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THESIS FOR THE DEGREE OF LICENTIATE OF ENGINEERING

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ADRIAN ANDREOIU

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Department of Electric Power EngineeringCHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden2002

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Genetic Algorithm Based Design of Power System Stabilizers

ADRIAN ANDREOIU

© ADRIAN ANDREOIU, 2002

Technical Report no. 431LDepartment of Electric Power EngineeringChalmers University of TechnologySE-412 96 GöteborgSweden

Telephone: +46 (0)31 772 1632Fax: +46 (0)31 772 1633http://www.elteknik.chalmers.se

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Chalmers Bibliotek, ReproserviceGöteborg, Sweden 2002

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For large scale power systems comprising many interconnected machines, theproblem of power system stabilizer (PSS) parameter tuning is a complex exercisedue to the presence of several poorly damped modes of oscillation. The problem isfurther complicated by continuous variation in power system operating conditions.In the simultaneous tuning approach, exhaustive computational tools are requiredto obtain optimal parameter settings for the PSS, while in case of sequential tuning,although the computational burden is lesser, evaluating the tuning sequence is anadditional requirement. There is a further problem of eigenvalue drift.

Genetic Algorithms (GA) are global search techniques providing a powerful toolfor optimization problems by miming the mechanisms of natural selection andgenetics. These operate on a �� of potential solutions (� �� )applying the principle of �� to produce better and betterapproximations to a solution. In each generation, a new set of approximations iscreated by selecting the individuals according to their level of fitness in theproblem domain, and breeding them together using operators borrowed fromnatural genetics. Thus, the population of solutions is successively improved withrespect to the search objective by replacing least fit individuals with new ones(offspring of individuals from the previous generation), better suited to theenvironment, just as in natural evolution.

This thesis proposes a novel approach using the powerful properties of GA, tosimultaneously optimize the PSS parameter settings. The classical Lyapunov’sparameter optimization method using an Integral of Squared Error Criterion hasbeen integrated with the GA search objective function. The method has been usedfor tuning of lead-lag-, derivative- and PID-PSS, and has been found to performvery satisfactorily.

The thesis also examines classical approaches to tuning of lead-lag and derivativePSS that consider one nominal operating condition. Investigations reveal that theclassical approach does provide satisfactory performances for operating conditionsup to the nominal but deteriorated responses when the load increases. However, the

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classically tuned PSS fails to stabilize the system at certain operating conditions.The proposed GA based method on the other hand, provides the option of includingany operating point within its tuning domain, thus ensuring system stability over alarge domain, and in particular, the tuning domain.

�� genetic algorithm, power systems, small-signal stability, power systemstabilizers, design

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The work presented in this thesis has been carried out at the Department of ElectricPower Engineering of Chalmers University of Technology. This research projecthas started in 1999 and has been funded through the Elektra program of Elforsk.The financial support is gratefully acknowledged.

I wish to express here my sincere gratitude to my supervisor, Assoc. Prof. KankarBhattacharya, whose precious advice and friendly encouragement made this workgo smoothly throughout this time. Many thanks go to my examiner, Professor JaapDaalder, for his support and useful discussions.

I gratefully acknowledge the constructive criticism, the guidance and valuableinputs that Dr. Bo Eliasson of Malmö Högskola and Dr. Sture Torseng of ABBPower Systems have contributed with during the course of this work, as membersof the Project Reference Group.

The colleagues at the department in general, and the Power Systems Group inparticular create a supportive and friendly working environment. They made thecoffee break worth more than a coffee during the break.

I owe my deepest gratitude to my family, and to Corina especially, for their love,understanding and support that gave me the comfort I needed to fully concentrateon my work.

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1. A. Andreoiu, K. Bhattacharya - "A Lyapunov’s method Based GeneticAlgorithm for a Robust PSS Tuning", Euro Conference on Risk Management inPower Systems Planning and Operation in Market Environment, Porto, Portugal,September 8-11, 2001.

2. A. Andreoiu, K. Bhattacharya - "Lyapunov’s Method Based Genetic Algorithmfor Multi-machine PSS Tuning", IEEE PES Winter Meeting, New York City, NewYork, January 27-31, 2002.

3. A. Andreoiu, K. Bhattacharya - "Robust Tuning of Power System StabilizersUsing Lyapunov Method Based Genetic Algorithm", IEE Proceedings – Gene-ration, Transmission and Distribution – accepted for publication.

4. A. Andreoiu, K. Bhattacharya - "Genetic Algorithm Based Tuning of PIDPower System Stabilizers", The 14th Power Systems Computation Conference,Sevilla Spain, June 24-28, 2002.

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1.1 BACKGROUND........................................................................................................... 11.2 BRIEF REVIEW OF LITERATURE ON PSS DESIGN....................................................... 31.3 GENETIC ALGORITHMS IN PSS TUNING .................................................................... 71.4 OUTLINE OF THE THESIS ........................................................................................... 9

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2.1 BACKGROUND......................................................................................................... 112.2 GENERAL APPROACH............................................................................................... 112.3 SMALL PERTURBATION DYNAMIC MODEL OF THE SYSTEM .................................... 132.4 SYSTEMS INVESTIGATED......................................................................................... 152.5 SINGLE MACHINE INFINITE BUS (SMIB) SYSTEM .................................................. 172.6 MULTI-MACHINE SYSTEM ...................................................................................... 192.7 CONCLUDING REMARKS ......................................................................................... 22

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3.1 GENERAL CHARACTERISTICS OF LEAD-LAG POWER SYSTEM STABILIZERS ............ 23&'!'! ��(��)�*�� ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' $&&'!'$ ��(��+�� ,�� -��(�� ''''''' $.

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3.4 CONCLUDING REMARKS ......................................................................................... 39

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4.1 GENERAL ASPECTS ................................................................................................. 417'!'! ��(�� '''''''''''''''''''''''' 7$

4.2 MATHEMATICAL MODEL OF THE SYSTEM............................................................... 444.3 SINGLE MACHINE CONNECTED TO INFINITE BUS .................................................... 457'&'! �� ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 7/7'&'$ ��(��(��(�� ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 6&

4.4 MULTI-MACHINE POWER SYSTEM........................................................................... 574.5 CONCLUDING REMARKS ......................................................................................... 63

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5.1 INTRODUCTION ........................................................................................................656'!'! ��(�� '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''.4

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5.3 MULTI-MACHINE POWER SYSTEM ANALYSIS ..........................................................725.4 CONCLUDING REMARKS ..........................................................................................76

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6.1 SALIENT FEATURES OF THE PRESENT WORK.............................................................786.2 SCOPE FOR FUTURE WORK IN THIS AREA ..................................................................78

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8.1 APPENDIX I ..............................................................................................................854'!'! 9��:;� �� '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''464'!'$ �(��-�� (�''''''''''''''''''''''''''''''''''''''''''''4/4'!'& ��(��+��+�(�� ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''444'!'7 �-�� ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''<"

8.2 APPENDIX II - SYSTEM DATA ..................................................................................934'$'! ��( ��'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''<&4'$'$ ��( ��&-(��(=>�++&-(��#<-��? '''''''''''''''''''''''<7

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� state (plant) matrix of the system� control matrix� damping coefficient� perturbation matrix� active power@ reactive power2 weighing matrixA inertia constant��#�� direct and quadrature components of armature current� inertia coefficient� rotor angle� denotes small perturbation or deviation of a variable from its

steady-state value� angular speedB�#B� synchronous reactances in and 3 axes, respectivelyB�5 direct axis transient reactanceB� quadrature axis reactance8�� equivalent excitation voltage (field circuit voltage)8�5 internal voltage behind transient reactance B�C� frequencyD performance index��#2� AVR gain and time constant, respectively�� lead-lag PSS gain�d derivative PSS gain��#��#�� proportional, derivative and integral components of PID-PSS2 Washout time constant

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2#2� mechanical and electrical torque, respectively2��5 time constant of excitation circuit2 sampling timeE�#E� direct and quadrature components of terminal voltage� admittance matrix2�#F2� lead-lag PSS time constantsx state vectorp perturbation vector� Laplace operator� time� stabilizing signal� transpose

ac alternative currentANN Artificial Neural NetworksAVR Automatic Voltage Regulatordc direct currentGA Genetic AlgorithmGEP Generator Exciter Power systemISE Integral of Squared ErrorMIMO Multi Input Multi OutputOLS Orthogonal Least SquaresPSS Power System Stabilizer(s)PID Proportional Integral DerivativeRBF Radial Basis FunctionRLS Recursive Least SquaresSMIB Single Machine to Infinite BusSISO Single Input Single OutputTGR Transient Gain ReductionVSC Variable Structure ControlVSCPSS Variable Structure PSS

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��34��&%& Power System stability classification [1]

a) �� - representing the ability of the system to maintain synchronism;b) E�� - representing the ability of the system to maintain steady

acceptable voltage.

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This work focuses on the angle stability category, with particular reference to thesmall-signal stability problem.

Amongst the different types of angular stability problems, the transient stabilityproblem is related to the short term or transient period, which is usually limited tothe first few seconds following the disturbance. It is concerned with the systemresponse to a severe disturbance, such as transmission system fault. Much of theelectric utility effort and interest related to system stability have been concentratedon the short-term response, and therefore the system is designed and operated so asto meet a set of reliability criteria concerning transient stability. Well-establishedanalytical techniques and computer programs exist for the analysis of transientstability.

Small signal stability on the other hand is concerned with the system response tosmall changes and is a fundamental requirement for the satisfactory operation ofpower systems. Usually, the problem is one of ensuring sufficient damping ofsystem oscillations. Small signal stability can be analyzed by linearizing the systemabout an equilibrium point represented by a steady state operating condition. Thisallows the use of powerful analytical tools of linear systems to determine thestability characteristics, which aid in the design of corrective controls.

In the past, many utilities took small signal stability for granted and carried out nostudies at all to reveal problems related to small signal performance. This wasprimarily because in the past, a system that remained stable for the first fewseconds following a severe disturbance was sure to be stable for smallperturbations about the post fault system condition. This is not true for present daysystems. As power systems have developed, the need for small signal studies andmeasures to ensure sufficient stability margins has being recognized.

Attention has been focused on the effect of excitation control on the damping ofoscillations, which characterizes the phenomena of stability. In particular it hasbeen found useful and practical to incorporate transient stabilizing signals derivedfrom speed, terminal frequency or accelerating power superimposed on the normalvoltage error signal to provide for additional damping to these oscillations. Suchdevices are known as Power System Stabilizers (PSS).

Power system stabilizers have been extensively used as supplementary excitationcontrollers to damp out the low frequency oscillations and enhance the overallsystem stability. The PSS extends the system stability limits by modulatinggenerator excitation to provide damping to the oscillations of synchronous machinerotors relative to one another. They produce a component of torque in phase withrotor speed deviations, in order to enhance the system damping.

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�� Over the last four decades, a large number of research papers have appeared in thearea of PSS. Research has been directed towards obtaining such a PSS that canprovide an optimal performance for a wide range of machine and systemparameters. However, as noted in [2], a universally applicable stabilizing functionis not practically feasible. Various control strategies and optimization techniqueshave found their applications in this area as also various degrees of systemmodeling have been attempted. While it is difficult to bring out a detaileddiscussion of the historical development of PSS and its applications, a modestattempt has been made in this section to discuss the most significant works in thearea.

Heffron and Phillips [3] analyzed the effect of modern amplidyne voltageregulators on under-excited operation of large turbine generators. They were thefirst to present the small perturbation model in terms of ��-�� constants of amachine-infinite bus system. Their investigations revealed that the use of moderncontinuously acting regulators greatly increased the steady-state stability limit ofturbine generators in the under-excited region. And that the trend towards loweringthe short circuit ratio of large turbine generators was sound from steady statestability standpoint, provided a modern continuously acting voltage regulator wasused.

DeMello and Concordia [2] examined the case of a single machine connected to aninfinite bus through external reactance. Their analysis developed insights into theeffects of thyristor-type excitation systems and established understanding of thestabilizing requirements for such systems. These stabilizing requirements includedthe voltage regulator gain parameters as well as the PSS parameters. They exploredthe effect of a variety of machine loading, inertia and system external impedance(length of the transmission line) on damping characteristics of voltage or speedfollowing a small perturbation in mechanical torque. They developed someunifying concepts that explained the stability phenomena of concern and predicteddesirable phase and magnitude characteristics of stabilizing functions.

Larsen and Swann [4] presented application of PSS utilizing either speed,frequency or power input signals. Guidelines were presented for tuning PSS thatenable the user to achieve desired dynamic performance with limited effort. Theneed for torsional filters in the PSS path for speed input PSS was also discussed.

Kundur �� ' [5] described the details of a ��-�-)(�� PSS design forgenerating units in Ontario Hydro. Two alternate excitation schemes wereconsidered, one with and the other without Transient Gain Reduction (TGR). It wasshown that with appropriate selection of PSS parameters, both schemes provided

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satisfactory performance. Appropriate choice of washout time-constant, PSS outputlimits and phase-lead compensation circuit parameters was demonstrated.

Yu and Siggers [6] presented the application of state-feedback optimal PSS whileMoussa and Yu [7] proposed an eigenvalue shifting technique for determining theweighing matrix in the performance index. The technique involved shifting of thedominant eigenvalue to the left, on the �-plane until a satisfactory shift is made orthe controller’s practical limit is reached. The optimal state-feedback controllerswere also applied to a multi-machine system. However, in spite of thepowerfulness of optimal control theory, the controllers so achieved, failed to appealto utilities because their realization was difficult, cumbersome and costly.

A lot of work has also been reported on coordinated tuning of PSS for multi-machine systems. DeMello ��' [8] presented an eigenvalue-eigenvector analysisto identify the most effective generating units to be equipped with PSS in multi-machine systems that exhibit dynamic instability and poor damping of severalinter-area modes of oscillations.

Fleming �� ' [9] proposed a sequential eigenvalue assignment algorithm forselecting the parameters of stabilizers in a multi-machine power system. Insequential tuning, the stabilizer parameters are computed using repeatedapplication of single-input/single-output (SISO) analysis. The suggested approachenables the selection of parameters of stabilizers such that a specified improvementin the damping ratio of each poorly damped mode can be realized approximately.The stabilizers are applied sequentially at different locations as ascertained bymodal analysis outlined by DeMello ��' [8]. However, it should be noted that thesequential addition of stabilizers disturbs the previously placed eigenvalues tosome extent.

Abdalla ��'[10] also presented a procedure for the selection of the most effectivemachines for stabilization. They suggested the addition of a damping term to eachmachine’s equation of motion, one at a time. An eigenvalue-based measure ofrelative improvement in the damping of oscillatory modes is implemented and usedas a criterion to find the best candidate machine for stabilizer application.

The sequential tuning methods discussed in [8]-[10] are computationally simplecompared to the simultaneous tuning methods, but they incur eigenvalue driftwithin the sequence. The eigenvalue drift problem does not arise with simultaneoustuning methods, and thus they provide the true optimal solution, but, on the otherhand, these methods are computationally expensive.

Doi and Abe [11] proposed the coordinated design/tuning of PSS in multi-machinesystem by combining eigenvalue sensitivity analysis and linear programming. ThePSS parameters are determined by minimizing a performance index, which is the

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sum of all PSS gains. This method is simultaneously able to select generatorswhere PSS can be effectively applied and to synthesize the adequate transferfunction of the PSS for these generators.

Lim and Elangovan [12],[13] presented a method for designing decentralizedstabilizers in a multi-machine system using complex frequency domain approach.Using this approach, the PSS parameters are obtained so that some or all of thesystem mechanical mode eigenvalues may be placed at the prescribed locations inthe �-plane. The problem of exact eigenvalue assignment is transformed to that ofsolving iteratively a set of equivalent characteristic equations, the solution of whichare the desired stabilizer parameters.

In all the above works we discussed, the PSS structure was considered to be fixedand were tuned considering a set of nominal operating conditions and systemparameters.

However, such a fixed structure optimum PSS would provide sub-optimumperformance under variations in system parameters and operating conditions.Several modern control strategies such as self-tuning control, variable-structurecontrol (VSC), fuzzy-logic systems (FLS), artificial neural networks (ANN),genetic algorithms (GA), ��' have been reported in the recent literature, aiming todevelop robust PSS configurations.

The applicability, advantages and disadvantages of minimum variance, poleassigned, linear quadratic and pole shifting adaptive controllers for power systemswere examined in detail by Ghosh �� ' [15]. They presented a comparison ofsystem dynamic performances obtained using three alternate PSS, �'�', adaptivepole-shifting, adaptive linear quadratic and a conventional PSS. Their studies showthat the adaptive pole-shifting PSS provides the best performance.

Cheng �� ' [16] presented an adaptive PSS using a self-searching pole-shiftingcontrol algorithm. The adaptive PSS so designed is effective in damping systemoscillations under both small as well as large perturbations. Cheng �� [17] furtherproposed a dual-rate adaptive PSS based on self-searching pole-shifting algorithmfor damping multi-mode oscillations. In this algorithm the system parameters areidentified every 80 msec while the control signal is updated every 20 msec.

Lim [18] proposed a method for designing a self-tuning PSS based on theminimization of a quadratic performance index. The effectiveness of the self-tuning PSS for either excitation or governor control under different disturbancesand over a wide range of operating conditions has been demonstrated.

As an alternative to self-tuning PSS, Variable Structure PSS (VSPSS) has beenproposed in the literature in order to counteract the problem of variation of system

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parameters and operating condition. The VSC are insensitive to system parametervariations and can easily be realized using microcomputers. A systematicprocedure for the selection of the proper switching vector is though very importantfor their design.

Chan and Hsu [23] proposed an optimal VSPSS for a machine-infinite bus systemas well as for a multi-machine system. The proposed VSPSS is optimal in the sensethat the switching hyperplane is obtained by minimizing a quadratic performanceindex, in the sliding mode operation. The resulting switching vector and hence theswitching hyperplane depends on the weighing matrices associated with theperformance index.

Kothari ��' [24] have presented the design of a VSPSS with desired eigenvaluesin the sliding mode, where the switching hyperplane is obtained using a poleplacement method. This has been further extended in [25] to apply a radial pole-shifting technique for design of VSPSS in the discrete-mode.

A fuzzy set theory based PSS was reported by Hsu and Cheng in [26]. Theproposed stabilizer adopted a decentralized output feedback control law thatrequired only local measurements within each generating unit, thus providingscope for further implementation.

In [27], Hoang and Tomsovic proposed a systematic approach to fuzzy logiccontrol design, where the controller parameters are either calculated off-line orcomputed in real time in response to system changes. In this design approach it wasshown that the controller is insensitive to the precise dynamics of the system.

Artificial neural network is based on the concept of parallel processing and hasgreat ability in realizing complicated non-linear mappings from the input space tothe output space, thus providing an extremely fast processing facility forcomplicated non-linear problems.

Zhang �� ' [28] presented a PSS design approach that employs the multi-layerperceptron with error back-propagation training method. The ANN was trainedwith the training data group generated by an adaptive power system stabilizer.

In [29], Segal ��' presented a new approach for real-time tuning of conventionalPSS using a radial basis function network, which is trained using an orthogonalleast squares (OLS) learning algorithm.

Abido and Abdel-Magid presented in [30] a PSS design that combines numerical(ANN) and linguistic (FLS) information in a uniform fashion, thus providing amodel-free description of the control system and overcoming the ANN and FLSweaknesses and facilitates on-line implementation.

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�� !��Genetic Algorithms are global search techniques providing a powerful tool foroptimization problems by miming the mechanisms of natural selection andgenetics. These operate on a population of potential solutions applying theprinciple of �� to produce better and better approximations to asolution. In each generation, a new set of approximations is created by selecting theindividuals according to their level of fitness in the problem domain and breedingthem together using operators borrowed from natural genetics [31]. Thus, thepopulation of solutions is successively improved with respect to the searchobjective by replacing least fit individuals with new ones (offspring of individualsfrom the previous generation), better suited to the environment, just as in naturalevolution.

According to Goldberg [31], GA are different from other optimization and searchprocedures in four ways:

1. GA work with a coding of the parameter set, not the parameters themselves.2. GA search from a population of points, not a single point.3. GA use payoff information, not derivatives or other auxiliary knowledge4. GA use probabilistic transition rules, not deterministic rules.

Figure 1.2 shows a schematic diagram of a genetic algorithm. The processcommences with random generation of a pool of possible solutions, �'�' thepopulation and the individuals that form it. Each individual in the population, alsocalled ��(��(� is represented by a string, which is formed by a number of sub-strings equal to the number of the problem’s variables. Each variable is coded in asuitable coding system (binary, integer, real-valued, ��). The population size andthe chromosome size are kept constant during the whole search process.

The performance of each individual in the population is evaluated through anobjective function, which models the dynamic problem and has as output a ��' The fitness value is a measure of how good the respective individual is withrespect to the problem objective.

Individuals will be selected in accordance with their fitness value to take part in thegenetic process. The purpose of �� is to keep the best well-fit individuals andincrease the number of their offspring in the next generation, on the account of theleast fit individuals.

The ��(�� process consists in the grouping of the selected individuals inpairs (�'�' parents) in which they exchange genetic information forming two newindividuals (�'�' children, or offspring). This process helps the optimization searchto escape from possible local optima and search different zones of the search space.

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A (�� genetic operator that replaces allele of genes is implemented to increasethe probability of complete search, by allowing the investigations in vicinity oflocal optima.

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;�� is the process in which children will populate the next generation byreplacing parents. Reinsertion can be made partially or completely, uniformly(offspring replace parents uniformly at random) or fitness-based.All genetic operators are implemented with a certain predefined probability.

Power System Stabilizers designed using the GA based search and optimization aremore likely to converge to a global optima than a conventional optimization basedPSS, since they search from a population of possible solutions, and are based onprobabilistic transition rules.

In the recent literature, applications of genetic algorithm to tune the parameters ofPSS have been reported [32]-[35]. A GA based optimization method has been usedin [33] to tune the parameters of a rule-based PSS. This way, the advantages of the

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rule-based PSS such as its robustness, less computational burden and ease ofrealization are maintained. Introduction of GA helps obtain an optimal tuning forall PSS parameters simultaneously, which thereby takes care of interactionsbetween different PSS.

In [34], simultaneous tuning for all the PSS in the system using a GA basedapproach has been developed. The GA seeks to shift all eigenvalues of the systemwithin a region in the stable domain.

Zhang and Coonick [35] proposed a GA based computational procedure to selectPSS parameters simultaneously in multi-machine power systems, by solving a setof inequalities that represent the objectives of optimization problem.

��"� #��!��This thesis examines the application of Genetic Algorithms to PSS tuning in orderto determine a globally optimum PSS parameter set that will ensure a stable androbust operation of a multi-machine power system, for each operating point withina wide range.

+��! introduces the problem of small-signal stability in power systems, withemphasis on the low frequency oscillation phenomena occurring due to smalldisturbances and its mitigation by means of PSS. A review of literature discussesthe relevant work in this area of tuning of PSS and lays down the motivations andobjectives of the work. The field of Genetic Algorithms is also introduced, as themethod chosen to perform the simultaneous tuning of PSS parameters.

+��$ presents the small-signal stability models of single machine connected toinfinite bus (SMIB) and multi-machine power systems. The mathematicalformulations have been detailed in Appendices for interested readers. The modelshave been formulated in state-space form and their open-loop (uncontrolled mode)characteristics have been examined.

+�� & presents the classical optimization method based on Lyapunov’sparameter optimization to tune the parameters of the lead-lag PSS. Phasecompensation characteristics of the lead-lag PSS have been examined and amethod of PSS tuning using the exact phase compensation approach has beendeveloped. Further, the Lyapunov’s parameter optimization method has beenextended to PSS tuning in multi-machine systems.

+�� 7 presents the application of a GA search on the classical Integral ofSquared Error (ISE) based method for tuning of PSS. The method ensures that forany operating condition within a pre-defined domain, the system remains stable

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when subjected to small perturbations. The optimization criterion employs aquadratic performance index that measures the quality of system dynamic responsewithin the tuning process. The solution thus obtained is globally optimal androbust. The proposed method has been tested on different PSS structures- theconventional lead-lag and the derivative type. System dynamic performances withPSS tuned using the proposed technique are satisfactory for different loadconditions and system configurations

+�� 6 considers the optimum tuning of fixed structure proportional plusintegral plus derivative (PID) PSS for the single-machine infinite bus and mublti-machine power systems. A GA based tuning technique is developed and tested forSMIB and multi-machine power systems. The tuning scheme proposed in thischapter uses a genetic algorithm (GA) based search that integrates a classicalparameter optimization criterion based on Integral of Squared Error (ISE). Thismethod succeeds in achieving a robust, simultaneously tuned and globally optimalPID-PSS parameter set, while maintaining the simplicity of the classicaloptimization method. The tuning method implicitly builds-in an increasedrobustness through an objective function, that depends on the operating domain.The system is represent in a discrete state-space form and the influence of thesampling time on the PSS parameter tuning is investigated.

+��. highlights the significant contributions of the present work and drawsthe scope for future work in this area.

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�� Small-signal stability is the ability of the power system to maintain synchronousoperation when subjected to small disturbances. Since the disturbance is consideredto be small, the equations that describe the resulting dynamics of the system maybe linearized. Instability that may result can be of two types:

a) steady increase in generator rotor angle due to lack of synchronizing torque;b) rotor oscillations of increasing amplitude due to lack of sufficient dampingtorque.

In today’s practical power systems, the small-signal stability problem is usually oneof insufficient damping of system oscillations.

For the analysis of small-signal stability, linearized models are generallyconsidered to be adequate for representation of the power system and its variouscomponents.

�� $$��The state-space representation is concerned not only with input and outputproperties, but also with its complete internal behavior. In contrast, the transferfunction representation specifies only the input/output behavior. If state-spacerepresentation of a system is known, the transfer function is uniquely defined. Inthis sense, the state-space representation is a more complete description of thesystem, and it is ideally suited for the analysis of multi-variable MIMO systems.

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In the development of a dynamic model for a multi-machine power system(classical stability model), the following assumptions are usually made [36]:

a) Mechanical power input is constant.b) Damping or asynchronous power is negligible.c) Constant-voltage-behind-transient-reactance model for the synchronous

machines is valid.d) The mechanical rotor angle of a machine coincides with the angle of the

voltage behind the transient reactance.e) Loads are represented by passive impedances.

To study the dynamic behavior of a system, the following data are needed:• System data (lines, buses, transformers, machines);

• Normal load-flow data (�� and

�E - the complex power and voltage at

generator nodes, respectively).

Each machine model is first expressed in its own -3 frame, which rotates with itsrotor. For the solution of interconnecting network equations, all voltages andcurrents must be expressed in a common reference frame. The real-axis of onemachine, rotating at synchronous speed, is used as the common reference. Axistransformation equations are used to transform between the individual machine ( -3) reference frames and the common (�-@) reference frame. The real-axis of thecommon reference frame is used as the reference for measuring the machine rotorangle. For a machine represented in detail, including dynamics of rotor circuit(s),the rotor angle is defined as the angle by which the machine 3-axis leads/lags thereal axis. Under dynamic conditions, the angle δ changes with rotor speed [37].

The following calculations are essential in order to prepare the system for astability study:

1. All system data are converted to a common base.

2. The loads are converted to equivalent admittances. The needed data is takenfrom a load-flow study. The equivalent shunt admittance at the bus is given by:

222

*

�

�

�

�

�

�

�

�

�

E

@*

E

�

E

�

E

�G −=== (2.1)

where ��@�E and,,, are the voltage, active power, reactive power and

current, respectively, corresponding to a load admittance ��

*��G += .

��

___________________________________________________________________________________________

13

3. The internal voltages of a generator 8�∠ δ�� are calculated from a load-flowrun. The internal angle of the generator during transients (δ5) is computed fromthe pre-transient terminal voltages E∠ �.

4. The admittance matrix � of the network is calculated'� is a � matrix,where is the total number of buses.

5. Obtain the admittance matrix for the reduced network ( �� ) by eliminatingall the nodes that are not internal generator nodes. All nodes except for theinternal generator nodes should have zero injection currents, and this property isused to obtain the network reduction [36].

Note that �� is a � � � dimension matrix, where � is the number of generators.In Appendix I, Section 8.1.1 provides details of the above.

�� %��&�� ' ��&�� The transfer function block diagram (Figure 2.1) describes the dynamics of the �th

machine in a multi-machine system [38]. This is a generalization of the extensivelyused single machine connected to infinite bus transfer function block diagram [2]and takes into account the interaction between machines via / matrices, which aresquare matrices of order �. The diagonal elements of the /�, … , /� matricesdetermine the machine’s dynamics, while the off-diagonal elements model thedynamic interactions between machines. Observe that in this block diagram thePSS is not represented, for convenience. The number of state variables is ��× �,where � is the number of state variables used to model a machine and its excitationsystem.

For the calculation of /�,�/�, … /� matrices, the armature current components ��and ��, and the terminal voltage components E� and E� of each machine areexpressed with respect to the common frame of reference.

During low-frequency oscillations, the current induced in a damper winding isnegligibly small; hence the damper windings are completely ignored in the systemmodel. As for the and 3-axis armature windings of the synchronous machine,their natural oscillating frequency being extremely high, their eigenmodes will notaffect the low-frequency oscillations, and hence can be described simply byalgebraic equations [38]. What is left is the field winding circuit of the machine,which is described by a differential equation, not only because of its loweigenmode frequency, but also because it is connected directly to the excitationsystem to which the supplementary excitation control is applied. The excitation

��

____________________________________________________________________________________________

14

system itself must be described by differential equations. Finally, the torquedifferential equation of the synchronous machine is included in the model.

+�

�π2

��

��

��

�

+−

1

�� ∆−∆ �

ω∆ ][��

δ∆

��∆

�� −∆

��∆

++

’��

�∆ ��∆

_

_

K1ii

K4ii K5iiK2ii

K6ii

��

��

��

�

3

3

’1+

_

K5ij

K6ij

+

�� +

1

K1ij

K2ij

�δ∆

’��∆

K4ii3ijK

1

�δ∆

’��

�∆

’��

�∆ �δ∆

_ _

+

+

��−

_

_

�� Transfer function block diagram representation of a multi-machine systemfor small-signal stability analysis

The complete mathematical formulation of the multi-machine dynamics, as well asthe subsequent small perturbation model has been discussed in Appendix I. Basedon the transfer function block diagram (Figure 2.1), the system dynamics can beexpressed by a set of linear differential equations in the small-perturbationvariables �� as follows:

( )

( )

( )

( )��

��

��

��

��

�

��

��

��

��

��

��

��

�

��

�

��

��

��

��

�

��

�

��

�

��

��

��

��

�

��

�

��

��

��

��

��

��

��

��

�

�

�

�

�

��

⋅+∆⋅−∆⋅⋅

−∆⋅⋅

=∆

∆+⋅

−∆−=∆

∆⋅⋅⋅=∆

∆⋅+∆⋅−∆⋅−∆⋅−=∆

1

11’

2

1

’65

’3

’’

4

’21

δ

δ

ωπδ

δωω

(2.2)

� ��1��

��

___________________________________________________________________________________________

15

It is to be noted here that when ��1, the above set of equations reduce to thewell-known SMIB system representation.

Using vector-matrix notation, the set of equations (2.2) can be represented in state-space form as follows:

( ) ( ) ( ) ( )��

pUxx ⋅+⋅+⋅= �� (2.3)

In equation (2.3), � � � and � are the state, control and perturbation matricesrespectively, and x(), U() and p() are state, control and perturbation vectors,respectively.

State matrix � is a function of the system parameters and operating conditions,while the perturbation matrix � and control matrix � depend on system parametersonly.

For the system operating conditions and parameters considered (see Appendix II),the system eigenvalues are obtained by solving the �� of thesystem.

The stability characteristic of the system is dependent on the eigenvalues of thestate matrix as follows:

a) A �� corresponds to a non-oscillatory mode. A negative realeigenvalue represents a decaying mode, while a positive real eigenvaluerepresents aperiodic instability.

b) A pair of �� represents an oscillatory mode. The realcomponent of the eigenvalue gives �� , and the imaginary componentgives �� . A negative real part represents a dampedoscillation whereas a positive real part represents oscillation of increasingamplitude.

�� The systems considered for analysis in this thesis are the single machine connectedto infinite bus through a double circuit transmission line (Figure 1.2), and the well-known nine-bus power system [36], which has three generators and three loads(Figure 2.3).

��

____________________________________________________________________________________________

16

In all generators, IEEE Type ST-1 excitation systems have been considered.System parameters and operating data for both systems investigated have beenprovided in Appendix II.

EtV0

XIg

Eq

Y = G + jB

Ish

RI

RI X

�� Single machine connected to an infinite bus system

Adrian Andreoiu

2 3

1

�

� �

�

� �

�

�

230 kV 230 kV

230 kV

16.5 kV

18 kV 13.8 kV

Load A

Load C

Load B

�� 3-machine system [36]

��

___________________________________________________________________________________________

17

�� The small-perturbation transfer-function block diagram of a SMIB system is shownin Figure 2.4.

�� Small-perturbation block diagram of SMIB system

The state-space model can be expressed as follows:

( ) ( ) ( ) ( )�� ⋅+⋅+⋅= bxx � (2.4)

where

−−−

−−

−−−

=

��

�

�

�

��

��

��

�

��

�

�

�

�

�

��

10

’

1

’

1

’0

0002

0

65

3

4

21

π�

[ ]�� ∆∆∆∆= ’x δω

��

____________________________________________________________________________________________

18

�

�

�

�

�

= 000b

�

�

= 000

1

Note that the control signal �, which is actually the PSS output, would act on thesumming junction of the terminal voltage reference of the AVR-excitation systemFigure 2.4. Also note that � is now a scalar, since we deal with a SISO system. Theperturbation considered consists of a step increase of 1% in the mechanical torque�� of the synchronous generator.

Table 2.1 shows the open-loop eigenvalues of the SMIB system for the nominaloperating point and system parameters considered. Evidently, the system isunstable under small perturbations and require a stabilizing signal from the PSS.The system dynamics without PSS are shown in Figure 2.5 and Figure 2.6.

�� Open-loop SMIB system eigenvalues

��[p.u.]

��

��!��"[Hz]

P = 0.8Q = 0.6

0.1028 � j5.5022-6.3710

-14.2975

-0.018711

0.875--

��# Rotor angle deviation for SMIB system under small perturbation

��

___________________________________________________________________________________________

19

��$ Rotor speed deviation for SMIB system under small-perturbation

�� As discussed in Section 2.4, the three-generator, nine-bus system [36] has beenconsidered for our analysis of multi-machine systems. The same set of statevariables as that used for SMIB system has been used to describe each machine ofthe multi-machine system behavior through state-space modeling approach. Thestate-space model for the three-machine system is expressed as follows:

( ) ( ) ( ) ( )��

puxx ⋅+⋅+⋅= �� (2.5)

where x(t), U(t) and p(t) are state, control and perturbation vectors, respectivelyand they are expressed as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) �

��

��

��

��

]

[x

3333

22221111

∆∆∆∆

∆∆∆∆∆∆∆∆=

δω

δωδω �

( ) ( ) ( ) ( )[ ]�� 321u ∆∆∆=

( ) [ ]��

321p ∆∆∆=

��

____________________________________________________________________________________________

20

� � � and � are thestate, control and perturbation matrices, respectively. Theyconstructed by extending the SMIB model to the 3-machine model and are given inAppendix III Section 8.3.1.

This system is analyzed in a similar manner as the SMIB system. Figure 2.7 andFigure 2.8 show the time response of angular speed- and rotor angle deviations ofall the machines of the system without PSS, when a step perturbation in mechanicalinput of 1% occurs at the shaft of Generator 1.

��% Angular speed deviations of all machines when Generator 1 is perturbed

��

___________________________________________________________________________________________

21

��& Rotor angle deviations for 3-machines open-loop system

Table 2.2 presents the eigenvalues and corresponding- damping factors and naturalfrequencies of oscillations. As also suggested by the figures is now evident that thesystem is unstable under small perturbations on generator 1.

�� Open-loop MM system eigenvalues

��[p.u.]

��

��!��"[Hz]

0.0524 � j8.0334 -0.0065 1.27860.0479 � j6.3195 -0.0076 1.0058

-0.0447 � j15.3935 -0.0029 2.4500-10.3506 � j2.9481 0.9618 1.7129-10.3957 � j6.6547 0.8156 2.0716

-4.8010 1.0000 -

P1 = 0.7160P2 = 1.6300P3 = 0.8500

Q1 = 0.2700Q2 = 0.0670Q3 = -0.1090 -15.5527 1.0000 -

��

____________________________________________________________________________________________

22

�� !�"#�This chapter presents the details of mathematical models required for the analysisof small signal stability for both single machine connected to infinite bus andmulti-machine systems. The mathematical models are in the state-space form,thereby making the application of linear analysis possible. As has beendemonstrated, both systems are unstable under small perturbations and requireadditional stabilizing control from the power system stabilizer, the design of whichwill be treated in the following chapters.

��

___________________________________________________________________________________________

23

� ��

��

Two distinct types of system oscillations are usually recognized in interconnectedpower systems [5]. One type is associated with units at a generating stationswinging with respect to rest of the power system. Such oscillations are referred toas "local mode" oscillations and have a frequency in the range of 0.8 to 2.0 Hz. Theterm local is used because the oscillations are localized at one power plant. Thesecond is associated with swinging of many machines in one part of the systemagainst machines in another part. These are "inter-area mode" oscillations, andhave frequencies in the range of 0.2 to 0.7 Hz.

The basic function of a power system stabilizer (PSS) is to add damping to thegenerator rotor oscillations by controlling its excitation using auxiliary stabilizingsignal(s). To provide damping, the PSS must produce a component of electricaltorque in phase with the rotor speed deviations.

$�%� &�"�� "��"�� '�� ( )"� ��*��+"�

$�%�%� ("� "��,*-�� (��The overall excitation control system (including PSS) is designed to [5]:a) Maximize the damping of the local plant mode as well as inter-area mode

oscillations without compromising the stability of other modes;

��

____________________________________________________________________________________________

24

b) Enhance system transient stability;c) Not adversely affect system performance during major system upsets which

cause large frequency excursions;d) Minimize the consequences of excitation system malfunction due to

component failures.

Since the purpose of a PSS is to introduce a damping torque component, the speeddeviation represents an appropriate signal to be used as input for the PSS. Inpractice, both generator and its exciter exhibit frequency dependent gain and phasecharacteristics, �!"�#. Hence, the PSS transfer function should have appropriatephase-lead circuits to compensate for the phase lag between the exciter input andthe electrical torque [37].

For large values of �� and the usual range of constants,� the composite transfer-function for �!"�# and the corresponding phase-lag characteristic can be writtenas follows [2]:

( )( )

( )�

�

��

�

�

��

��

� �!

��

��

�� !

⋅+

⋅⋅

=∠

⋅+⋅

⋅⋅+⋅

=

−− ωω 1

6

’1

6

’

6

2

tantan

11

(3.1)

�� Phase-lag characteristic of �!"�# for a SMIB system

��

___________________________________________________________________________________________

25

Figure 3.1 shows a plot of the phase-lag introduced by �!"�#��as a function of thefrequency. It can be seen that for the frequency range of interest (0.2 to 2.0 Hz), thephase lag due to �!"�# is between 25� to 105�.

The transfer function of the lead-lag PSS on the th machine is shown in equation(3.2) and the corresponding block diagram is shown in Figure 3.2.

( ) ( )( )

⋅+

⋅+⋅

⋅+

⋅+⋅

⋅+

⋅⋅=

∆∆

=�

�

�

�

��

��

��

�

�

��

��

��

��

��

��

�

��

4

3

2

1

1

1

1

1

1ω(3.2)

�� Transfer function block diagram for Lead-lag PSS

The corresponding phase-lead characteristic of the lead-lag PSS is given by:

( ) ( ) ( )[ ]21

111 tantan2tan

2�� ⋅−⋅⋅+⋅−=∠ −−− ωωωπ

(3.3)

Note that the phase angle of PSS signal was expressed in (3.3) under theassumption that �� = �� and �� = ��for the purpose of simplification.

�� PSS phase lead characteristics for different time constants, ��=�� and��=��=0.05 sec

��

____________________________________________________________________________________________

26

Figure 3.3 shows the phase-lead characteristics of the PSS for several values of ��.In the figure, we note that for the frequency range under consideration, the lead-lagPSS can provide up to about 90� phase compensation, with �� = 0.3 seconds.

The parameters of the lead-lag PSS are required to be tuned optimally, in order toobtain the best performance of the system.

$�%�� ("��"�� "�� "� �� '�� (��(�"��"�

In principle, the lead-lag PSS (also regarded as the conventional1 PSS) consists ofthree blocks: a phase compensation block, a signal washout block and a gain block(refer Figure 3.2).

The �� block provides the appropriate phase-lead characteristicto compensate for the phase lag between the exciter input and the generatorelectrical torque �!"�#. The phase characteristic to be compensated changes withthe system conditions, therefore a characteristic acceptable for a range offrequencies (normally 0.1 to 2.0 Hz) is sought. This may result in less thanoptimum damping at any one frequency. The required phase lead can be obtainedby choosing the appropriate values of time constants ��.

The signal $�� block functions as a high-pass filter, which allows the dcsignals to pass unchanged, thus avoiding terminal voltage variation due to steadychanges in speed. The washout time constant �� should be long enough to passstabilizing signals at the frequencies of interest unchanged, but not so long that itleads to undesirable generator voltage excursions during system-island conditions[37].

The stabilizing gain �� determines the amount of damping introduced by the PSS,and, ideally, it should be set to a value corresponding to maximum damping.However, in practice the gain is set to a value that results in satisfactory dampingof the critical system modes without compromising the stability of other modes, ortransient stability, and that does not cause excessive amplification of PSS inputsignal noise.

In order to restrict the level of generator terminal voltage fluctuation duringtransient conditions, limits are imposed on PSS outputs.

The PSS parameters to be optimized are the time constants, ��, �� and gain��. �� = 10 seconds is chosen at all machines in order to ensure that the phase-lead and gain contributed by the washout block for the range of oscillation

1 In this thesis, the terms "lead-lag PSS" and "conventional PSS" are used interchangeably

��

___________________________________________________________________________________________

27

frequencies normally encountered is negligible [5]. The number of PSS parametersto be optimized is reduced by considering the PSS to be comprising two identicalcascaded lead-lag networks. Therefore, �� = �� and �� = ��. Also, �� = �� = 0.05seconds is assumed fixed from physical realization considerations [2]. Thus, theoptimization problem reduces to determining �� and �� ( = 1, …, �) only.

$�� .�� )�� '��(��

$��%� � / �� Figure 3.4 shows the composite transfer-function block-diagram of the SMIBsystem equipped with a lead-lag PSS.

�� Small perturbation block diagram of a single machine to infinite bus systemequipped with a lead-lag PSS

The linear dynamic model of the open-loop power system ( %�% without PSS) wasdiscussed in detail in Chapter 2. Following the same convention, the representationof the above composite system inclusive of the PSS can be described by:

( ) ( ) ( )��

pxx ⋅+⋅=� (3.4)

��

____________________________________________________________________________________________

28

� and � are thestate and perturbation matrices, respectively, x() the state vector,and p() is the perturbation. The state vector x() is given as follows:

( ) [ ]�� &&�� ∆∆∆∆∆∆∆= 21’x δω (3.5)

Note that the two additional intermediate variables of the PSS, namely �&� and�&��appear in (3.4) as state variables, as does the PSS output signal ��.

The state matrix � of this system is given by:

−−−−−

−−−−

−−−

−⋅−⋅−

⋅−−

⋅⋅

−−

=

�

��

��

��

�

�

��

�

�

�

��

��

�

��

��

��

��

��

��

�

��

��

��

��

��

��

��

��

��

�

��

�

��

��

�

��

�

�

�

�

�

1110

011

0

001

0

001

0

00011

0

0000002

00000

442

3

442

312

42

311

42

3

442

3

442

312

42

311

42

3

22

12

2

11

2

65

’’3

’4

21

π

�

$�� ,/��+�� (�� (�"��"�� 0�� (�� /�� 1��2�

The phase compensation technique is based on the objective of tuning the PSSparameters to fully compensate for the phase lag introduced through the exciter andgenerator characteristics �!"�#, such that the torque changes provided by the PSSare in phase with the rotor speed deviations. The following step by step approach isused:

a) Find the natural frequency of oscillation �n of the electromechanical mode.Neglecting the damping, the characteristic equation of the mechanical loop maybe written as:

02 12 =⋅⋅⋅+⋅ �� π (3.6)

and the solutions are:

��

___________________________________________________________________________________________

29

��'��

/2 1, ⋅⋅⋅=±= πωω (3.7)

where � is the inertia constant in seconds is the system frequency in Hz�� is the synchronizing torque coefficient

For the present system investigated and the operating conditions described inSection 8.2, �� = 0.9538, and the natural frequency of oscillation of theelectromechanical mode is �� = 5.474 rad/sec.

b) Find the phase lag between �� and �� of the electrical loop using therelationship established in (3.1). For �� = 5.474 rad/sec we find that the phase-lagintroduced by �� equals 69.87�.

c) Obtain using (3.3) the PSS phase-lead time constants that exactlycompensate for the system phase-lag. In the given system, to exactly compensatefor the phase-lag of 69.87�, it was found that the phase-lead time constantrequired is � = � = 0.216 sec. Figure 3.5 shows the phase-lag characteristic of�� vis-à-vis the phase-lead characteristic of the lead-lag PSS for � = � =0.216 sec. It can be seen that the so tuned PSS closely compensates for the phase-lag in a frequency range up to 1 Hz and compensates accurately for theelectromechanical mode.

�� Open-loop SMIB system and PSS phase characteristics

��

____________________________________________________________________________________________

30

d) Tune the gain �� of the PSS using any suitable criterion. In this work, theLyapunov’s parameter optimization method, which will be discussed in detail inthe next section, was used to obtain��. The optimum �� thus found was equal to26.0.

��

3.2.3.1 �� The choice of a suitable performance index is extremely important for the design ofPSS. In this thesis, a performance index as given in (3.8), where x is the statevector and � is a weighing matrix has been used:

� � �∫∞

⋅⋅=0

)xx( � (3.8)

The performance index � can be evaluated using the relation:

( ) ( )0x0x ⋅⋅= �� (3.9)

where x(0) is the initial state of the state vector, and � is a positive definitesymmetric matrix obtained by solving the Lyapunov’s equation:

� �� −=⋅+⋅� (3.10)

where is the state matrix of the system.

By appropriate choice of � matrix elements, various penalization weights can beassigned to the state variables (which in this case are deviations from steady-stateconditions) and a desirable dynamic performance for the system can be achieved.

3.2.3.2 ��As described in the previous section, by an appropriate choice of �, theperformance criterion, and hence the optimal PSS parameters can be manipulated.In this work, we choose an �� (ISE) criterion that seeks tominimize the squared of the deviation of power angle deviation (��) from itssteady-state value. Thus the state variable �� is assigned a high weight andpenalized for deviations and the PSS parameters are obtained accordingly.

Mathematically, this can be written as,

��

___________________________________________________________________________________________

31

( )∫∑∞

∆−∆=0

2� ��

δδ (3.11)

It can be seen that in this case, � = ��[0 1 0 0 0 0 0].In order to obtain the optimal values of �� and �, the following procedure hasbeen used:

1. Choose a set of PSS parameters for which the state matrix of the compositesystem is stable.

2. Fix the value of � and vary �� over a wide range of values and determine theperformance index (using (3.9)). It is seen that for a fixed �, when �� isincreased, the performance index � decreases continuously, attains a minimum(say ��) and then increases with further increase in ��.

3. Carry out Step-2 for various values of � and determine the ��(say, ��*)

Figure 3.6 shows the plot of variation of � as a function of �� for different valuesof �. From the figure, we note that � attains the overall minimum for �� = 38.65and � = 0.11 seconds, which are the optimal settings of the lead-lag PSS.

�� Performance index as a function of PSS gain ��, for different �

Figure 3.7 shows a comparison of the system dynamic performances with theoptimal lead-lag PSS, but designed using two different techniques: the phasecompensation approach and the Lyapunov’s method. It can be seen that both thedesign approaches provide satisfactory performances, though the phase

��

____________________________________________________________________________________________

32

compensation approach requires somewhat more settling time, which is evidentfrom a comparison of the values of performance indices � also (Table 3.1).

�� Performance indices corresponding to different PSS settings

��

� ��!��

0.21626

0.1138.65

��"��#��$!�� 1.7905�10-5 1.4452�10-5

��% Performance of SMIB system with optimal lead-lag PSS

Further, we examine the phase compensation characteristics (Figure 3.8) drawnfrom the two optimal PSS settings previously obtained. While the phasecompensation technique closely compensates for the �� phase lag up to afrequency of 1 Hz, the system phase-lag is considerably under-compensated whenthe optimal lead-lag PSS is used.

��

___________________________________________________________________________________________

33

��& Phase characteristics for systems and different PSS designs

�� !��!��"��

��#� $��The representation of the multi-machine system without PSS has been discussed inChapter 2, and a state-space model was developed. In this section, the developmentof the state-space model for the same system is presented in the line of Chapter 2,considering that all the generators are equipped with lead-lag PSS.

For the sake of continuity in understanding, the state-space model of the multi-machine system without PSS is re-stated below:

( ) ( ) ( ) ( ) � �

pUxx ⋅+⋅+⋅= ' (3.12)

, ' and � are the state, control and perturbation matrices and have been describedin Section 8.3.1. The associated state, control and perturbation vectors are givenbelow:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ��

��

� � �

� � �

]

[x

33332

2221111

∆∆∆∆∆

∆∆∆∆∆∆∆=

δω

δωδω �

��

____________________________________________________________________________________________

34

( ) ( ) ( ) ( )[ ]� � � � 321u ∆∆∆=

( ) [ ]��

321p ∆∆∆=

The control vector U( ) is a vector of stabilizing signals that represents the PSSoutput at different machines.

The dynamic equations of the PSS in state-space form as obtained from the transferfunction block-diagram is given below:

( )( ) ( ) ( )

( )( ) ( ) ( ) ( )( )( )( ) ( ) ( )( ) ( )

�

��

�

�

��

��

�

��

� �

� �

� ��

�

� �

� � ��

�

� �

� ��

4232

211212

11

1

1

⋅

∆−∆⋅+∆=∆

⋅

∆⋅+∆−=∆

∆−∆⋅=∆ ω

(3.13)

� ��= 1,2,3

where �� and �� are the state-variables associated with each PSS, � is the�� time constant, �� are the !��"�� time constants and �� is thestabilizer ��.

Equations (3.13) may be arranged in standard vector-matrix form as in (3.14), andrepresent the state-space model of the PSS at all machines:

( ) ( ) p1PSSPSS xx)(x ⋅+⋅+⋅= ()

� �

(3.14)

where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]� � � � � � � 323132221212111PSS uuux ∆∆∆∆∆∆∆∆∆=

), ( and �1 are the matrices associated with the PSS model with appropriatedimensions, and are given in the Section 8.3.2.

By defining an augmented state-vector ( ) ( ) ( )[ ]� PSSC xxx = , the state-space

model of the closed-loop system becomes:

��

___________________________________________________________________________________________

35

( ) ( ) pxx CCCC ⋅+⋅= � �

(3.15)

where

=

=

1C

1C and

()

'

'1 is a re-defined control matrix given as:

'1 = [0 0 b1 0 0 b2 0 0 b3]

where b1, b2, b3 are the column vectors of the control matrix '.

The perturbation term in (3.15) can be eliminated, by applying a coordinatetransformation in the state-space, as follows:

( ) ( ) ( )∞−= CC xx’x (3.16)

Hence, (3.15) reduces to the standard state-variable form:

( ) ( ) � �

’x’x C ⋅= (3.17)

where

( ) ( ) px0’x C1

C ⋅⋅−=∞−= −

is the initial state of x’( ), which is also the steady-state value of x( ).

�� "� �� !��

Section 3.2.3 provides the details of the method of PSS parameter optimizationusing Lyapunov’s method. In this section, the analysis is extended in a similarmanner, to find the optimal parameters of PSS in a multi-machine system.

One important aspect associated with PSS tuning in multi-machine systems is theproblem of siting of PSS on appropriate machines. This is required in order to findthose critical machines where a PSS optimally tuned would damp out specificmodes. This helps reduce the computational burden, particularly in case of large

��

____________________________________________________________________________________________

36

systems. A lot of work has been reported in the literature addressing the sitingproblem [8],[10],[11],[40] In this chapter, and in the thesis as well, this issue hasnot been addressed.

Conventionally, the PSS tuning methods used for multi-machine systems haveeither used a sequential approach or a simultaneous approach. The sequentialtuning approach is computationally simple but introduces eigenvalue drift as thesequence progresses, while the simultaneous tuning approach though being verycomplex to handle, particularly for large systems, does provide the optimalsolution.

In the following analysis, the Lyapunov’s method was applied to multi-machinePSS tuning using the sequential approach.

The weighing matrix � is now the sum of the squares of each machine’s powerangle deviation from their respective steady-state value.

Thus, we have

� = ��[0 1 0 0 0 1 0 0 0 1 0 0]

and

( ) � ��

��∫∑

∞

=∆−∆=

0

3

1

2δδ (3.18)

Using the approach described in Section 3.2.3 we obtain the optimal parameters ofeach PSS sequentially. Various combinations of tuning sequences were tried outand the system performance was found to be the best for a sequence: machine-1-machine-2-machine-3, and therefore we report the results obtained with thissequence only. Table 3.2 provides a detailed report of the behavior of the PSSparameters, tuned using the above sequence, and shows the corresponding systemeigenvalues as the tuning sequence progresses.

It can be seen that the system eigenvalues keep changing as the sequenceprogresses and this is an undesirable phenomenon. Thus, there is a case forexamining and determining the PSS parameters simultaneously while also lookingfor ways on how to address the problem of increased computational burden arisingfrom such an approach.

��

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

__

37

��

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Sequ

enti

al tu

ning

of

mul

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ne p

ower

sys

tem

s

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Step

1: s

ub-o

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tep

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dSt

ep 3

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tune

dS

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achi

ne 1

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nd 3

tune

d

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[s]

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T1

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10.

103

0.11

730.

1173

0.11

73

20.

103

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1615

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30.

103

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30.

103

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enva

lues

Dam

ping

Eig

enva

lues

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enva

lues

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ping

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enva

lues

Dam

ping

-0.0

555 �

j15.

6009

-0.1

674 �

j0.4

693

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908 �

j8.2

613

-10.

1813

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.337

3-1

0.33

32 �

j7.6

062

-16.

2746

� j1

.600

7-1

7.90

45-2

2.19

60-2

3.08

25-2

3.94

50-0

.100

0-0

.100

1-0

.100

2-4

.682

3-1

5.78

24

0.00

360.

0259

0.02

310.

9503

0.80

530.

9952

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

00

-0.1

357 �

j15.

4128

-0.3

618 �

j7.5

701

-2.2

198 �

j11.

1289

-10.

2190

� j3

.334

1-1

0.44

67 �

j7.3

496

-10.

5846

� j4

.220

5-0

.100

0-0

.100

2-0

.102

3-2

.748

1-1

5.76

79-1

7.60

73-2

2.78

11-2

3.92

23-3

1.23

42

0.00

880.

0477

0.19

560.

9507

0.81

790.

9289

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

00

-0.0

555 �

j15.

6009

-0.1

674 �

j6.4

693

-0.1

908 �

j8.2

613

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.337

3-1

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32 �

j7.6

062

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2746

� j1

.600

7-0

.100

0-0

.100

1-0

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2-4

.682

3-1

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24-1

7.90

45-2

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25-2

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0.00

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1.00

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1.00

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j5.0

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j12.

1247

-10.

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.016

0-0

.100

1-0

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5-2

.722

3-6

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25-3

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0.35

390.

1622

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020.

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001.

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1.00

001.

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001.

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1.00

00

��

____________________________________________________________________________________________

38

Figure 3.9 shows the system dynamic performances with the initial PSS settings(�� = 3.0, �� = 0.1). This was chosen as a �� since, the open-loopsystem being unstable, it was required to obtain a stable system to proceed with theoptimization (Table 3.2).

�� System dynamic performance with the ��

The system dynamic performance obtained using the optimally tuned PSS is shownin Figure 3.10.

�� System dynamic performance with the optimal PSS

��

____________________________________________________________________________________________

39

�� This chapter presents the development of a composite state-space model of thesystem including the lead-lag power system stabilizer (PSS). Phase characteristicsof the system as well as the PSS have been investigated and a method for tuning ofPSS parameters based on exact phase compensation has been presented. It is foundthat for desirable system performance, a full phase compensation may not benecessary.

Subsequently, a method based on Lyapunov’s parameter optimization has beenpresented for tuning of lead-lag PSS. This method makes use of the Integral ofSquared Error criterion with an objective of minimizing the power angle deviationfrom its steady-state value. System dynamic performance for single machine toinfinite bus, as well as a multi-machine system show that this method providessuperior responses as compared to the phase-compensation technique based PSS.

Due to the eigenvalue drift phenomenon and the high computational burdenrequired for simultaneous type of tuning approach, Genetic Algorithm basedsimultaneous tuning methods have a promising scope and could address many ofthe concerns raised in large scale PSS parameter tuning. This issue will beaddressed in the following chapters.

��

____________________________________________________________________________________________

41

� ��

��

�� Fixed-structure lead-lag type of power system stabilizers discussed in Chapter 3have found practical applications and these generally provide acceptable dynamicperformances [5]. There have been arguments that these controllers, being tunedfor one nominal operating condition, provide sub-optimal performance when thereare variations in system operating load or system configuration. To address thisissue, PSS parameter tuning methods that incorporate robustness within theoptimization scheme and are applicable to fixed structure stabilizers are desirable.To ensure a sufficiently robust behavior, a wide operating domain need beconsidered in the tuning process.

In the present chapter we propose the application of a Genetic Algorithm (GA)search based on the classical Integral of Squared Error (ISE) criterion for tuning ofPSS, as discussed in [39]. The proposed method ensures that for any operatingcondition within a pre-defined domain, the system with the optimally tuned PSSremains stable when subjected to small perturbations. The optimization employs aquadratic performance index that measures the quality of system dynamicperformance within the tuning process. The solution obtained is globally optimaland robust. The proposed method has been tested on two different PSS structures-the conventional lead-lag type and the derivative type.

The proposed method makes use of the classical Lyapunov parameter optimizationtechnique within a GA search framework to determine the "degree of goodness" ofan individual within a population of possible solutions. The performance of eachindividual in the population is then evaluated through an objective functiondynamically modeling the problem to solve and assess a �� The selectedindividuals are then modified through the application of � � �� , in order

��

____________________________________________________________________________________________

42

to obtain the next generation. The globally optimal PSS parameter set thusobtained, is robust over a wide operating range.

�� The proposed Lyapunov method based genetic algorithm is initiated by generatingrandomly an initial population of binary coded individuals, where each individualrepresents a possible solution for the PSS parameters.

�� Lyapunov Method Based Genetic Algorithm for PSS Tuning

��

____________________________________________________________________________________________

43

A basic requirement for obtaining a feasible solution to the Lyapunov equation isthat the state-matrix ��should be stable. Fulfillment of this condition is ensured by"stability screening". The entire population of individuals in each generation isscreened (Figure 4.1) in order to ensure that only those individuals (each of themrepresenting a PSS parameter set) that provide a stable system over the wholeoperating domain �, are allowed to proceed further in the optimization process.This also brings about significant reduction in the computational burden.Individuals resulting in unstable systems for an operating point within the domain� ("bad individuals") are assigned a very high value of ��, where �� is given asthe mean value of performance indexes over the operating domain �, and given by(4.1). The bad individuals are gradually phased out from the population within afew generations.

( ) ��

��

�

�� ∈∀

⋅⋅⋅⋅= ∑ ∫

∞

1

, 0

� (4.1)

Every individual (�� ) of the current population is evaluated for �� anda basis for the biased selection process is then established. To avoid prematureconvergence and speeding up of the search when the convergence is approached,the objective values obtained for each individual are mapped into �� through a ranking process. The rank-based fitness assignment overcomes thescaling problems of the proportional fitness assignment. The individuals will beranked in the population in descending order of their fitness with respect to theproblem domain. The higher the individual’s fitness is, the higher is its chance topass-on genetic information to successive generations.

The next generation will be populated with offspring, obtained from selectedparents. The � � �� is a process used to determine the number of trials for oneparticular individual used in reproduction. The selection process uses the �� method, a single-phase sampling algorithm with minimum�� , zero ��and time complexity in the order of the number of individuals(��).

Recombination of the selected individuals is carried out with pairs of individualsfrom the current population using a�� process having a certainprobability�� The individuals in the pairs will exchange genetic information witheach other, thereby creating two new individuals, the offspring. After that, eachindividual in the population will be �� with a given probability, through arandom process of replacing one allele of a gene with another to produce a newgenetic structure.

The GA employed in this study uses an �� !�� in which the offspring iscreated with a � � �� of 80% and reinserted in the old population by

��

____________________________________________________________________________________________

44

replacing the least fit predecessors. Most fit individuals are allowed to propagatethrough successive generations and only a better individual may replace them.

The GA stops when a pre-defined maximum number of generations is achieved orwhen the value returned by the objective function, being below a threshold,remains constant for a number of iterations.

�� Two types of PSS have been considered for analysis- (a) the conventional lead-lagnetwork (4.2) with gain ��, time-constants �� and �, and wash-out filter time-constant ��, and (b) the derivative network (4.3) with gain �� and time constant �.

( )��

��

��

��

��

��

�

�

� ω∆⋅

++

⋅

++

⋅+

⋅=2

1

2

1

1

1

1

1

1)( (4.2)

( ) ( ) ( )��

��

�� ∆⋅

+⋅

+⋅=

11)( (4.3)

As discussed in Chapter 3, �� is the washout time-constant, which is used towashout dc signals and without it, steady changes in speed would modify theterminal voltage. To guarantee the lead characteristic of the control signal, � iskept to minimum physically achievable (� = 0.05 seconds). Thus, the lead-lag PSSparameters to be optimized are �� and ��.

It is important to note here the difference in input signals to the two PSS. Thephase-lead PSS is based on the commonly used rotor speed deviation input signalas shown in (4.2). On the other hand, the derivative PSS (4.3) is based on electricaltorque deviation signal �� . The parameters to be tuned in the later case are thederivative gain, �� and the time constant, �.

The linear dynamic model of the composite system inclusive of excitation systemand PSS on each generator, can be obtained in a similar manner to the one outlinedin Chapter 3, and state-space representation is given in:

( )p)(x

x ⋅+⋅= � ��

��(4.4)

� and � are the state and perturbation matrices and depend on the systemconfiguration and operating conditions, while x and p are state and perturbationvectors, respectively.

��

____________________________________________________________________________________________

45

In order to eliminate the perturbation term in (4.4) and reduce the system model tothe standard closed-loop state-space form, a coordinate transformation in the state-space is applied as given by:

)(xx’x ∞−= (4.5)

The resulting state-transformed system model thus obtained is given by:

’x’x ⋅= ��

(4.6)

where ( ) ( ) px0’x 1 ⋅⋅−=∞−= −� is the steady state value of x’(t).

�� !��"�The proposed genetic algorithm based approach applying the classical Lyapunovstability and parameter optimization technique is now used to determine theoptimal parameters of a PSS that is robust over a wide operating domain. For thepurpose of these study cases we consider the operating domain � {�� ∈ �"#$�%�&�%'��"# $�%��&�$'�� }, with a step-size of 0.1 per-units in each case.Therefore, we have a total of 154 operating points on the � plane.

�� Small perturbation transfer function model of SMIB system equipped withlead-lag type PSS

��

____________________________________________________________________________________________

46

Figure 4.2 shows the small perturbation transfer function model of the single-machine infinite bus system with the lead-lag PSS.

The state variable vector x and state matrix � for the SMIB system equipped withthe lead-lag PSS of (4.2) have been provided in Section 3.2.1.

Figure 4.3 shows the small perturbation transfer function model of the SMIBsystem equipped with a derivative type PSS.

�� Small perturbation transfer function block-diagram of single-machine infinitebus system equipped with derivative type PSS

The state vector of the system can be defined as:

( ) [ ]�� ((�� ∆∆∆∆∆= ’δω (4.7)

The system matrix � with the derivative type PSS of (4.3) is described by:

��

____________________________________________________________________________________________

47

−−

−

−−⋅

−⋅

−

⋅−−

−−

=

��

�

�

��

�

�

��

�

��

��

�

��

�

�

)

�

)

�

��

��

�

�

��

�

�

�

��

1100

01

00

01

0

0011

0

000002

0000

21

21

65

’’3

’4

21

π

� (4.8)

�� The optimal parameters for both lead-lag and derivative types of PSS obtained withthe proposed GA based method are compared with the optimal parameters settingobtained by applying the ISE Technique earlier reported in [39], that was obtainedconsidering one nominal operating condition.

Additionally, for the derivative PSS, the set of optimal parameters obtained withthe proposed GA based method is also compared with an earlier reportedeigenvalue shift based GA method [32].

�� Performance index as function of PSS gain ��, for different time constants �

��

____________________________________________________________________________________________

48

Figure 4.4 depicts the tuning process of a derivative type PSS in a SMIB systemperformed using the ISE Technique, in a similar manner as described in Section3.2.3.2 for the lead-lag PSS.

Table 4.1 provides a summary of the converged PSS parameters for the lead-lagPSS, along with the corresponding system eigenvalues, damping factors � andnatural frequencies �� for oscillatory modes. The ISE technique based PSS [39]achieves a fairly robust optimum parameter set that is very close to the globaloptimal set obtained using the GA based method. �� and �� are thecorresponding values of � and ��, respectively, ��* � with respect to thebest, and provide a quantitative measure of the quality of dynamic performancewith a particular type of PSS and PSS settings.

�� Optimal parameter setting for lead-lag PSS obtained with two different methods

��

�!��"�#�$��%�&

��#��#��"�#��#�

_ 33.980.12

38.650.11

!��#'��#��#��#��#�(�(��[Hz]

-14.298,-6.371

0.103 ± j 5.5

-30.964-3.888 ± j7.773-8.472 ± j5.003

-4.777-0.102

-0.430.86

--

-1.3801.566

--

-30.461-3.196 ± j8.471-9.901 ± j3.928

-3.806-0.102

-0.350.93

--

-1.4411.695

--

��#��#��#��#�

_ 1.008 ��

�� _ �� 1.01

Table 4.2 shows the optimal settings for derivative PSS and the corresponding-eigenvalues and performance indices. The performance indices are normalized tothe best (see Table 4.1). The proposed GA based method, though with �� of4.72, does provide a fairly satisfactory dynamic performance, while the eigenvalueshifting method based PSS with �� of 24 provides a much worsened response.Note that when tuned with classical ISE Technique, the system is not stable for theentire operating domain � considered, although for the nominal operating pointbehaves better than the genetically tuned systems.

��

____________________________________________________________________________________________

49

�� Optimal parameter setting for derivative PSS obtained with three differentmethods

��

��!��#'��)�

��"�#�%��&

�!��"�#�$��%�&

��#��#��"�#��#�

_ 2.030.30

6.470.34

1.550.38

!��#'��#��#��#��#�(

-14.298,-6.371

0.103 ± j 5.5

-16.823-1.357 ± j5.33-2.756 ± j4.62

-2.081

-19.417-1.959 ± j9.21-0.726 ± j2.892

-1.56

-16.225-0.877 ± j5.759-2.981 ± j3.248

-1.7857��#��#��#��#�

_ 4.97 21.33 4.69

�� _ 4.72 24 -

Figure 4.5 and Figure 4.6 show the distribution of �� for individuals in thepopulation and their convergence to the global optimum during the search process,for two different PSS types, �� the conventional lead-lag PSS and the derivativePSS, respectively.

�� * Distribution of the solution during the search process for conventional lead-lag type PSS

It is seen that the best individual from a generation progresses towardsconvergence, corresponding to the minimum of �� and thereby providing the

��

____________________________________________________________________________________________

50

optimal values of PSS parameters. The figures provide an image of the distributionof possible solutions cumulatively obtained during the genetic search, thusemphasizing the algorithm’s convergence towards the global optimum andintuitively pinpointing the same.

It should be noted that, the vertical axis denoting the performance ��, appearswith a different scale for the two PSS cases- the derivative PSS scale being of theorder of 10-4, while the conventional lead-lag PSS is of the order of 10-5.

�� + Distribution of the solution during the search process for derivative type PSS

Figure 4.7 shows that �� decreases monotonously with time, and inapproximately 50 generations, the optimization process finds a solution thatremains unchanged thereafter, and �� reaches a steady minimum value. Theseresults were obtained with a genetic process over 160 generations and having apopulation of 50 individuals.

��

____________________________________________________________________________________________

51

�� , �� of best individual for each generation

Figure 4.8 shows the corresponding PSS parameters variation during the geneticprocess, and indicates their convergence to the optimal value.

�� - PSS parameter variation during genetic process, for each generation

Figure 4.9 and Figure 4.10 shows the area covered by the imaginary positive partof the system eigenvalues over the entire operating domain �, for optimumparameter settings of lead-lag PSS and derivative type PSS, respectively. It can be

��

____________________________________________________________________________________________

52

seen that the conventional lead-lag PSS provides a more stable system since itseigenvalues are further away from the unstable axis vis-à-vis the derivative PSS.

�� Distribution of system eigenvalues with optimal lead-lag PSS over theoperating domain

�� Distribution of system eigenvalues for optimal derivative-type PSS over theoperating domain

The optimum parameters obtained using our proposed method, which is based onmeasurement of system dynamic performance in the time domain, is nowcompared with an earlier reported GA based PSS [32] where the principle is to usea frequency-domain approach that applies eigenvalue shifting technique.

��

____________________________________________________________________________________________

53

Understandably, the addition of an optimally tuned PSS enlarges the stabilityregion on the P-Q plane considerably. The conventional lead-lag PSS tuned usingthe proposed GA based method provides the largest stability region (the entiremarked region) (Figure 4.11). It is to be noted that this stability region isconsiderably larger than the domain actually considered for the GA basedoptimization (��+��!� �� ). Also shown in Figure 4.11 are thecorresponding stability regions with the derivative type PSS tuned using theproposed GA based method (��+�� !� � ��,�) and when tuned using theeigenvalue shifting method of [32] (��+��!�� ,�).

�� Stability regions on the � plane with different PSS

�� # ��Performance of the PSS with optimum parameters obtained using the proposedmethod was examined through dynamic analysis for various system loadingconditions (heavy, nominal and light), small perturbations, as well as large faults.Figure 4.12 and Figure 4.13 show the comparative performances of conventionallead-lag and derivative type PSS, respectively, for different load conditions whensubjected to one per cent change in mechanical torque. The system behavedsatisfactorily with both PSS for light and nominal load conditions. However,during heavy load, the settling time of the oscillations is considerably shorter andthe overall performance is better with the conventional lead-lag PSS.

��

____________________________________________________________________________________________

54

�� System performance under small perturbation for lead-lag PSS tuned usingthe proposed GA based method

�� System performance under small perturbation for derivative type PSS tunedusing the proposed GA based method

Further, comparing dynamic performance of the lead-lag PSS obtained using theproposed GA based method with two different PSS parameter sets of derivative

��

____________________________________________________________________________________________

55

type PSS, one obtained using our proposed method (PSS 1 in�Figure 4.14) and theother obtained in [32] (PSS 2 in�Figure 4.14), we note that the proposed GA basedlead-lag PSS provides considerably superior responses. This behavior was alsoexplained through the comparison of �� values in Table 4.2.

�� Dynamic behavior of conventional and derivative type PSS at nominaloperating point

Subsequent studies on dynamic performance analysis reported in this chapter arecarried out considering the lead-lag PSS only. The system is now tested for acombination of events, commencing with a small perturbation at time �"0,followed by a three-phase short circuit on one of the two parallel lines, very closeto the generator bus, at time �"3 seconds. The short circuit is cleared by theprotection system after 0.1 seconds, �� , at time �"3.1 seconds, by disconnectingthe faulted line.

The system dynamic response and the associated PSS output signal with the lead-lag PSS tuned using the proposed GA based method are shown in Figure 4.15,Figure 4.16 and Figure 4.17. It is seen that following the disturbance, the systemrecovers very satisfactorily, while reaching a new steady state in approximately 6seconds. It is also to be noted that in order to avoid large variations in terminalvoltages, a signal limiter has now been applied to restrict the PSS output withincertain pre-decided limits, [-0.1 p.u., 0.2 p.u.] in the present case.

��

____________________________________________________________________________________________

56

�� * Rotor angle deviation for the system with optimal GA based lead-lag PSSduring a small perturbation, followed by a three-phase short-circuit and removal of fault

�� + Angular speed deviation for the system with optimal GA based lead-lag PSSduring a small perturbation, followed by a three-phase short-circuit and removal of fault

��

____________________________________________________________________________________________

57

�� , Optimal GA based lead-lag PSS output signal during a small perturbation,followed by a three-phase short-circuit and removal of fault

�� $��%�� The proposed GA based technique incorporating Lyapunov’s parameteroptimization criterion is now used to determine the optimal parameters of PSS onthe three-machine system described in Section 2.6. In order to bring aboutrobustness, a set of six operating points (OP-1 to OP-6) is considered by varyingthe system loads, Load-A, Load-B and Load-C (refer Figure 2.3) in steps. The setof system load conditions considered for the GA based method is presented inTable 4.3.

�� The set of six load conditions (in p.u.)

./0� ./0� ./0� ./0 ./0* ./0+Load-A 0.83+j0.55 0.91+j0.61 0.99+j0.67 1.1+j0.73 1.21+j0.81 1.33+j0.97Load-B 0.44+j0.33 0.48+j0.36 0.53+j0.40 0.59+j0.44 0.64+j0.48 0.71+j0.59Load-C 0.55+j0.39 0.61+j0.42 0.67+j0.47 0.73+j0.51 0.81+j0.56 0.89+j0.68

Based on the above load conditions, an �� + �� + -./0 with"minimization of losses" as the criterion, is run for each load configuration. Theoptimal generation schedule so obtained for each unit for each load configurationconsidered is shown in Table 4.4. This OPF solution is used as the initial operatingcondition for the multi-machine system for PSS tuning using the proposed GAbased method.

��

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58

�� The OPF solution providing the initial operating conditions

1�0� 1�0� 1�0� 1�0 1�0* 1�0+��#0� 0.93+j0.16 1.02+j0.24 1.12+j0.33 1.23+j0.43 1.36+j0.54 1.49+j0.67��#0� 0.51-j0.04 0.56+j0.02 0.62+j0.08 0.68+j0.14 0.76+j0.22 0.84+j0.31��#0� 0.38-j0.14 0.42-j0.10 0.47-j0.05 0.52+j0.01 0.57+j0.07 0.63+j0.15

2� 1.04∠ 0° 1.04∠ 0° 1.04∠ 0° 1.04∠ 0° 1.04∠ 0° 1.04∠ 0°2� 1.025∠ -0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0°2� 1.025∠ 0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0° 1.025∠ 0°2� 1.07∠ -5.6° 0.99∠ -6.2° 0.99∠ -6.8° 0.98∠ -7.6° 0.97∠ -8.4° 0.95∠ -9.3°2� 1.02∠ -4.5° 1.02∠ -4.9° 1.01∠ -5.4° 1.0∠ -5.96° 0.99∠ -6.6° 0.98∠ -7.3°2� 1.02∠ -4.8° 1.01∠ -5.3° 1.01∠ -5.9° 1.00∠ -6.5° 0.99∠ -7.2° 0.99∠ -7.9°

Table 4.5 shows the optimal PSS parameters obtained using the proposed GAbased method. For the sake of comparison, the optimal parameters obtained usingthe ISE technique described in [41] are also presented. The eigenvalues of theclosed loop matrix �, for the optimal PSS parameter settings, are also provided. Itshould be noted that the ISE technique uses a sequential approach to tune theparameters while in the present GA based approach, all PSS have been tunedsimultaneously.

�� * Optimal PSS parameters using the proposed GA based technique as compared tothose obtained using ISE Technique

��0�� !0��

34�(��5!��#'��

34�(��5!��#'��

Generator-1:(45.06, 0.17)

Generator-1:(73.0, 0.11)

Generator-2:(45.52, 0.06)

Generator-2:(15.0, 0.10)

Generator-3:(2.13, 0.44)

-5.93 ± j16.37-1.54 ± j15.14-5.78 ± j12.63-2.19 ± j8.753-10.72 ± j2.78-4.52 ± j5.215

-35.834-34.191-24.167-18.238-5.0434-3.1556-0.1024-0.1000-0.1009

Generator-3:(12.5, 0.29)

-3.33 ± j23.48-1.42 ± j13.11-2.85 ± j10.41-9.085 ± j6.63-10.53 ± 4.055-1.983 ± j5.48

-41.7615-31.12-26.75

-14.866-2.69

-6.4069-0.1025-0.1001-0.1007

The average performance index, �� (given by (4.1)), of the "best" individual ineach generation is selected and plotted over generations to show its convergence

��

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59

rate. Figure 4.18 is an accurate representation of all GA based optimizationprocesses performed during this study, and presents the convergence rate evolutionof a population of 40 individuals, during a genetic process of 460 generations. Thevalues of the solution and performance index are presented in Table 4.6, case 7.

�� - Plot of JAVG for the ‘best’ individual for each generation

The corresponding PSS (a) time constants and (b) gains’ variations during thegenetic process are depicted in Figure 4.19, for 460 generations.

�� Parameters (�� and ��, � =1,2,3) of best individual, for each generation

��

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60

The convergence rate of the performance index is also reflected in the variation ofthe best individual of each generation during the entire genetic process.

Table 4.6 shows the performance indexes �� of the Lyapunov’s optimizationmethod based GA search solutions for different configurations and in differentstages of the genetic search process. It can be seen that the GA search provides thebest solution in Case 8, for �� = 50 over 500 generations.

�� + Performance of the proposed GA based PSS tuning method for different geneticconfigurations

��3��5/��

��#0� ��#0� ��#0�

��)��#"� #��6��

� 10-6

1. 50 50 (44.63, 0.18) (40.7, 0.06) (2.32, 0.45) 3.1606

2. 100 30 (44.29, 0.18) (49.38, 0.06) (2.48, 0.39) 3.1546

3. 100 50 (45.2, 0.17) (42.66, 0.06) (2.64, 0.37) 3.1546

4. 150 30 (44.84, 0.17) (46.15, 0.06) (3.16, 0.35) 3.1538

5. 200 30 (43.21, 0.17) (49.89, 0.06) (3.61, 0.35) 3.1545

6. 200 50 (42.03, 0.18) (43.94, 0.07) (3.42, 0.297) 3.1639

7. 460 40 (44.56, 0.17) (46.24, 0.06) (1.96, 0.46) 3.1526

8. 500 50 (45.06, 0.17) (45.52, 0.06) (2.13, 0.44) 3.1524

Usually, the solution is not reached in 50 generations and for reliable result,approximately 100 generations are required. A higher number of individuals inpopulation will increase the probability of finding the optimum solution in asmaller number of generations, but it will also increase the computational timeneeded to complete the evaluation of one generation. The simulations performedshow that, very often, a population with 30 individuals would suffice to find anoptimum within 150 generations ( �� Table 4.6, case 4).

However, of a big importance in the GA based optimization process, is theconvergence criterion, whose inadequate setting may cause premature terminationof the process, far away from the global optimum.

In order to test the robustness of the GA based PSS, three different operatingconditions were considered: a light load, a nominal load and a heavy load, as givenin Table 4.7. It might be noted that these load conditions at buses #5, #6 and #8, aresame as those used in [34], except that the corresponding generation levels areobtained here using an OPF simulation with "minimizing losses" as objective.

��

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61

�� , Three different loading conditions for examining the performance of the GAbased PSS

1��#��"�#��#��3�#��#��5Gen-1 Gen-2 Gen-3.��#�

P Q P Q P Q��#�� 0.71 0.28 1.63 0.07 0.85 -0.117��'8 2.77 1.20 1.38 0.50 1.26 0.36.�� 0.81 0.14 0.44 -0.11 0.36 -0.20

The dynamic responses are plotted for rotor speed deviation of generator-1following a 0.01 per unit step change in mechanical torque on the same generator.The responses are plotted for both the GA based PSS and the ISE technique basedPSS.

Figure 4.20 shows the plot for the nominal operating condition. The GA based PSShas a lower peak off-shoot and smaller oscillations and an overall better dampedresponse.

�� Rotor speed deviation on generator-1 with GA based PSS and ISE techniquebased PSS for nominal load condition

Figure 4.21 shows the plot of rotor speed deviations for the heavy operating loadcondition. It is evident that the GA based PSS performs distinctly better comparedto the ISE technique based PSS.

��

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�� Rotor speed deviations on generator-1 with GA based PSS and ISE techniquebased PSS for heavy load condition

�� Rotor speed deviations on generator-1 with GA based PSS and ISE techniquebased PSS for light load condition.

��

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63

Figure 4.22 shows the comparison of dynamic performances under a light loadoperating condition. In this case, the GA based PSS and the ISE technique basedPSS both do provide satisfactory responses.

Evidently, it can be said that the conventionally tuned PSS provides satisfactoryperformance at light loads and up to the nominal operating point, at which it istuned. However, when the system load increases beyond the nominal point, theperformance deteriorates. The GA based PSS, on the other hand, continues toperform well for all operating loads and hence has a higher level of robustness.

��&� �� This chapter presents a novel approach to tuning of Power System Stabilizers(PSS) using Genetic Algorithms (GA) based search process that incorporates theclassical Lyapunov optimization criterion. The advantage of using a GA basedsearch is that within these global search techniques a wide operating range can betaken in consideration in the tuning process. In contrast, the conventional tuningapproaches are based on one nominal operating condition. Furthermore, theproblems associated with eigenvalue drift arising from sequential tuning in multi-machine PSS is avoided.

The advantage of the proposed Lyapunov method based GA over other earlierreported GA methods is that the proposed method takes into consideration thedynamics of the system in the time-domain and is hence much more convenient tounderstand. The Integral of Squared Error criterion also provides an exactquantification of the system performance as against other methods, which primarilyuse the eigenvalue shift approach and measuring the damping factors. The optimalPSS obtained using the proposed method provides considerably superior dynamicperformances under a wide range of operating conditions. The computationalburden of the proposed method is within practical limits.

Genetic algorithms represent a useful tool for large-scale optimization problems,but inappropriate selection of genetic search parameters may lead to prematuretermination, or even to the divergence of genetic process. However, during theinvestigations reported here, the final solution was always found to be in vicinity ofthe same location within the search space, depending on the desired accuracy (pre-specified in the convergence criterion).

The proposed method has been tested on two different PSS structures- theconventional lead-lag and the derivative type, and two different system models- thesingle machine to infinite bus (SMIB) and a multi-machine model. Investigationsreveal that the conventional lead-lag PSS provides performances superior to thederivative PSS.

��

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64

The simulations and tests performed showed that the operating range for which thesystem proposed GA based PSS withstands small perturbations is much bigger thanthe range considered within the objective function.

The dynamic responses were satisfactory for large variations in system loadconditions, for different system topologies and even for transient phenomenaoccurred due to severe faults in the system.

��

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65

� ��

��

��

&�� !��In this chapter, the tuning of fixed structure proportional plus integral plusderivative (PID) PSS for the single-machine infinite bus and multi-machine powersystems has been considered. PID controllers have found applications in powersystem control problems for their simplicity and ease of realization. In [42] a pole-shifting self-tuning PID controller has been designed for damping of low frequencyoscillations in multi-machine systems. The PID controller gains are adapted in real-time to track the system conditions in order to provide robustness to the system. In[43], a fuzzy rule-base is used to tune the gain settings of a PID stabilizer. Theintroduction of fuzzy logic to tune the PID gains makes the PID control structureinherently non-linear. On-line tracking of the error signal and their time derivative(difference) is used to evaluate the gains. Genetic algorithm based PID controllershave been proposed in [44] for controller design to improve the transient stabilityof ac-dc lines after faults. The PID controller is applied to the HVDC controlsystem, both on the rectifier side as well as the inverter side and the gains are tunedsuch that the disturbance from a fault is minimum.

The tuning scheme proposed in this chapter uses a genetic algorithm (GA) basedsearch that integrates a classical parameter optimization criterion based on Integralof Squared Error (ISE). This method succeeds in achieving a robust,

�� !��

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66

simultaneously tuned and globally optimal PID-PSS parameter set, whilemaintaining the simplicity of the classical optimization method. The tuning methodimplicitly builds-in an increased robustness through an objective function, thatdepends on the operating domain.

PID controllers have been used for power system stabilization for their simplicityand ease of realization. They are feedback controllers whose output is generallybased on the error between a user-defined set point, ω� and the measured variableωt. Each element of the PID controller refers to a particular action taken on theerror for example, the �� is an adjustable amplifier that isusually responsible for system stability. The �� is responsible fordriving the error to zero, while the derivative gain �� is responsible for systemdamping.

In all practical implementations of PSS, the input signals are available in discreteform since digital instruments are used to measure the system variables such asspeed, voltages, terminal power, current, �� Therefore, it is important to capturethe effect of discrete inputs on PSS parameter settings, which understandably willbe affected. A proper selection of the sampling time is important because, though asmall sampling time would be desirable, it would nevertheless increase thecomputational burden significantly. On the other hand, a large sampling time willmiss significant system information on the dynamics while achieving fastcomputation. A proper compromise selection of �� is thus critical and shall bediscussed later in this chapter.

Tuning of a PID controller involves the adjustment of its gains ��, ��, and �� toachieve some user-defined "optimal" character of system response. The structure ofa PID PSS with rotor speed deviation as input can be represented as

∫ ⋅∆⋅+∆⋅+∆⋅= ��

��-�0��

ωωω )()( (5.1)

The above control logic can be expressed in discrete-mode as follows:

( ) ∑=

− ⋅∆⋅+∆−∆⋅+∆⋅=�

��

11 ωωωω (5.2)

In equation (5.2), the PSS parameters to be optimized are ��, �� and ��.

The small perturbation dynamic model of the multi-machine system without PSSwas discussed in detail in Chapter 2, and the transfer function block diagram

��

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67

representation is given in Figure 2.1. The PSS output signal �-�0�shall be acting onthe voltage regulator summing junction of each machine.

The general structure of the PID-PSS is shown in Figure 5.1 where ��, �� and ��

are the ��, � �� and �� gains respectively. Note that the inputto the PSS comprises discrete samples of the speed deviation signal ∆ω obtainedwith a sampling time ��.

��*� The general structure of a PID-PSS with discrete input signal

The small perturbation transfer function model for the above systemrepresentations can be expressed in state-space form as follows:

pu(t)xx(t) ⋅+⋅+⋅= 9��

(5.3)

In (5.3), � is the state matrix, 9 is the control matrix and � is the perturbationmatrix and depend on the system parameters and operating conditions, x(t) is thestate-vector defined in (5.4) and p is the perturbation vector. �� is the number ofgenerators.

�� (��(

�� ] [x ∈∀∆∆∆= δω (5.4)

The linear dynamic model of the composite system inclusive of excitation systemand the PID-PSS can be represented in state-space form as in (5.5). �� is thecorresponding composite system matrix.

p(t)xx(t) ⋅+⋅= ��

�(5.5)

The discrete mode equivalent of (5.3) can be expressed as:

��

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68

puxx Dkk1k ⋅+⋅+⋅=+ 7� (5.6)

�, 7 and �D are discrete-mode equivalents of �, 9 and � respectively and aredefined as follows:

( )( ) �

9� 7

�

�

�

�

⋅⋅−=

⋅⋅−=

=

−

−

1D

1

��

��

��

(5.7)

&�� The GA employed in this study uses an �� !�� in which the offspring iscreated with a � � �� of 90% and reinserted in the old population byreplacing the least fit predecessors. Most fit individuals are allowed to propagatethrough successive generations and only a better individual may replace them.

Each individual of a generation is a Gray coded binary string of search variables,each variable using a 30-bit representation. The selection process uses the�� method, a single-phase sampling algorithm withminimum �� , zero �� and time complexity in the order of the number ofindividuals ��. Recombination is performed using a� �� process with a probability of 0.7 and mutation is applied with a low probability of0.03.

Within the genetic search, the evaluation process is performed by an objectivefunction, which is a measure of the system’s behavior under a small perturbation.The average performance index �� is calculated as follows:

��

��

�

�

��

��

� ∈∀

⋅∆⋅= ∑ ∑

=,

1

, 1

2δ (5.8)

where �� is the number of operating points in the considered domain �.� is the sampling time�� is the rotor angle deviation

Simulations with population sizes ranging between 30 to 200 individuals have beenperformed. Very often, a population with 30 individuals would suffice to find anoptimum, and the number of generations required being proportional to the numberof variables. As the population size increases, the probability of finding the globaloptimum increases, while also increasing the simulation time required for eachgeneration. The results presented in this chapter have been obtained with

��

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69

population sizes of 40 and 60 individuals for single- and multi-machine systemsrespectively.

The convergence criterion is of critical importance and it determines the requirednumber of generations to complete the genetic process. Improper setting of thecriterion may lead to premature termination of the process, far away from theglobal optimum.

Figure 5.2 shows the workin scheme of the proposed GA based method for tuningof PID PSS using the ISE criterion.

��*� Proposed GA based tuning scheme for PID - PSS

��

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70

&�� !��"�� The proposed GA based tuning scheme is applied to a single-machine infinite bussystem operating over a wide operating domain. For this study case an operatingdomain �� -�� "� #$�%�� &�%'� �� "� # $�%�� &�$'0 comprising 154 operatingpoints was considered. Figure 5.3 shows the variation of the performance index ofthe best individual in current generation of the GA based search process and Figure5.4 shows the variation of the corresponding PID controller gains over thegenerations and their convergence towards the optimal solution.

��*� Genetic search performance for SMIB PID-PSS tuning

��* PID-PSS parameter variation and convergence during genetic process forSMIB

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It can be noted that the process reaches the optimum solution in about 25generations, after which the performance index reaches a value that remains steadyover the remaining search process, and genetic operators do not affect the bestindividual in consequent generations.

The optimal gains of the PID-PSS are shown in Table 5.1. Also shown in the table,are the corresponding optimum parameters of a lead-lag PSS (Table 4.1) tunedusing the GA based scheme discussed in Chapter 4. The values of �� for PID-PSS and lead-lag PSS are very close, indicating that both provide very goodperformance for the single-machine system considered.

��*� Optimum PSS parameters and performance index for GA based PID and lead-lagPSS

�� :0�� 0��

�� 38.28 473.84 -0.3 1.557x 10-6 33.98 0.12 1.598x 10-6

A comparison of the dynamic behavior of the SMIB system equipped withoptimally tuned PID and lead-lag PSS is shown in Figure 5.5. The system issubjected to a 1% step change in mechanical torque under heavy load operatingconditions (�"�&�1��"�&�&��). We can see that both the PSS show gooddynamic performance even for a load condition that is outside the operatingdomain � that was considered for the PSS tuning.

��** Rotor angle deviation of SMIB equipped with PID and Lead-lag PSS

��

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&�� '�� !#$��

For the design of the PID-PSS, the sampling time � plays an important role andTable 5.2 shows the dependence of optimum PSS setting and correspondingperformance index on �. We observe from Table 5.2 that a gradual deterioration inperformance takes place as � increases. In order to achieve a high degree ofaccuracy, a very small sampling time is desirable, which however increases thecomputational burden. The performance index is lowest for � = 0.001 secondswhich however will have a very high computing burden. As we progressivelyincrease � the performance index deteriorates gradually while reducing thecomputational burden. In our perspective, the best trade-off between accuracy andcomputational burden is at � = 0.01 seconds.

Now we examine if the chosen sampling period of � = 0.01 sec is the optimalchoice. The last column of Table 5.2 shows the performance index for different �

with ��. The �� works well up to � = 0.05 sec while beyondthis, the performance index is very high thus implying that re-tuning is required.

��*� Effect of TS on GA based PID-PSS design

1�� :0��#��#��#��#��

��"��

��(�10-5)

��(�10-5)

with ��

�� - *� 0�+� �*�� 3.84120.005 40.60 957 -1.02 1.5420 1.67150.01 38.28 473.84 -0.30 1.5568 1.55680.05 34.28 126.36 0.06 1.6922 2.45820.1 23.26 76.64 0.35 2.0325 very high

In all further investigations in this paper, we consider a sampling time of � = 0.01seconds.

&�� $��%�� The proposed GA based technique is now used to determine the optimal parametersof the PSS on the three-machine system (� � ��Figure 2.30. In order to bring aboutrobustness, a set of six operating points, as discussed in Section 4.4 is consideredby varying the system loads, Load-A, Load-B and Load-C, in steps. Based on theseload conditions, an �� + �� + -./0 with minimizing losses as thecriterion is run for each load configuration. Thus, the generation level for each unit

2 � �� is the optimum PSS obtained with TS = 0.01 sec ( � �KP=38.28, KD=473.84, KI=-0.30)

��

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73

in each load configuration considered is obtained and represents the initialoperating point for the system.

As in the case of SMIB system, Figure 5.6 shows the current generation bestindividual’s performance index and Figure 5.7 shows the convergence of PID-PSSparameters.

��*+ Genetic search performance for multi-machine PID PSS tuning

��*, Solution variation during genetic search for multi-machine PID-PSS tuning

��

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74

We note that the optimum solution is achieved in approximately 115 generationswhen �� attains a steady-state. However, the derivative and integral gains changeslightly even afterwards, while on the other hand, �� remains unchanged. Thisindicates a strong relationship between �� and �� and a weak dependencebetween the other two gains (�� and ��,) and ��.

Table 5.3 shows the optimal solutions obtained with the proposed technique forPID and lead-lag types of PSS (4.1). In this case, just by comparing the values ofthe corresponding performance indices it can be concluded that the PID-PSSperforms better.

��*� Optimum PSS solution and performance index for GA based PID and Lead-lagtypes of stabilizer

;�"��#� ��9�� :�� <��[x10-6]

��9��0��

<��[x10-6]

��

1 56.11 44.01 -0.24 45.06 0.172 12.92 2.75 4.91 45.52 0.063 29.93 29.81 28.92

2.552.13 0.44

3.15

In order to test the robustness of the GA based PSS we consider three differentoperating conditions, a light load, a nominal load and a heavy load, as given inTable 5.4. It might be noted that these load conditions at buses #5, #6 and #8, aresame as those used in [34], except that the corresponding generation levels areobtained here using an OPF simulation with minimizing losses as objective.

��* Loading conditions used to test the robustness of GA based PID-PSS

1��#��"�#��#��3�#��#��5

��#0� ��#0� ��#0�.��#�

� � � ��#�� 0.71 0.28 1.63 0.07 0.85 -0.117��'8 2.77 1.2 1.38 0.5 1.26 0.36.�� 0.81 0.14 0.44 -0.11 0.36 -0.2

Figure 5.8, Figure 5.9 and Figure 5.10 show the dynamic responses correspondingto PID and lead-lag PSS plotted for rotor speed deviation of generator-1 followinga 1% step change in mechanical torque on the same generator for nominal, lightand heavy load conditions, respectively. The PID-PSS has a lower peak offshootand a well-damped response for all cases investigated.

��

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75

��*- Rotor speed deviation on generator 1 for multi-machine system with PID orlead-lag stabilizer under nominal load conditions

��* Rotor speed deviation on generator 1 for multi-machine system with PID andlead-lag stabilizer under light load conditions

��

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76

��*�� Rotor speed deviation on generator 1 for multi-machine system with PID andlead-lag stabilizer under heavy load condition

&�� This chapter presents the design of a proportional plus integral plus derivative(PID) power system stabilizer (PSS) that uses the Integral of Squared Error (ISE)criterion to determine the performance of each individual during the searchprocess.

The advantage of using PID-PSS is that these are easy to realize and being indiscrete mode, their associated computations are lesser. Genetic algorithmsapplications to such PID-PSS provide an increased benefit, that of robustness.

Analytical studies show that the proposed GA based method for tuning PID-PSSprovides very satisfactory dynamic performances over a wide operating domainand that their performances are comparable with the conventional lead-lag PSS.

��

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77

� ��

This thesis attempts to apply the powerful properties of a genetic algorithm (GA)based search and optimization method to tuning of power system stabilizers (PSS).One of the primary requirements of a good tuning method is that the resulting PSSis robust enough to wide variations in system parameters, while also beingcomputationally manageable. In this respect, the proposed GA based tuningmethod provides satisfactory results.

The work presented here deals with the design of lead-lag, derivative type andproportional plus integral plus derivative (PID) types of PSS. Parameteroptimization based on Lyapunov’s method incorporating Integral of Squared Error(ISE) criterion has been used within the GA process in the proposed PSS tuningapproaches.

The thesis also examines classical approaches to tuning of lead-lag and derivativePSS that consider one nominal operating condition. Investigations reveal that theclassical approach does provide satisfactory performances for operating conditionsup to the nominal but deteriorated responses when the load increases. Morever, theclassically tuned PSS fails to stabilize the system at certain operating conditions.The proposed GA based method on the other hand, provides the option of includingany operating point within its tuning domain, thus ensuring system stability over alarge domain, and in particular, the tuning domain.

Genetic Algorithms represent a powerful tool to solving optimization problems.They possess an intrinsic flexibility and the freedom to choose desirable optimaaccording to design specifications. Therefore the objective function, being the partof the algorithm that models the actual dynamic problem we attempt to solve, playsa crucial role towards finding the global optimum.

!��"��#�

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78

Although the design method is meant to merely cope with small signal stabilityphenomena, when tested for transients, the system behaved satisfactorily.

(�� %��

� The thesis provides a broad-ranging overview of the research work carried outin the area of PSS tuning over the past two decades and brings out the mainresearch issues that have been addressed.

� The thesis provides a detailed description of the development of the systemmathematical models, both for single-machine infinite bus as well as the multi-machine system under small perturbations. These models are generic enoughand can be applied to large sized power systems.

� The thesis provides an exhaustive analysis of classical tuning methodsapplicable to lead-lag and derivative type PSS and brings out a comparison ofperformances achieved by systems having PSS designed using these methods.

� The thesis proposes a novel approach to tune lead-lag PSS using GA byapplying the Lyapunov’s method of parameter optimization. The main featureof this method is that it is a time-domain approach and uses a performancecriterion that accurately quantifies the dynamic performance of the systemunder perturbation. The genetic process further uses this in the individuals’fitness assignment stage as a measure of one’s quality, thereafter creating asound basis to finding the best individual in the population.

� The thesis develops a novel GA based optimization approach for tuning of PIDPSS. The main feature of this approach is that, the PID PSS acts in discretemode and thus, the system model has been developed in discrete domain. Anoptimal sampling period has been determined considering the conflictingrequirements of computation time vis-à-vis accuracy of information on systemdynamics, due to discretization.

(�� %��Tuning of PSS for large interconnected power systems has been a challengingproblem for power engineers and though a lot of work has been reported in thisarea, several issues remain unresolved. Based on the work reported in this thesis,we briefly layout some of the issues that need to be addressed within the sameframework discussed here, in order to gain an exhaustive understanding of theproblem of PSS tuning and its characteristics.

��

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79

� The system investigated has been limited up to a three generator, nine bussystem. It would be desirable to examine GA based PSS tuning for larger andmore realistic systems. Based on the experience accumulated duringsimulations and due to the development of both the system model and the GAprogram in a generic manner, the extension of the work could be done withoutdifficulties.

� As mentioned in one of the earlier chapters, siting of PSS is an important issue,more so, when the system size increases considerably. It is thus important toexamine the GA based PSS tuning method while incorporating the PSS sitingissues.

� The systems considered in the thesis assume that the loads are constantimpedance loads. It would be of interest to the designer to understand how thedynamics of the system will be affected by the load dependence on voltage andconsequently, how the optimal PSS parameters will be affected.

� The powerful properties of GA based optimization can be further exploited toexamine various other controller structures and determine their globallyoptimal settings.

� Test and implement different genetic algorithm strategies ( �� multi-population, multi-objective) in an attempt to achieve a less time consumingprocess and gain better understanding of genetic algorithms applicability tovarious power system phenomena.

��

____________________________________________________________________________________________

81

� ��

[1] P. Kundur, "A Course on Power Stability and Control", ABB T&D University,Ludvika, Sweden, April 2000.

[2] F.P. DeMello and C. Concordia, "Concepts of synchronous machines stability asaffected by excitation control", IEEE Trans. on Power Apparatus and Systems, Vol.PAS-88, Apr.’69, pp. 316-329.

[3] W.G. Heffron, R.A. Phillips, "Effect of modern amplidyne voltage regulator onunder-excited operation of large turbine generators", AIEE Transactions on PowerApparatus and Systems, Vol. PAS-71, Aug.’52, pp. 692-697.

[4] E.V. Larsen and D. A. Swann, "Applying power system stabilizers. Part I: Generalconcepts; Part II: Performance objectives and tuning concepts; Part III: Practicalconsiderations", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, June ’81, pp. 3017-3046.

[5] P. Kundur, M. Klein, G. J. Rogers and M. S. Zywno, “Application of power systemstabilizers for enhancement of overall system stability”, IEEE Transactions onPower Systems, Vol. 4, May 1989, pp. 614-626.

[6] Y.N. Yu and C. Siggers, "Stabilization and optimal control signals for a powersystem", IEEE Transactions on Power Apparatus and Systems, Vol.PAS-90,Jul/Aug.'71, pp.1469-1481.

[7] H.A.M. Moussa and Y.N. Yu, "Optimal power system stabilization throughexcitation and/or governor control", IEEE Transactions on Power Apparatus andSystems, Vol.PAS-91, May/June'72, pp.1166-1174.

[8] F.P. DeMello, P.J. Nolan, T.F. Laskowski and J.M. Undrill, "Co-ordinatedapplication of stabilizers in multi-machine power systems", IEEE Transactions onPower Apparatus and Systems, Vol. PAS-99, May/June'80, pp.892-901.

[9] R.J. Fleming, M.A. Mohan and K. Parvatisam, "Selection of parameters ofstabilizers in multi-machine power systems", IEEE Transactions on PowerApparatus and Systems, Vol.PAS-100, May'81, pp.2329-2333.

[10] O.H.Abdalla, S.A.Hassan and N.T.Tweig, "Co-ordinated stabilization of a multi-machine power system", IEEE Transactions on Power Apparatus and Systems, Vol.PAS-103, Mar.'84, pp. 483-494.

[11] A. Doi and S. Abe, "Coordinated synthesis of power system stabilizers inmultimachine power systems", IEEE Transactions on Power Apparatus andSystems, Vol.PAS-103, June'84, pp.1473-1479.

[12] C.M. Lim and S. Elangovan, "Design of stabilizers in multi-machine powersystems", Proceedings of IEE, Part-C, Vol.132, May'85, pp.146-153.

[13] C.M. Lim and S.Elangovan, "New approach to power system stabilizer design",Electric Power Systems Research, Vol.8, 1985, pp.285-292.

[14] S. Elangovan and C.M. Lim, "Efficient pole-assignment method for designingstabilizers in multi-machine power systems", Proceedings of IEE, Part-C, Vol.134,Nov.'87, pp.383-384.

[15] A. Ghosh, G. Ledwich, O.P. Malik and G.S. Hope, "Power system stabilizer basedon adaptive control techniques", IEEE Transactions on Power Apparatus andSystems, Vol. PAS-103, Aug. '84, pp. 1983-1989.

"��

____________________________________________________________________________________________

82

[16] S.J. Cheng, Y.S. Chow, O.P. Malik and G.S. Hope, "An adaptive synchronousmachine stabilizer", IEEE Transactions on Power Systems, Vol.PWRS-1, Aug.’86,pp.101-109.

[17] S.J. Cheng, O.P. Malik and G.S. Hope, "Damping of multi-modal oscillations inpower systems using a dual-rate adaptive stabilizer", IEEE Transactions on PowerSystems, Vol.PWRS-3, Feb. ’88, pp. 101-108.

[18] C.M. Lim, "A self-tuning stabilizer for excitation or governor control of powersystems", IEEE Transactions on Energy Conversion, Vol. 4, June’89, pp. 152-159.

[19] C.M.Lim and T.Hiyama, "Self-tuning control scheme for stability enhancement ofmulti-machine power systems", Proceedings of IEE, Part-C, Vol.137, July’90,pp.269-275.

[20] C.M.Lim and T.Hiyama, "Application of a self-tuning control scheme to a powersystem with multi-mode oscillations", Electric Power Systems Research, Vol.24,1992, pp.91-98.

[21] T. Hiyama, "Application of rule-based stabilizing controller to electric powersystem", Proceedings of IEE, Part-C, Vol.136, May’89, pp.175-181.

[22] T. Hiyama, "Rule-based stabilizer for multi-machine power system", IEEETransactions on Power Systems, Vol.PWRS-5, May ’90, pp.403-411.

[23] W.C. Chan and Y.Y. Hsu, "An optimal variable structure stabilizer for power systemstabilization", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102,June’83, pp. 1738-1746.

[24] M.L. Kothari, J. Nanda, K. Bhattacharya, “Design of variable structure powersystem stabilizers with desired eigenvalues in the sliding mode”, IEE Proceedings-C(Generation, Transmission and Distribution), Vol. 140, No. 4, July 1993, pp. 263-268.

[25] K. Bhattacharya, M.L. Kothari, J. Nanda, “Design of discrete-mode variablestructure power system stabilizers”, Electrical Power & Energy Systems, Vol. 17,No. 16, 1995, pp. 399-406.

[26] Y.-Y Hsu and C.-H Cheng, "Design of fuzzy power system stabilisers formultimachine power systems", IEE Proceedings, Vol. 137, Pt. C, No. 3, May 1990.

[27] P. Hoang and K. Tomsovic, “Design and analysis of an adaptive fuzzy power systemstabilizer”, IEEE Transactions on Energy Conversion, Vol. 11, Dec. 1996, pp. 455 -461.

[28] Y. Zhang, G.P. Chen, O.P. Malik, G.S. Hope, "An artificial neural network basedadaptive power system stabilizer", IEEE Transactions on Energy Conversion, Vol. 8,No. 1, March 1993.

[29] R. Segal, M. L. Kothari and S. Madnani, “Radial basis function (RBF) networkadaptive power system stabilizer”, IEEE Transactions on Power Systems, Vol. 15,May 2000, pp. 722-727.

[30] M.A. Abido and Y.L. Abdel-Magid, "A hybrid neuro-fuzzy power system stabilizerfor multimachine power systems", IEEE Transaction on Power Systems, Vol. 13,No. 4, November 1998.

[31] D. E. Goldberg, 2 � ��3�� .��*��)�� 4 ��,Addison-Wesley Publishing Company Inc., January 1989.

[32] Y. L. Abdel-Magid, M. Bettayeb, M. M. Dawoud, “Simultaneous stabilization ofpower systems using genetic algorithm”, IEE Proceedings-Generation, Transmissionand Distribution, Vol. 144, No. 1, January 1997.

��

____________________________________________________________________________________________

83

[33] M. A. Abido and Y. L. Abdel-Magid, "Hybridizing rule-based power systemstabilizers with genetic algorithms", IEEE Transactions on Power Systems, Vol.14,May 1999, pp. 600-607.

[34] Y.L. Abdel-Magid, M.A. Abido, S. Al-Baiyat and A.H. Mantawy, "Simultaneousstabilization of multi-machine power systems via genetic algorithms", IEEETransactions on Power Systems, Vol. 14, Nov. ’99, pp. 1428-1439.

[35] P. Zhang and A.H. Coonick, "Coordinated Synthesis of PSS Parameters in Multi-Machine Power Systems Using the Method of Inequalities Applied to GeneticAlgorithms", IEEE Transactions on Power Systems, Vol. 15, No. 2, May 2000.

[36] P. M. Anderson and A. A. Fouad, �+ �� !�� !, Iowa StateUniversity Press 1977.

[37] P. Kundur, "�+ ��!�� !��5��", McGraw-Hill Inc. 1994.[38] Yao-Nan Yu, (� ��+ ��!�� !��, Academic Press 1983.[39] K. Bhattacharya, J. Nanda and M. L. Kothari, “Optimization and performance

analysis of conventional power system stabilizers”, International Journal ofElectrical Power & Energy Systems, Vol. 19, 1997, pp. 449-458.

[40] Y.N. Yu and Q.H. Li, "Pole-placement power system stabilizers design of anunstable nine-machine system", IEEE Transactions on Power Systems, Vol. PWRS-5, May '90, pp.353-358.

[41] K. Bhattacharya, M. L. Kothari, J. Nanda, M. Aldeen and A. Kalam, “Tuning ofpower system stabilizers in multi-machine systems using ISE technique”, ElectricPower Systems Research, Vol. 46, 1998, pp. 119-131.

[42] C. J. Wu and Y. Y. Hsu, "Design of self-tuning PID power system stabilizer formulti-machine power systems", IEEE Transactions on Power Systems, Aug. 1988,pp. 1059-1064.

[43] P. K. Dash, A. C. Liew and B. R. Mishra, "An adaptive PID stabilizer for powersystems using fuzzy logic", Electric Power Systems Research, Volume 44, March1998, pp. 213-222.

[44] Y. P. Wang, N. R. Watson and H. H. Chong, "Modified genetic algorithm approachto design of an optimal PID controller for AC-DC transmission systems",International Journal of Electrical Power & Energy Systems, Volume 24, Jan 2002,pp. 59-69.

��

____________________________________________________________________________________________

85

� ��

)�� *�!

)�� +��%��This property is used to obtain the network reduction as shown below.

Let

VI ⋅= = (8.1)

where V is the bus voltage vector, = is the admittance matrix of the system and is the bus current vector. Therefore,

=

�

� (8.2)

Now the matrices � and � are partitioned accordingly to get

=

�

�

��

��

�

�

��

��

�

�(8.3)

where the subscript � is used to denote generator nodes and the subscript � is used

for the remaining nodes (Figure 8.1). Thus for the analyzed system �

� , which now

��

____________________________________________________________________________________________

86

is the generator terminal voltage vector, has the dimension (� × 1) and �

� has thedimension (� × 1).

1��

2��

1�1�

1’��

11 δ∠�

Transmission network

Reference node

2�2�

2’��

+

_22 δ∠�

��

�1’�

��

�� δ∠

+

_

+

_

�� Multi-machine system representation

Expanding equation (8.3),

rn

rnn

VV0

VVI

⋅+⋅=

⋅+⋅=

��

��

��

��(8.4)

from which we eliminate V , to find

n1

n V)(I ⋅⋅⋅−= −��

�� (8.5)

If m1 )( �� =⋅⋅− −

��, then (8.5) becomes:

nmn VI ⋅= � (8.6)

The matrix �

� is the desired reduced matrix of � �It has the dimensions (�×�),

where � is the number of generators.

��

____________________________________________________________________________________________

87

��

The phasor diagram of the �th machine of a multi-machine system may be shown asin Figure 8.2. While

� and

� are the coordinates for the �th machine alone, D and

Q are the common coordinates for all machines in the system. The phase-angledifference between

� and D, or

� and Q, is denoted by ��, which is constantly

changing and could be positive or negative.

D-axis

Q-axis

qi-axis

di-axis

��

��

��

�� ’ �� )’( −

�� ’

�δ

�δ

�� Phasor diagram of the��th machine

� �� The upper bar stands for complex values. Since we look only into thereduced network (having only the generator nodes), we leave out the subscript �standing for generator nodes. Therefore, the terminal voltage

�� of the �th machine

of the system in common coordinates becomes:

��

��

��

��

�� δδ −− ⋅⋅−+⋅⋅−⋅= )( ’’)90(’ (8.7)

Note that

( ) ��

��

��

��

��

� �� δδ −− ⋅⋅=⋅= ,90’’(8.8)

For � machines of an �-machine system, equation (8.7) may be written in matrixform

��

____________________________________________________________________________________________

88

( )[ ] [ ] [ ] [ ] ��

�� ⋅⋅−+⋅⋅−⋅= −− δδ ’’90 ’ (8.9)

where the coefficients ( )[ ] [ ] [ ] [ ]δδ �

��

� �� −− − and’,’,90 should be read as

diagonal matrices and �� and,,, are column vectors of size �.

��

Substituting the solution of � of (8.9) in (8.6) and solving for � gives

( )[ ] [ ] [ ] [ ]( )�� ⋅⋅−+⋅⋅−⋅⋅= −− δδ �

�� ’’90 (8.10)

where

[ ]( ) 11m ’

−− ⋅+=�� (8.11)

Note that the admittance matrix � in equation (8.11) refers to the reduced network,therefore differs from Y matrix from (8.1).

For the �th machine of an �-machine system in �� coordinates, the current has �terms

∑=

⋅−−

⋅⋅−+⋅=

�

��

��

��

��

�� 1

’’)90()(

δδ(8.12)

including the term of ��.

In ��coordinates,

=⋅= ��

�� δ ∑

=

+⋅−++⋅

⋅⋅

−+⋅

�

��

��

��

��

��

�� 1

)(’’)90( δβδβ(8.13)

where

��

�

��

�� δδδβ −=⋅= , (8.14)

Therefore

��

____________________________________________________________________________________________

89

∑

∑

=

=

⋅⋅

−+⋅⋅==

⋅⋅

−+⋅−⋅==

�

��

��

��

��

�

��

��

��

��

��

��

1

’’

1

’’

)Im(

)Re(

(8.15)

where

( ) ( )��

�� δβδβ +=+= sin,cos (8.16)

Let the deviation of �� be defined by

( )�

��

�� ∆⋅+∆=⋅+∆ (8.17)

Here �’s stand for currents in individual machine coordinates. Same rule will beapplied for �’s.

From (8.15) and for � machines, we will have

’qqqqq

qd’qddd

EI

IEI

⋅+⋅=⋅

⋅+⋅+⋅=

��

��(8.18)

where

��

��

��

��

��

��

��

��

�

��

��

�

��

��

�

��

��

�

��

��

��

��

��

��

��

��

,...,1

1

,...,1,,

,

’

’

’

ij

q’’

q’’

=⋅

−⋅=

⋅

−⋅−=

≠⋅

−⋅−=

==⋅−=

−=−=

⋅⋅

−−⋅⋅=

⋅⋅

−+⋅⋅=

∑∑≠≠

(8.19)

��

____________________________________________________________________________________________

90

The solutions for �Id and �Iq of (8.18) become

EI

EI

q’qqq

d’qdd

⋅+⋅=

⋅+⋅=

��

��(8.20)

where

q1

qqq1

qq

qdddqddd

,

,

��

��

⋅=⋅=

⋅+=⋅+=−− (8.21)

�� Having found the � and � current components including the transmission relation,�� will be expressed using the electrical torque expression, internal voltageequation, and from the terminal voltage relation, as it will be shown below.

An electric torque approximately equals an electric power when the synchronousspeed is chosen as the base speed. For the �th machine,

( ) ( )�

��

��

�� ⋅−⋅+⋅=⋅≅ ’’*Re (8.22)

After linearization and for � machines, (8.22) becomes:

’q21e ET ⋅+⋅= �� (8.23)

where

q0dtqt2

dtqt1

��

��

+⋅+⋅=

⋅+⋅=(8.24)

in which �, �� and �� are diagonal matrices with the elements as it follows

��

��

��

��

��

��

��

��

��

00

’00

’

0’

=

+⋅

−=

⋅

−=

(8.25)

��

____________________________________________________________________________________________

91

The component of torque given by K1 is in phase with ��, hence representing asynchronizing torque component. The second term of ( ) represents the componentof torque resulting from variations in field flux linkage.

The internal voltage equation for � machines may be written

( ) [ ] dfdqdo I’E’E’ ∆⋅−−∆=∆⋅⋅+�� (8.26)

where ��is the unity matrix and �� a diagonal matrix. Substituting �Id of (8.20) in(8.26) and shifting terms gives

( )

∆⋅−∆⋅−∆⋅=∆⋅⋅⋅+ ∑∑

=≠

�

�

��

�

��

��

��

��1

3’

33

’3

’ 11 δ (8.27)

where

( )( )( )( )( )( )

��

��

��

��

��

��

⋅−=

⋅−=

⋅−+=−

−

’4

1’3

1’3 1

(8.28)

The terminal voltage relation could be written as

[ ][ ] d

’qq

qd

I’EV

IV

∆⋅−∆=∆

∆⋅=∆

�

�

�

�(8.29)

Furthermore

qq00dd01

0 VV ∆⋅⋅+∆⋅⋅=∆ −− �� (8.30)

which can be written as

’q65 EV ∆⋅+∆⋅=∆ �� (8.31

where

��

____________________________________________________________________________________________

92

[ ] [ ][ ] [ ] vdvqv6

dvqv5

’

’

��

��

+⋅⋅−⋅⋅=

⋅⋅−⋅⋅=

��

��

��

��(8.32)

and

01

0v01

0v , �� ⋅=⋅= −−

In these equations, ��,��,��, �, ��, [xq] and [xd’] should be read as diagonalmatrices.

��

____________________________________________________________________________________________

93

��

�� All data are in p.u., except when specified.

= 0.8� = 0.6� = 1.0� = 0.0� = 0.2� = 0.0� = 0.0�� = 1.60�� = 1.55��’ = 0.32 = 10.0 [s]�� = 6.0 [s]� = 50 [Hz]�� = 50�� = 0.05 [s]�� = 0.01�� = 0.05 [s]�� = 0.05 [s]�� = 10.0 [s]

��

____________________________________________________________________________________________

94

�� !"�##�� $�%�&'�(

�� Generator data

�� Rated MVAVoltage, kVPower factor

TypeSpeed, r/min

��, pu��, pu�� pu�� pu�� pu

��, sec��, secH, sec

247.516.51.0

hydro180

0.14600.06080.09690.09690.0336

8.960.0

23.64

192.018.00.85

steam3600

0.89580.11980.86450.19690.0521

6.000.5356.40

128.013.80.85

steam3600

1.31250.18131.25780.250

0.07425.890.603.01

�� Network data

�� ! �"��#�[pu]

$"!%��#�[pu]

� ��%�[kV/kV]

1-4 trafo 0.0 + �0.0576 0.0 + �0.0 16.5/2302-7 trafo 0.0 + �0.0625 0.0 + �0.0 18/2303-9 trafo 0.0 + �0.0586 0.0 + �0.0 13.8/2304-5 line 0.010 + �0.085 0.0 + �0.088 -4-6 line 0.017 + �0.092 0.0 + �0.079 -5-7 line 0.032 + �0.161 0.0 + �0.153 -6-9 line 0.039 + �0.170 0.0 + �0.179 -7-8 line 0.0085 + �0.072 0.0 + �0.0745 -8-9 line 0.0119 + �0.1008 0.0 + �0.1045 -

��

____________________________________________________________________________________________

95

��

�� ')�� *�� '��+��

−−−−−−−

−−−−−−

−−−−−−−

−−−−−−−

−−−−−−

−−−−−−−

−−−−−−−

−−−−−−

−−−−−−−

=

33

3353

3

3353

3

3263

3

3253

3

3163

3

3153

3

’

3333

’

3

’334

3233

’

3

’324

3133

’

3

’314

3

332

3

331

3

3

3

322

3

321

3

312

3

311

2

2362

2

2352

22

2262

2

2252

2

2162

2

2152

2332

’

2

’234

2

’

2232

’

2

’224

2132

’

2

’214

2

232

2

231

2

222

2

221

2

2

2

212

2

211

1

1361

1

1351

1

1261

1

1251

11

1161

1

1151

1331

’

1

’134

1231

’

1

’124

1

’

1131

’

1

’114

1

132

1

131

1

122

1

121

1

112

1

111

1

1

100000

1100

100

10

000200000000

00000

001

000

01

011

001

0

000000020000

00000

00001

0

01

001

011

0

000000000002

00000

��

�

�

�

�

�

�

�

�

�

�

�

��

�

�

�

�

��

�

�

�

�

�

�

�

��

�

�

�

�

�

�

�

�

��

�

�

�

��

��

��

�

��

�

��

�

��

�

��

�

��

��

�

��

�

��

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

��

�

��

��

��

�

��

�

��

�

��

��

�

��

�

��

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

��

�

��

�

��

�

��

��

��

�

��

��

�

��

�

��

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

π

π

π

�

�

�

�

�

�

�

�

�

�

�

�

�

�

=

3

3

2

2

1

1

00000000000

00000000000

00000000000

�

�

�

�

�

=

0002

100000000

00000002

10000

000000000002

1

3

2

1

��

____________________________________________________________________________________________

96

�� ')�� *�� +��

Let us consider the following notations:

43

3333

23

1333

42

3222

22

1222

41

3111

21

1111

;

;

;

�

��

�

��

�

��

�

��

�

��

�

��

�

�

�

⋅=

⋅=

⋅=

⋅=

⋅=

⋅=

Thus, the matrices of the multi-machine power system introduced in Chapter 3become:

=

12,9311,9310,939,938,937,936,935,934,933,932,931,93

12,9311,9310,939,938,937,936,935,934,933,932,931,93

12,911,910,99,98,97,96,95,94,93,92,91,9

12,5211,5210,529,528,527,526,525,524,523,522,521,52

12,5211,5210,529,528,527,526,525,524,523,522,521,52

12,511,510,59,58,57,56,55,54,53,52,51,5

12,1111,1110,119,118,117,116,115,114,113,112,111,11

12,1111,1110,119,118,117,116,115,114,113,112,111,11

12,111,110,19,18,17,16,15,14,13,12,11,1

��

��

��

��

��

��

��

��

��

�

where �� are the elements of the $ matrix of the multi-machine system describedin Section 8.3.1.

�

��

��

��

��

��

��

��

�

��

��

��

��

�

��

��

��

��

�

=

43233

33133

233

133

3

42222

32122

222

122

2

41211

31111

211

111

1

1

222

1000000

000222

1000

000000222

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