputs of the SDC andthe PSS, respectively.

The open loop frequency response of output of the SDC toinput of the SDC is illustrated in Fig. 3. A 15-order reducedmodel is obtained through model reduction of the original47-order system and time delay is approximated by the Padeformula. Model reduction for each operating condition takesabout 0.082 s; all time consumption tests in this paper areconducted in a desktop computer with 2.66-GHz CPU and2G RAM. It is clearly seen that the approximated model canbe employed to accurately represent the full model within thefrequency range of interest.

Lower and upper boundaries LB and UB of controller param-eters are given in Table II. Here parameters ,and are corresponding to the SDC, while ,

Fig. 4. Searching process of SQP.

TABLE IICONTROLLER PARAMETERS

and are for the PSS. Initial as well as tuned values of theseparameters are also shown in the table. The searching processfor solution of (26)–(28) is illustrated in Fig. 4 and the time costfor finding this solution is about 1.74 s. It is clearly seen that theSQP is quite efficient in solving the proposed controller tuningproblem.

Eigenvalues of cost matrices are calculated when the PSS andthe SDC are simultaneously tuned by the proposed method andare sequentially tuned by the conventional method. The first foureigenvalues for each operating condition considered in the de-sign are depicted in Fig. 5. The first is the dominant one; itis much larger than the remaining. It is clear that this eigen-value is obviously reduced when the proposed controllers areinstalled, compared to when the system is equipped with se-quentially tuned controllers. Moreover, from (14), it is knownthat is defined as the sum of eigenvalues of the cost matrix,which means the proposed controllers will result in smallerthan that given by sequentially tuned controllers. Becauseis capable of indicating the system’s dynamic performance, it isnaturally inferred that simultaneously tuned controllers will leadto a better dynamic performance of the system than sequentiallytuned controllers. This is verified by computation of the closedloop system eigenvalues and the time domain simulations shownin the following.

The inter-area mode eigenvalues of the closed loop system forall operating conditions are presented in Table III. The compar-ison shows that the proposed controllers truly provide more ad-ditional damping to the inter-area mode than sequentially tunedcontrollers not only for operating conditions used for design butalso for the condition included in validation; damping of the


Fig. 5. Eigenvalues of cost matrices.

TABLE IIIEIGENVALUES OF INTER-AREA MODE

inter-area mode is enhanced considerably by the proposed con-trollers.

Because inter-area oscillations are adequately observed incan effectively measure the performance of these inter-

area oscillation dynamics according to the inference drawn inSection II. Minimization of suppresses inter-area oscilla-tions and thus inevitably increases damping of the inter-areamode. Accordingly, though the proposed index is directly re-lated to eigenvalues of the cost matrix, rather than the closedloop system, lowering it can actually enhance damping of closedloop system modes strongly associated with . This is verifiedas reduction of accords well with increase of damping of theinter-area mode shown above.

A three-phase short circuit fault occurs at Bus 6 when thesystem is in operating condition 5 and the fault is cleared 50 mslater. The power angle oscillations between Generators 2 and 4are shown in Fig. 6. It is seen that inter-area oscillation decaysquite fast when the system is equipped with simultaneouslytuned controllers, compared to the marginally stable systemwithout controllers and the system with sequentially tuned con-trollers. Outputs of PSS and SDC are also optimized when theyare coordinately designed by the proposed method (Fig. 7). Thelimits for outputs of PSS and SDC are set to and ,respectively. Therefore, together with the above eigen-analysis,these results indicate that the proposed index is an effectivemeasurement of performance of system dynamics as well ascontrol efforts. Furthermore the proposed simultaneous tuningmethod based on this index for coordination of PSS and SDCcan well damp inter-area oscillation under multiple operatingconditions.

Fig. 6. Power angle oscillations (solid line: proposed; dot line: conventional;dash line: no controller).

Fig. 7. Control signals (solid line: proposed; dot line: conventional).

Fig. 8. New England and New York interconnected system.

B. New England and New York Interconnected System

The equivalent (16-machine five-area) model of New Eng-land and New York interconnected networks (Fig. 8) is em-ployed to demonstrate performance of the proposed controllerdesign including multiple inter-area modes. This system is amodified version of [12]: the quite large mechanical dampingcoefficients in the original model have been removed from somegenerators, and local PSSs are then installed in some generatorsto damp their local mode oscillations. A thyristor controlled se-ries capacitor (TCSC) is installed in transmission line 50-18 tocompensate 50% of its reactance in the steady state. The allow-able TCSC dynamic compensation is from 10% to 90% of linereactance. The dynamic model of the TCSC is shown in Fig. 9[11], where time constant is chosen to be 10 ms. A com-bination of constant impedances (50%) and induction motors(50%) is employed to model the loads. Thus, there are 96 statevariables of generators (the same six-order model as in the firstexample), 64 state variables of excitation systems, 18 state vari-ables of local PSSs, one state variable of TCSC, and 57 state


Fig. 9. Dynamic model of TCSC.

TABLE IVOPERATING CONDITIONS FOR THE 16-MACHINE FIVE-AREA SYSTEM

variables of induction motors in this system. By changing net-work configurations and transmitting different levels of powerfrom area A1 to A2, nine typical operating conditions (Table IV)are considered in this study.

Eigen-analysis shows that for all operating conditions, thereare two quite poorly damped inter-area modes in this system:M1 and M2. M1 with frequency at about 0.65 Hz is dominatedby the oscillation between generators in areas A1 and A2, whileM2 with frequency at about 0.32 Hz depicts oscillations of gen-erators in areas A1 and A2 with respect to generators in the restof the system.

To provide additional damping to the two inter-area modes,an SDC equipped in TCSC is designed. Meanwhile since Gen-erator 13 greatly participates in both inter-area modes, a PSS isinstalled in Generator 13 to work in coordination with the SDCfor damping control. Furthermore according to residue analysis,active power in transmission line 13–17 is selected as controlinput for both PSS and SDC. Time delay is assumed to be 80ms for transmitting the signal to the remote SDC. Participationfactor analysis shows that Generators 13, 5, and 6 greatly partic-ipate in M1, while Generators 14, 15, and 13 greatly participatein M2. Therefore output vectors are formed as follows:

(47)

(48)

where , and are power angles of Generators5, 6, 13, 14, and 15, respectively; and and are outputs ofSDC and PSS, respectively. The first six operating conditions inTable IV are used for design, while the last three are applied forvalidation. It costs about 3.29 s to reduce the 236-order openloop power system model to a 28-order reduced model whichcan lead to satisfactory results of controller tuning by the pro-posed method.

The lower and upper boundaries, the initial values, and thetuned values of controller parameters are shown in Table V.Here parameters , and are for the SDC,while , and are for the PSS. Specifically, it

TABLE VCONTROLLER PARAMETERS

Fig. 10. Eigenvalues of cost matrices.

takes about 15 iterations and 6.18 s for the SQP method to con-verge. It is found that the controller parameter searching processfor this much higher order system is still as efficient as thatin the first example, although the time cost (3.29 s) for modelreduction in this system is relatively larger, compared to that(0.082 s) in the first small system. This means that the proposedtuning scheme can be applied to large scale power systems dueto the efficient reduced-order model-based controller parametersearching process.

Eigenvalues of cost matrices for the first six operatingconditions are illustrated in Fig. 10. It is obvious that theyare reduced by optimization. According to the verificationshown in the first example, the proposed controllers performbetter than the sequentially tuned controllers. This is indeedconfirmed again by facts shown in Table VI, that the proposedcontrollers provide more damping to both inter-area modes thanthe sequentially tuned controllers. The two inter-area modesare sufficiently damped for all operating conditions when theproposed controllers are installed.

A three-phase short circuit fault occurs at Bus 60 when thesystem is in operating condition 1. The fault is cleared 100 mslater by tripping one of tie-lines between Bus 60 and 61. Oscil-lations of relative power angles and active power in key tie-linesare depicted in Figs. 11 and 12, respectively. The compensationpercentage provided by the TCSC and the output of the PSSduring the dynamics are illustrated in Fig. 13. It is clear that si-multaneously tuned controllers outperform sequentially tunedcontrollers, and the two inter-area oscillations are well damped


TABLE VIEIGENVALUES OF INTER-AREA MODE

Fig. 11. Power angle oscillations (solid line: proposed; dot line: conventional;dash line: no controller).

Fig. 12. Oscillations of active power in key tie-lines (solid line: proposed; dotline: conventional; dash line: no controller).

by coordinately tuned PSS and SDC with optimum control out-puts.

Fig. 13. Dynamics of controllers (solid line: proposed; dot line: conventional).

VI. CONCLUSION

A novel eigenstructure-based performance index is proposedto solve structurally constrained control problems. This indexcan measure performance of system dynamics as well as controlefforts. Calculation of this index has no bearing on control struc-tures; it is based only on eigenstructure of the closed loop systemand the design parameters. This index has been applied to aproposed optimization-based tuning scheme used for coordina-tion of structurally constrained PSSs and SDCs to damp inter-area oscillations and to optimize their control efforts under mul-tiple operating conditions. Applications of the proposed con-trol design method on a four-machine two-area system and theNew England and New York interconnected system have beendemonstrated. Results show that the proposed index is an ef-fective way of measuring system’s dynamic performance andreducing this index can improve the system dynamics. Struc-turally constrained PSSs and SDCs simultaneously tuned by theproposed method can effectively damp inter-area oscillationsand their control efforts are also optimized.

Future works will include applying the proposed dampingcontroller tuning scheme to real large scale power networks inMainland China, and also extending application of the proposedindex and control design method to other control systems al-though this paper focuses only on application of the proposedmethod to PSSs and SDCs in power systems.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions to this research.

REFERENCES

[1] I. Kamwa, R. Grondin, and G. Trudel, “IEEE PSS2B versus PSS4B:The limits of performance of modern power system stabilizers,” IEEETrans. Power Syst., vol. 20, no. 2, pp. 903–915, May 2005.

[2] C. Y. Chung, K. W. Wang, C. T. Tse, and N. Riu, “PSS design byprobabilistic sensitivity indices,” IEEE Trans. Power Syst., vol. 17, no.3, pp. 688–693, Aug. 2002.

[3] H. G. Far, H. Banakar, P. Li, C. Luo, and B. T. Ooi, “Damping interareaoscillations by multiple model selectivity method,” IEEE Trans. PowerSyst., vol. 24, no. 2, pp. 766–775, May 2009.


[4] U. P. Mhaskar and A. M. Kulkarni, “Power oscillation damping usingFACTS devices: Modal controllability, observability in local signals,and location of transfer function zeros,” IEEE Trans. Power Syst., vol.21, no. 1, pp. 285–294, Feb. 2006.

[5] R. A. Ramos, A. C. P. Martins, and N. G. Bretas, “An improvedmethodology for the design of power system damping controllers,”IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1938–1945, Nov. 2005.

[6] A. C. Zolotas, B. Chaudhuri, I. M. Jaimoukha, and P. Korba, “A studyon LQG/LTR control for damping inter-area oscillations in power sys-tems,” IEEE Trans. Control Syst. Technol., vol. 15, no. 1, pp. 151–160,Jan. 2007.

[7] B. Chaudhuri, S. Ray, and R. Majumder, “Robust low-order controllerdesign for multi-modal power oscillation damping using flexible ACtransmission system devices,” IET Gen. Transm. Distrib., vol. 3, no. 5,pp. 448–459, 2009.

[8] R. V. de Oliveira, R. Kuiava, R. A. Ramos, and N. G. Bretas, “Au-tomatic tuning method for the design of supplementary damping con-trollers for flexible alternating current transmission system devices,”IET Gen. Transm. Distrib., vol. 3, no. 10, pp. 919–929, 2009.

[9] D. Dotta, A. S. Silva, and I. C. Decker, “Wide-area measurements-based two-level control design considering signal transmission delay,”IEEE Trans. Power Syst., vol. 24, no. 1, pp. 208–216, Feb. 2009.

[10] R. A. Ramos, L. F. C. Alberto, and N. G. Bretas, “A new methodologyfor the coordinated design of robust decentralized power systemdamping controllers,” IEEE Trans. Power Syst., vol. 19, no. 1, pp.444–454, Feb. 2004.

[11] B. Chaudhuri and B. C. Pal, “Robust design of multiple swing modesemploying global stabilizing signals with a TCSC,” IEEE Trans. PowerSyst., vol. 19, no. 1, pp. 499–506, Feb. 2004.

[12] B. Pal and B. Chaudhuri, Robust Control in Power Systems. NewYork: Springer, 2005.

[13] Z. Wang, C. Y. Chung, K. P. Wong, C. T. Tse, and K. W. Wang, “Ro-bust power system stabilizer design under multioperating conditionsusing differential evolution,” IET Gen. Transm. Distrib., vol. 2, no. 5,pp. 690–700, 2008.

[14] I. Kamwa, G. Trudel, and L. Gerin-Lajoie, “Robust design and coor-dination of multiple damping controllers using nonlinear constrainedoptimization,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 1084–1092,Aug. 2000.

[15] D. Z. Fang, S. Q. Yuan, Y. J. Wang, and T. S. Chung, “Coordinatedparameter design of STATCOM stabilizer and PSS using MSSA algo-rithm,” IET Gen. Transm. Distrib., vol. 1, no. 4, pp. 670–678, 2007.

[16] L. J. Cai and I. Erlich, “Simultaneous coordinated tuning of PSS andFACTS damping controllers in large power systems,” IEEE Trans.Power Syst., vol. 20, no. 1, pp. 294–300, Feb. 2005.

[17] N. Kshatriya, U. D. Annakkage, F. M. Hughes, and A. M. Gole, “Op-timized partial eigenstructure assignment-based design of a combinedPSS and active damping controller for a DFIG,” IEEE Trans. PowerSyst., vol. 25, no. 2, pp. 866–876, May 2010.

[18] A. J. A. S. Costa, F. D. Freitas, and H. E. Pena, “Power system stabilizerdesign via structurally constrained optimal control,” Elect. Power Syst.Res., vol. 33, no. 1, pp. 33–40, Apr. 1995.

[19] A. J. A. S. Costa, F. D. Freitas, and A. S. e Silva, “Design of decentral-ized controllers for large power systems considering sparsity,” IEEETrans. Power Syst., vol. 12, no. 1, pp. 144–152, Feb. 1997.

[20] J. J. Sanchez, V. Vittal, M. J. Gibbard, A. R. Messina, D. J. Vowles, S.Liu, and U. D. Annakkage, “Inclusion of higher order terms for small-signal (modal) analysis: Committee report-task force on assessing theneed to include high order terms for small-signal (modal) analysis,”IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1886–1904, Nov. 2005.

[21] P. Kundar, Power System Stability and Control. New York: McGraw-Hill, 1994.

[22] F. D. Freitas, J. Rommes, and N. Martins, “Gramian-based reductionmethod applied to large sparse power system descriptor models,” IEEETrans. Power Syst., vol. 23, no. 3, pp. 1258–1270, Aug. 2008.

[23] I. Kamwa, G. Trudel, and L. Gerin-Lajoie, “Low-order black-boxmodels for control system design in large power systems,” IEEE Trans.Power Syst., vol. 11, no. 1, pp. 303–311, Feb. 1996.

[24] M. G. Safonov and R. Y. Chiang, “A Schur method for balancedmodel reduction,” IEEE Trans. Autom. Control, vol. AC-34, no. 7, pp.729–733, Jul. 1989.

[25] J. Nocedal and S. J. Wright, Numerical Optimization. New York:Springer, 2006.

[26] M. H. Haque, “Improvement of first swing stability limit by utilizingfull benefit of shunt FACTS devices,” IEEE Trans. Power Syst., vol. 19,no. 4, pp. 1894–1902, Nov. 2004.

[27] C. W. Taylor, D. C. Erickson, K. E. Martin, R. E. Wilson, and V.Venkatasubramanian, “WACS-Wide-area stability and voltage controlsystem: R&D and online demonstration,” Proc. IEEE, vol. 93, no. 5,pp. 892–906, May 2005.

[28] J. W. Stahlhut, T. J. Browne, G. T. Heydt, and V. Vittal, “Latencyviewed as a stochastic process and its impact on wide area powersystem control signals,” IEEE Trans. Power Syst., vol. 23, no. 1, pp.84–91, Feb. 2008.

D. P. Ke received the B.S. and M.S. degrees inelectrical engineering from Huazhong Universityof Science and Technology, Wuhan, China, in 2005and 2007, respectively. Currently, he is pursuing thePh.D. degree in electrical engineering at The HongKong Polytechnic University, Hong Kong, China.

His research interests are in power system dy-namics and control.

C. Y. Chung (M’01–SM’07) received the B.Eng. de-gree (with First Class Honors) and the Ph.D. degreein electrical engineering from The Hong Kong Poly-technic University, Hong Kong, China, in 1995 and1999, respectively.

After his Ph.D. graduation, he worked in theElectrical Engineering Department at the Universityof Alberta, Edmonton, AB, Canada, and PowertechLabs, Inc., Surrey, BC, Canada. Currently, he is theConvenor of the Power Systems Research Groupand an Associate Professor in the Department of

Electrical Engineering, The Hong Kong Polytechnic University. His researchinterests include power system stability/control, planning and operation,computational intelligence applications, and power markets.

Dr. Chung was the Chairman of the IEEE Hong Kong Joint Chapter ofPES/IAS/PELS/IES in 2007–2009. During his tenure as chairman, the JointChapter received the 2008 IEEE PES Outstanding Small Chapter Award andthe 2009 IEEE IAS Outstanding Small Joint Chapter Award. He was theTechnical Chairman of IET APSCOM2009 International Conference, andHonorary Secretary of IEEE DRPT2004 International Conference and IEEEIAS 2005 Annual Meeting. Currently, he is the Chairman of the IEEE HongKong Section.

Yusheng Xue (M’87) is an academician of the Chi-nese Academy of Engineering (CAE). He is also theHonorary President of State Grid Electric Power Re-search Institute (SGEPRI or NARI), China. He holdsthe positions of Adjunct Professor in many universi-ties in China.

Mr. Xue is a member of the PSCC Council and hasbeen the Chairman of Technical Committee of Chi-nese National Committee of CIGRE since 2005.

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