Definition and Classification of power system stability.pdf
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO.2, MAY 2004 1387 Definition and Classification of Power System Stability IEEE/CIGRE Joint Task Force on Stability Terms and Definitions Prabha Kundur (Canada, Convener), John Paserba (USA, Secretary), Venkat Ajjarapu (USA), Göran Andersson (Switzerland), Anjan Bose (USA) , Claudio Canizares (Canada), Nikos Hatziargyriou (Greece), David Hill (Australia), Alex Stankovic (USA), Carson Taylor (USA), Thierry Van Cutsem (Belgium), and Vijay Vittal (USA) Abstract—The problem of defining and classifying power on the subject by CIGRE and IEEE Task Forces –. These, system stability has been addressed by several previous CIGRE however, do not completely reflect current industry needs, ex- and IEEE Task Force reports. These earlier efforts, however, periences, and understanding. In particular, definitions are not do not completely reflect current industry needs, experiences and understanding. In particular, the definitions are not precise precise and the classifications do not encompass all practical in- and the classifications do not encompass all practical instability stability scenarios. scenarios. This report is the result of long deliberations of the Task Force This report developed by a Task Force, set up jointly by the set up jointly by the CIGRE Study Committee 38 and the IEEE CIGRE Study Committee 38 and the IEEE Power System Dynamic Performance Committee, addresses the issue of stability definition Power System Dynamic Performance Committee. Our objec- and classification in power systems from a fundamental viewpoint tives are to: and closely examines the practical ramifications. The report aims to define power system stability more precisely, provide a system- • Define power system stability more precisely, inclusive of atic basis for its classification, and discuss linkages to related issues all forms. such as power system reliability and security. • Provide a systematic basis for classifying power system Index Terms—Frequency stability, Lyapunov stability, oscilla- stability, identifying and defining different categories, and tory stability, power system stability, small-signal stability, terms providing a broad picture of the phenomena. and definitions, transient stability, voltage stability. • Discuss linkages to related issues such as power system reliability and security. I. INTRODUCTION Power system stability is similar to the stability of any dynamic system, and has fundamental mathematical under- P OWER system stability has been recognized as an important problem for secure system operation since the 1920s , . Many major blackouts caused by power system instability have pinnings. Precise definitions of stability can be found in the literature dealing with the rigorous mathematical theory of stability of dynamic systems. Our intent here is to provide a illustrated the importance of this phenomenon . Historically, physically motivated definition of power system stability which transient instability has been the dominant stability problem on in broad terms conforms to precise mathematical definitions. most systems, and has been the focus of much of the industry’s The report is organized as follows. In Section II the def- attention concerning system stability. As power systems have inition of Power System Stability is provided. A detailed evolved through continuing growth in interconnections, use of discussion and elaboration of the definition are presented. new technologies and controls, and the increased operation in The conformance of this definition with the system theoretic highly stressed conditions, different forms of system instability definitions is established. Section III provides a detailed classi- have emerged. For example, voltage stability, frequency stability fication of power system stability. In Section IV of the report the and interarea oscillations have become greater concerns than relationship between the concepts of power system reliability, in the past. This has created a need to review the definition and security, and stability is discussed. A description of how these classification of power system stability. A clear understanding terms have been defined and used in practice is also provided. of different types of instability and how they are interrelated Finally, in Section V definitions and concepts of stability from is essential for the satisfactory design and operation of power mathematics and control theory are reviewed to provide back- systems. As well, consistent use of terminology is required ground information concerning stability of dynamic systems in for developing system design and operating criteria, standard general and to establish theoretical connections. analytical tools, and study procedures. The analytical definitions presented in Section V constitute The problem of defining and classifying power system sta- a key aspect of the report. They provide the mathematical un- bility is an old one, and there have been several previous reports derpinnings and bases for the definitions provided in the earlier sections. These details are provided at the end of the report so Manuscript received July 8, 2003. that interested readers can examine the finer points and assimi- Digital Object Identifier 10.1109/TPWRS.2004.825981 late the mathematical rigor. 0885-8950/04$20.00 © 2004 IEEE Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. Downloaded on March 04,2010 at 13:39:06 EST from IEEE Xplore. Restrictions apply. however. Restrictions apply. In an equilibrium set. a power system may be stable for a a small spherical neighborhood of radius result in the system given (large) physical disturbance. Often.. A stable equilibrium set thus the analytical definition directly correlates to the expected be- has a finite region of attraction. It must also logical changes occurred in the system). and definition which. if the system is unstable. and (b) at time the system trajec- are selected on the basis they have a reasonably high probability tory approaches the equilibrium point which corresponds to the of occurrence. Discussion and Elaboration elements or intentional tripping to preserve the continuity of op- The definition applies to an interconnected power system as a eration of bulk of the system. it is usually valid to as- system motion around an equilibrium set. on the initial operating condition as well as the nature of the Power systems are continually experiencing fluctuations disturbance.. a progres- The power system is a highly nonlinear system that oper. The response of the power system to a disturbance may in. Further. for assessing stability when Stability of an electric power system is thus a property of the subjected to a specified disturbance. motors may lose stability (run down and stall) state. through a single contiguous transmission system. Hence. NO. we provide a formal definition of power erator and transmission network voltage regulators. the system must be able to adjust to the rium or return to the original operating condition (if no topo- changing conditions and operate satisfactorily. the larger the region. it will state of operating equilibrium after being subjected to a reach a new equilibrium state with the system integrity pre- physical disturbance. a fault on a crit.e. or a pro- ates in a constantly changing environment. is impractical and uneconomical to design power systems to be the initial time which corresponds to all of the system state vari- stable for every possible disturbance. the various opposing condition. loads. It should be recognized generator. of small magnitudes. the more havior in a physical system. On the other hand. i. with practically all generators and loads connected so that practically the entire system remains intact. the stability of the system depends tion of the power system. DEFINITION OF POWER SYSTEM STABILITY cause variations in power flows. The region of attraction changes with the operating condition of the power III. network bus voltages. thereby weakening the system and possibly leading • Power system stability is the ability of an electric power to system instability. These requirements are be able to survive numerous disturbances of a severe nature. it will result without cascading instability of the main system. ering a given operating condition and the system being subjected Power systems are subjected to a wide range of disturbances. CLASSIFICATION OF POWER SYSTEM STABILITY system. It trajectory remaining in a cylinder of radius for all time . is easily understood and readily applied by power to varying degrees depending on their individual characteristics. As a result. system engineering practitioners. The intent is to provide a physically based ator speed variations will actuate prime mover governors. may also be intentionally split into two group of generators is also of interest. Proposed Definition equipment. the voltage variations will actuate both gen- In this section. directly correlated to the system-theoretic definition of asymp- such as a short circuit on a transmission line or loss of a large totic stability given in Section V-C-I. For instance.e. system. 1388 IEEE TRANSACTIONS ON POWER SYSTEMS. and unstable for another. Some gener- ators and loads may be disconnected by the isolation of faulted B. stability of particular loads or load areas human operators will eventually restore the system to normal may be of interest. The design contingencies ables being bounded. sive increase in angular separation of generator rotors. Downloaded on March 04. all initial conditions starting in At an equilibrium set. MAY 2004 II. . Interconnected systems. with most system variables bounded served i. gressive decrease in bus voltages. for cer- whole. one observes that to a specified disturbance scenario. Similarly. An unstable system condition puts and key operating parameters change continually. for a given initial operating condition. A typical modern power system is a high-order multivariable volve much of the equipment. 2. and ma- chine rotor speeds. Sta- Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. sume that the system is initially in a true steady-state operating erating condition. the stability of a particular generator or tain severe disturbances. in the sense of Lyapunov. Under these conditions we require small and large. Small disturbances in the form of load changes the system to either regain a new state of operating equilib- occur continually. Conformance With System—Theoretic Definitions slow cyclical variations due to continuous small fluctuations in In Section II-A. A remote generator may or more “islands” to preserve as much of the generation and lose stability (synchronism) without cascading instability of the load as possible. devices used to protect individual equipment may re- spond to variations in system variables and cause tripping of the A. we have formulated the definition by consid- loads or aperiodic attractors). generator out. However. to a physical disturbance. while conforming to definitions from system the voltage and frequency variations will affect the system loads theory. robust the system with respect to large disturbances. A large disturbance may lead to structural changes here that this definition requires the equilibrium to be (a) stable due to the isolation of the faulted elements.. VOL. to regain a If following a disturbance the power system is stable. When could lead to cascading outages and a shutdown of a major por- subjected to a disturbance. i. the initial op. for example. in a run-away or run-down situation.e. 19. process whose dynamic response is influenced by a wide array ical element followed by its isolation by protective relays will of devices with different characteristics and response rates. forces that exist in the system are equal instantaneously (as in the case of equilibrium points) or over a cycle (as in the case of C. The actions of automatic controls and possibly main system.2010 at 13:39:06 EST from IEEE Xplore. the gener- system stability. large-disturbance stability always refers equilibrium point being attractive. Classification. or between groups of machines. is essential for meaningful the system.Small-disturbance stability depends on the initial op- Rotor angle stability refers to the ability of synchronous ma. it helps to make by a decrease in power transfer such that the angular separation simplifying assumptions to analyze specific types of problems is increased further. Analysis of stability. synchronism can occur between one machine and the rest of gories .1 Rotor Angle Stability: of analysis . with synchronism practical analysis and resolution of power system stability maintained within each group after separating from each other. based on the following considerations : System stability depends on the existence of both components • The physical nature of the resulting mode of instability as of torque for each of the synchronous machines. bances are considered to be sufficiently small that lin- earization of system equations is permissible for purposes B. 1 gives the overall picture of the power system stability • Small-disturbance (or small-signal) rotor angle stability problem. Categories of Stability angle deviation. For any given situation. If one generator temporarily runs faster than another. . Restrictions apply. insufficient damping of oscillations. differences. Need for Classification chine will advance. As discussed in Section V-C-I. This tends to however.2010 at 13:39:06 EST from IEEE Xplore. the method of calculation and prediction of stability. the stability of the system including identifying key factors that contribute to instability depends on whether or not the deviations in angular positions and devising methods of improving stable operation. system operating condition output electromagnetic torque of each generator. identifying its categories and subcategories. and the time span that must be the nature of stability problems. Loss of facilitated by classification of stability into appropriate cate. The resulting angular difference transfers part of the load from the slow machine to the fast machine. The distur- phenomena. The aperiodic The rotor angle stability problem involves the study of the instability problem has been largely eliminated by use electromechanical oscillations inherent in power systems.KUNDUR et al. such classification is The change in electromagnetic torque of a synchronous entirely justified theoretically by the concept of partial stability machine following a perturbation can be resolved into two –. an increase in angular separation is accompanied ality and complexity of stability problems. It depends on the a nonoscillatory or aperiodic mode due to lack of syn- ability to maintain/restore equilibrium between electromagnetic chronizing torque. instability. For convenience in analysis and for gaining useful insight into • The devices. therefore. . • Damping torque component. may undergo cannot be properly understood and effectively The power-angle relationship is highly nonlinear. equilibrium between the input mechanical torque and the pending on the network topology.In today’s power systems. whereas lack of damping torque results in oscillatory • The size of the disturbance considered. . . fundamental factor in this problem is the manner in which however. depending on the power-angle relationship. this problem can still occur when generators the power outputs of synchronous machines vary as their operate with constant excitation when subjected to the rotor angles change. erating state of the system. is concerned with the ability of the power system to main- lowing are descriptions of the corresponding forms of stability tain synchronism under small disturbances. . the angular position of its rotor relative to that of the slower ma- A. this equilibrium may experience sustained imbalance leading to different forms is upset. If the system is perturbed. The fol. De. in phase with the speed de- The classification of power system stability proposed here is viation. Power system stability is essentially a single problem. in phase with rotor B. problems. Instability that may result chines of an interconnected power system to remain in synchro. different sets of opposing forces remains constant. which influences instability. Instability results if the system cannot using an appropriate degree of detail of system representation absorb the kinetic energy corresponding to these rotor speed and appropriate analytical techniques. Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. Lack of suffi- indicated by the main system variable in which instability cient synchronizing torque results in aperiodic or nonoscillatory can be observed. the system. is greatly of the rotors result in sufficient restoring torques . components: • Synchronizing torque component. Downloaded on March 04. angle stability in terms of the following two subcategories: Fig. body. can be of two forms: i) increase in rotor angle through nism after being subjected to a disturbance. processes. Instability that may result occurs in the form of in. we provide a systematic basis for of the machines according to the laws of motion of a rotating classification of power system stability. A of continuously acting generator voltage regulators.: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1389 bility is a condition of equilibrium between opposing forces. certain limit. Beyond a dealt with by treating it as such. small-disturbance rotor creasing angular swings of some generators leading to their loss angle stability problem is usually associated with of synchronism with other generators. resulting in acceleration or deceleration of the rotors of instability. the various forms of instabilities that a power system reduce the speed difference and hence the angular separation. there is actions of excitation limiters (field current limiters). Because of high dimension. In this section. and the speed and the form of disturbance. Under steady-state conditions. it is useful to characterize rotor taken into consideration in order to assess stability. or ii) rotor oscillations of increasing torque and mechanical torque of each synchronous machine in amplitude due to lack of sufficient damping torque. ability of the power system to maintain synchronism when subjected to a severe disturbance. a class of rotor angle stability. Their characteristics are very complex and sig. Stability (damping) of these oscillations depends on . or tripping of transmis- always occur as first swing instability associated with sion lines and other elements by their protective systems leading Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. as well as transient stability are categorized as short term They involve oscillations of a group of generators in one phenomena. Such oscillations are called interarea mode oscil. we recommend against its usage. In the North nificantly differ from those of local plant mode oscilla. sient stability. NO. bility studies is on the order of 10 to 20 seconds fol. or rise of voltages of some buses. . 1390 IEEE TRANSACTIONS ON POWER SYSTEMS. Restrictions apply. It depends on the ability to maintain/restore equilibrium be- turbance. It could also be a result of nonlinear power plant against the rest of the power system. Downloaded on March 04. it could be a result of superposition of be either local or global in nature.2010 at 13:39:06 EST from IEEE Xplore. Local problems a slow interarea swing mode and a local-plant swing involve a small part of the power system.2 Voltage Stability: on a transmission line. ularly. 1. tain steady voltages at all buses in the system after being sub- . . it has been used to lations. The resulting system response in- volves large excursions of generator rotor angles and is Voltage stability refers to the ability of a power system to main- influenced by the nonlinear power-angle relationship. of the term dynamic stability. transient instability may not instability is loss of load in an area. 1. However. 2. MAY 2004 Fig. . beyond the first swing. Classification of power system stability. have a major turbance stability in the presence of automatic controls (partic- effect on the stability of interarea modes . A possible outcome of voltage in large power systems. denote different phenomena by different authors. it has been used mostly to denote small-dis- tions. In- angular separation due to insufficient synchronizing stability that may result occurs in the form of a progressive fall torque.Transient stability depends on both the initial jected to a disturbance from a given initial operating condition. Load characteristics. classical “steady-state stability” with no generator controls . the generation excitation controls) as distinct from the .The time frame of interest in transient stability studies the strength of the transmission system as seen by the is usually 3 to 5 seconds following the disturbance. area swinging against a group of generators in another The term dynamic stability also appears in the literature as area. Such effects affecting a single mode causing instability oscillations are called local plant mode oscillations. American literature. Instability is usually in the form of aperiodic tween load demand and load supply from the power system. operating state of the system and the severity of the dis. In the European literature. . Since much confusion has resulted from the use • Large-disturbance rotor angle stability or transient sta. 19. . as it is commonly referred to.Small-disturbance rotor angle stability problems may a single mode. mode causing a large excursion of rotor angle beyond ally associated with rotor angle oscillations of a single the first swing . manifesting as first swing instability. bility.The time frame of interest in small-disturbance sta. is concerned with the as did the previous IEEE and CIGRE Task Forces . in particular. However. small-disturbance rotor angle stability large groups of generators and have widespread effects. such as a short circuit B. and are usu. It power plant. it has been used to denote tran- lowing a disturbance. with dominant inter-area swings.Global problems are caused by interactions among As identified in Fig. generator excitation control systems and may extend to 10–20 seconds for very large systems plant output . VOL. converters and capacitor commutated converters) have sig- The term voltage collapse is also often used. –. the exciter is a feature that has a positive influence on the limits stability occurs when load dynamics attempt to restore power for self-excitation. The driving force for voltage instability is usually the loads. the synchronous machine. this turbances such as system faults. Restrictions apply. Nor. rent limiters. the voltages near the electrical center rapidly os. and the connected demand at normal voltage that can initiate self-excitation are open ended high voltage lines is not met. the loss of synchronism may occur at rectifier or inverter stations. tap-changing transformers. and are associated of machines as rotor angles between two groups of machines with the unfavorable reactive power “load” characteristics approach 180 causes rapid drop in voltages at intermediate of the converters. nificant part of the power system .: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1391 to cascading outages. and thermostats. a phenomenon is relatively fast with the time frame of interest cillate between high and low values as a result of repeated “pole being in the order of one second or less.2010 at 13:39:06 EST from IEEE Xplore. De- their field or armature current time-overload capability limits. Voltage instability slips” between the two groups of machines. . Such so separated. It is the process nificantly increased the limits for stable operation of HVDC by which the sequence of events accompanying voltage insta. how the system voltages will respond to small load power. commutated converters. links in weak systems as compared with the limits for line bility leads to a blackout or abnormally low voltages in a sig. continuous controls. consumption beyond the capability of the transmission network As in the case of rotor angle stability. voltage stability into the following subcategories: A major factor contributing to voltage instability is the • Large-disturbance voltage stability refers to the system’s voltage drop that occurs when active and reactive power flow ability to maintain steady voltages following large dis- through inductive reactances of the transmission network. and discrete controls at a given ability of the combined generation and transmission system to instant of time. In contrast. with intentional and/or un. synchronous compensators from absorbing the excess reactive This form of stability is influenced by the characteristics of power. The power transfer and voltage system and load characteristics. at operate below some load level. In their attempt to restore this any instant. With appropriate assumptions. and . the system is not stresses it beyond its capability. The study period of interest may extend from bility also exists and has been experienced at least on one system a few seconds to tens of minutes. For example. distribution of change followed by a more gradual rise. equations can be linearized for analysis thereby allowing Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. Remaining load tends to be machine is too large. and the interactions of support are further limited when some of the generators hit both continuous and discrete controls and protections. transformer tap changers cause long-term voltage system changes. The HVDC link control strategies have a points in the network close to the electrical center . very significant influence on such problems. The overvoltages that result when generator load changes to ca- in response to a disturbance. Stable One form of voltage stability problem that results in uncon- (steady) operation at low voltage may continue after transformer trolled overvoltages is the self-excitation of synchronous ma- tap changers reach their boost limit. and shunt capacitors and filter banks from HVDC stations . since the active mally. This concept is useful in determining. voltage instability occurs. power consumed by the loads tends pacitive are characterized by an instantaneous rise at the instant to be restored by the action of motor slip adjustment. protective systems operate to separate the two groups and reactive power at the ac/dc junction are determined by of machines and the voltages recover to levels depending on the controls. the risk of overvoltage insta.KUNDUR et al. the type may also be associated with converter transformer tap-changer of sustained fall of voltage that is related to voltage instability controls. and generator field-cur- progressive drop of bus voltages. termination of large-disturbance voltage stability requires Voltage stability is threatened when a disturbance increases the the examination of the nonlinear response of the power reactive power demand beyond the sustainable capacity of the system over a period of time sufficient to capture the per- available reactive power resources. In this case. . chines. . Negative field current capability of ther voltage reduction. This latter rise de- voltage regulators. . back-to-back applications . however. which is a considerably slower phenomenon . formance and interactions of such devices as motors. loads. un- While the most common form of voltage instability is the derload transformer tap changers. or limits the capability of the transmission network for power circuit contingencies. Loss of synchronism of some generators Voltage stability problems may also be experienced at the may result from these outages or from operating conditions that terminals of HVDC links used for either long distance or violate field current limit . system instability. It is caused by a capacitive behavior of the network (EHV • Small-disturbance voltage stability refers to the system’s transmission lines operating below surge impedance loading) as ability to maintain steady voltages when subjected to small well as by underexcitation limiters preventing generators and/or perturbations such as incremental changes in system load. A run-down situation causing voltage in. If. This can arise if the capacitive load of a synchronous intentional tripping of some load. it is useful to classify and the connected generation . involves loads and may occur where rotor angle stability is not Recent developments in HVDC technology (voltage source an issue. They are usually asso- Progressive drop in bus voltages can also be associated with ciated with HVDC links connected to weak ac systems and rotor angle instability. loss of generation. Downloaded on March 04. pends on the relation between the capacitive load component Restored loads increase the stress on the high voltage network and machine reactances together with the excitation system of by increasing the reactive power consumption and causing fur. If the resulting loading on the ac transmission the post-separation conditions. the instability is associated with the in. Examples of excessive capacitive loads voltage sensitive. This ability is determined by the transfer and voltage support. 4 Frequency Stability: identifying factors influencing stability. to several minutes. cannot account for nonlinear effects such as maintain steady frequency following a severe system upset re- tap changer controls (deadbands. coupling is strong for underfrequency load shedding that unloads the system. or voltage stability studies. as reactive power flows. tections that are not modeled in conventional transient stability electronically controlled loads. the time frame of interest for voltage sta. loss of load. Therefore. voltage. 19. . this is similar to analysis In large interconnected power systems. cause undesirable operation of impedance relays. . The tections. static analysis voltage regulators. devices such as prime mover energy supply systems and load morning load increase). . and other system variables. Dynamic modeling of loads is most commonly associated with conditions following splitting often essential. frequency. or only triggered for extreme system seconds. of such problems are reported in references –. . On time-domain simulations . In fact.g. NO. librium with minimal unintentional loss of load. as identified in Fig. system differential equations. voltage magnitudes may coupling between variations in active power/angle and reac. The study frequency stability problems are associated with inadequacies period of interest may extend to several or many minutes. It is determined • Long-term voltage stability involves slower acting equip- by the overall response of the island as evidenced by its mean ment such as tap-changing transformers. Voltage stressed conditions and both rotor angle stability and voltage magnitude changes. especially for islanding conditions with tive power/voltage magnitude. Examples system dynamic performance . This linearization. Frequency stability refers to the ability of a power system to however. frequency . Stability in this case is a question of near loads are important. Restrictions apply. VOL.. Therefore. identify factors influencing stability. Instead. and generator current limiters. In an overloaded system.. 1392 IEEE TRANSACTIONS ON POWER SYSTEMS. and HVDC converters. 2. thermostatically frequency. when loads try to restore their power beyond the generation . It is recommended that the term whether or not each island will reach a state of operating equi- transient voltage stability not be used. a combination of linear and It depends on the ability to maintain/restore equilibrium be- nonlinear analyzes is used in a complementary manner tween system generation and load. time delays). tained frequency swings leading to tripping of generating units bility problems may vary from a few seconds to tens of min. discrete tap steps. and analysis requires solution of appropriate conditions. short circuits of systems into islands.3 Basis for Distinction between Voltage and or boiler/reactor protection and controls are longer-term phe- Rotor Angle Stability: nomena with the time frame of interest ranging from tens of It is important to recognize that the distinction between rotor seconds to several minutes . controls. more complex situations in which frequency instability is caused by steam turbine overspeed controls  B. An example of short-term frequency instability is a wide range of system conditions and a large number the formation of an undergenerated island with insufficient un- of scenarios. . frequency stability could be of concern for Instability is due to the loss of long-term equilibrium any disturbance causing a relatively significant loss of load or (e. 1. with minimum unintentional . utes. derfrequency load shedding such that frequency decays rapidly tant. . when underfrequency load shedding and generator controls and pro- a remedial action is applied too late) . rather than relative motion of machines. or a lack of attraction of seconds.. Stability is usually determined by the resulting outage of equip. and/or loads. These processes may be very slow. and screen nomenon. In iso- ment. MAY 2004 computation of valuable sensitivity information useful in B. Therefore. corresponding to the response of disturbance could also be a sustained load buildup (e. lated island systems. . low voltage may consequent instability is apparent.g. this should be complemented by quasi-steady-state causing blackout of the island within a few seconds . The study period of interest is in the order of several such as boiler dynamics. Instability that may result occurs in the form of sus- As noted above. In contrast to angle stability. the characteristic times of the generation). such as volts/Hertz protection tripping generators. in equipment responses. power flows. Where timing of control actions is impor. the distinction is based on High voltage may cause undesirable generator tripping by the specific set of opposing forces that experience sustained poorly designed or coordinated loss of excitation relays or imbalance and the principal system variable in which the volts/Hertz relays. In many cases.  can be used to estimate stability stability may be a short-term phenomenon or a long-term phe- margins.2010 at 13:39:06 EST from IEEE Xplore. Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. this type of situation is of rotor angle stability. which may be higher in percentage than stability are affected by pre-disturbance active power as well frequency changes.g. corresponding to the response of devices such as toward the stable post-disturbance equilibrium (e. the other hand. and sulting in a significant imbalance between generation and load. or insufficient generation reserve. capability of the transmission network and connected During frequency excursions. affect the load-generation imbalance. . Downloaded on March 04. rather than the severity of the initial disturbance. controlled loads. and pro- acting load components such as induction motors. . change significantly. poor coordination of control and pro- and long-term simulations are required for analysis of tection equipment. angle stability and voltage stability is not based on weak During frequency excursions. Generally. • Short-term voltage stability involves dynamics of fast thereby invoking the actions of processes. post-disturbance steady-state operating point processes and devices that are activated will range from fraction being small-disturbance unstable. . voltage stability may be either a short-term or Severe system upsets generally result in large excursions of a long-term phenomenon as identified in Figure 1. in a bulk power electric system. However. depends on the operating condition and the nature of the phys. and magnitude of adverse effects on consumer service. This implies that. Adequacy—the ability of the power system to supply the ag- cepts of power system reliability. satisfactory operation over the long run. it can only be judged by con- between different forms is important for understanding the un.” on the system operating condition as well as the contingent probability of disturbances. the system settles to aspects of power system performance: new operating conditions such that no physical constraints are 1) Reliability is the overall objective in power system violated. For a ical disturbance. we discuss the relationship between the con. kept in mind.  disturbances such as electric short circuits or nonanticipated loss Reliability of a power system refers to the probability of its of system components. NERC Definition of Reliability  tive and convenient means to deal with the complexities of the NERC (North American Electric Reliability Council) defines problem. sideration of the system’s behavior over an appreciable derlying causes of the problem in order to develop appropriate period of time. Restrictions apply. Section III. as discussed in Section II. Other alternative forms of definition of power system security Security of a power system refers to the degree of risk in have been proposed in the literature. applying suitable analysis because the consequences of instability are less severe. We will gregate electric power and energy requirements of the customer also briefly describe how these terms have been defined and at all times. security is defined in terms of satisfying a set of inequality without interruption of customer service. taking into account scheduled and unscheduled out- used in practice. the overall stability of the system should always be power system reliability as follows. There are two important components of security analysis. the system must survive the transition must be secure most of the time. damage to equipment such as an explosive • Static security analysis—This involves steady-state anal- failure of a cable.. ages of system components. . For example. in reference its ability to survive imminent disturbances (contingencies) . as systems fail one form of instability the other hand. and developing corrective measures. two systems may both be stable with equal We have classified power system stability for convenience in stability margins. distinguishing mance of the power system. 2) System security may be further distinguished from Stability analysis is thus an integral component of system se- stability in terms of the resulting consequences. Working Group/Task Force documents . the power system ditions being acceptable. It is essential to look at to which the performance of the elements of that system all aspects of the stability phenomenon. Downloaded on March 04. however. It denotes the ability to The above definitions also appear in several IEEE and CIGRE supply adequate electric service on a nearly continuous basis. B. with few interruptions over an extended time period. Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. It relates to robustness constraints over a subset of the possible disturbances called the of the system to imminent disturbances and. refers The analysis of security relates to the determination of the ro- to the continuance of intact operation following a disturbance. Conceptual Relationship .g. IV. the system to these conditions.KUNDUR et al. duration. depends “next contingency set. Analysis of Power System Security Stability of a power system. . As well. To be secure. a system may be stable no equipment ratings and voltage constraints are violated. Reliability. 3) Security and stability are time-varying attributes which tion. RELATIONSHIP BETWEEN RELIABILITY. For curity and reliability assessment. power system subjected to changes (small or large). fall of transmission towers due to ice ysis of post-disturbance system conditions to verify that loading or sabotage. in addition to the next operating con- design and operation. must be stable but must also be secure against other The above characterization of system security clearly high- contingencies that would not be classified as stability lights two aspects of its analysis: problems e.: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1393 B. and stability. design and operating procedures. The degree of reliability may be measured by the frequency. and at each aspect from results in power being delivered to consumers within ac- more than one viewpoint. following a contingency. it is impor- The following are the essential differences among the three tant that when the changes are completed. C. In any given situa. is the degree should not be at the expense of another. is a function of the time-average perfor- may ultimately lead to another form. While classification of power system stability is an effec.5 Comments on Classification: example. but one may be relatively more secure identifying causes of instability. It bustness of the power system relative to imminent disturbances. security. Security—the ability of the power system to withstand sudden A. This is particularly true in highly stressed systems system under a particular set of conditions. tools. AND STABILITY Reliability can be addressed by considering two basic func- tional aspects of the power systems: In this section.2010 at 13:39:06 EST from IEEE Xplore. To be reliable. Solutions to stability problems of one category • Reliability. SECURITY. yet insecure due to post-fault • Dynamic security analysis—This involves examining system conditions resulting in equipment overloads or different categories of system stability described in voltage violations. any one form of instability may not occur in its can be judged by studying the performance of the power pure form. cepted standards and in the amount desired. hence. on and for cascading events. such as generation tripping. While this is quite common in the system order. In addition. and controlled islanding. with distributed models . 1) The problem of defining stability for general nonau- tions that constrain various quantities. Power systems are also an example of constrained dynamical tify and manage risk. .g. It also does not give adequate consideration as to how likely evaluated on the level of single element—e. This type of consideration will introduce restricted stability re- gions in power system stability analysis. is of models that include partial differential equations.. Emergency controls. 2. as their state trajectories are restricted to a particular risk-based security assessment. however. etry is presented in  (for sufficient conditions see ). be studied effectively not be acceptable. load dynamic symmetrical components . This is mostly dictated by the fundamental intractability the study effort is minimized. Finally. with a diversity of participants Some qualitative aspects of fault propagation in spatially with different business interests. Consideration may be given to extreme An often useful approximation of the fast dynamics is based contingencies that exceed in severity the normal design contin. Its main limitation. and a set of differen. the use of or unlikely various contingencies are. sible with today’s computing and analysis tools. this desired region may either lead to structural changes (e. it is typ- components. 1394 IEEE TRANSACTIONS ON POWER SYSTEMS. topology changes due to switching in substations. or permitted) operating region . . This is usually referred to as surfaces on which rank conditions for the Jacobian of the alge- the criterion because it examines the behavior of an braic part do not hold are commonly denoted as impasse sur- -component grid following the loss of any one of its major faces. The algebraic part often arises from a say that a system to which the environment delivers singular perturbation-type reasoning that uses time separation square-integrable signals as inputs over a time interval between subsets of variables to postulate that the fast variables is stable if variables of interest (such as outputs) are have already reached their steady state on the time horizon of also square integrable. consider signals truncated in time. The double-. theory of dynamical systems.g. VOL. while a mathematical description may involve variations in the a state-space description). we can interest . it may not always be entirely nat. there often exist algebraic (implicit) equa. and the degree of exposure to system failure is esti. the non-local results are they have a significant likelihood of occurrence. While local results can be derived designed and operated to withstand a set of contingencies from the implicit function theorem that specifies rank conditions referred to as “normal contingencies” selected on the basis that for the Jacobian of the algebraic part. especially seldom in this form. system (e. or lead to unsafe operation. to explicit first-order form may. In today’s utility environment. The trajectories that exit mated. in the form of explicit first-order differential equations (i. We assume that the model of a power system is given physical description may include outages of system elements. the probability subset in the state space (phase space in the language of dy- of the system becoming unstable and its consequences are ex. Additional nitions of power system stability from a system-theoretic view. making the power system model nonautonomous (or A. MAY 2004 The general industry practice for security assessment has Proving the existence of solutions of DAE is a very chal- been to use a deterministic approach. 19. and to quan. lumped parameter models without a major loss of model fi- ably well in the past—it has resulted in high security levels and delity. One general approach to non-local study they are usually defined as the loss of any single element in a of stability of DAE systems that is based on differential geom- power system either spontaneously or preceded by a single-. The power system is lenging problem in general. . power system analysis framework in the following way: More importantly. interactions with the environment include disturbances whose point.g.2010 at 13:39:06 EST from IEEE Xplore. and it is necessary to include the models that are typically used to describe power systems are effects of control inputs (including their saturation).. and transformations required to bring them on longer time horizons. In practice. NO. Downloaded on March 04. and in the analysis of models of power systems. and by that it treats all security-limiting scenarios as having the same satisfactory behavior of lumped parameter models (when risk. breaker tripping in a power system). SYSTEM-THEORETIC FOUNDATIONS OF POWER Several additional issues are raised by the fact that the SYSTEM STABILITY power system interacts with its (typically unmodeled) envi- ronment. loss of load or cascading outages ically assumed that equilibrium sets of interest in stability anal- may not be allowed for multiple-related outages such as loss of ysis are disjoint from such surfaces . There is a need to account for the proba. may be used to cope with assumed that distributed nature of some elements of a power such events and prevent widespread blackouts. First-principle with numerous feedback loops. The trend will be to expand the use of systems. the deterministic approach may extended power systems can. transmission lines) can be approximated with The deterministic approach has served the industry reason. however. or three-phase fault. namical systems) denoted as the feasible (or technically viable. a double-circuit line. or the number of variables of interest. In a more general setup.e.. much harder to obtain. a power system is a controlled (or forced) system ural for physical systems such as power systems. bilistic nature of system conditions and events. This approach is computationally intensive but is pos. and one possible approach is to power system transients. introduce spurious The outlined modeling problems are typically addressed in a solutions . It is also typically shedding. V. multiple “ ” section models for a long transmission line). In this approach. Examples include load variations and network In this section.. we address fundamental issues related to defi. Restrictions apply. and denote the system Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. in general. on the concept of dynamic (time-varying) phasors  and gencies. amined. tonomous systems is very challenging even in the tial-algebraic equations (DAE) is often used in simulations of theoretical realm . Preliminaries time-varying). g.g. load variations) disturbances NP-hard).e. For example. it is maximum permissible duration of the fault (the so called clear that the task of determining stability of a power system will “critical clearing time”) for which the subsequent system be a challenging one. An event-type linear system models. some are balanced (either instantaneously. tractor. Downloaded on March 04. and trajectories in such sys- Before considering specifics about power system stability. is an attractor consisting ical model appropriate for the time-scales and phenomena of a single point in the phase space. boring trajectories converge. an equilibrium set. . denote equilibrium points. willingness) to utilize approximations. B. system initial condition belongs to a (known) starting set pects) would necessarily entail a reassessment of notions of sta. and repeat if same region of phase space. in that set. to a equilibrium set where system trajectories never con- 4) Review results in light of assumptions. Attractors therefore describe lowing steps: the long-term behavior of a dynamical system.g. It turns out. an equilibrium point or perhaps to minutes. a proper analysis framework is that of A typical scenario for power system stability analysis hybrid systems (for a recent review. however.. For a recent review on notions of stability in various types signals into signals with the same property. or an equilibrium point. In the case of 2) Next. attractors are non-periodic. the available algorithms are inherently in.KUNDUR et al. occur repeatedly.2010 at 13:39:06 EST from IEEE Xplore. tractor.g. the dual problem of designing stabilizing controls the equilibrium points are the sets of interest. A Scenario for Stability Analysis dictable)—e. Given the are described by their size (norm. see for example In general. typically chaotic (or aperiodic. event-type involves three distinct steps. the system dynamics is of event-type disturbances that we are analyzing. Unlike limit cycles.. we will assume cated structures like aperiodic attractors (which may be that the actions of all controllers are fully predictable in possible in realistic models of power systems). and (or incident-type) disturbance is characterized by a spe- can be answered efficiently in polynomial time . the various driving terms (forces) affecting system variables relevant notion of stability needs re-examination. the effects of market even a benign limit cycle in the state space). 2) The disturbances of interest will fall into two broad categories—event-type (typically described as outages of where is the state vector (a function of time. need to assess the required computational effort. compare with verge to a point or a closed curve. limit sets and more compli- 3) Given our emphasis on stability analysis. short circuit somewhere in the of nonlinear systems. switching) disturbances independent of and is said to be nonautonomous otherwise. but remain within the the engineering experience (“reality”). In the case cific fault scenario (e. A Limit cycle at- under study. and it is unclear if it is decidable at all .. is a . This portion of stability analysis reducing the computational complexity will be our ability (and requires the knowledge of actions of protective relaying. a disturbance acts on the system. We also observe that system described above is said to be autonomous if is in cases when event-type (e. on a longer time scale. or over a time leads about systems with distributed decision making interval). .. . in terms of various norms of signals). see . the stability question is decidable. but we omit ex- specific pieces of equipment) and norm-type (described plicitly writing the time argument ). The garding various phenomena in power systems (e. ious norms of signals—e.. one typically assumes that the variables at the interface with the environment are known (or pre. or an attractor. we tems are very sensitive to the initial conditions. or signal intensity). structures may become prominent. A Point at- 1) Make modeling assumptions and formulate a mathemat. studied with respect to a known post-disturbance equi- We also want to point out that a possible shift in emphasis re. tories of a system with a single scalar nonlinearity converge to while norm-type (described by their size in terms of var- the origin turns out to be very computationally intensive (i.. set of trajectories in the phase space to which all neigh- A typical power system stability study consists of the fol. terms of known system quantities (states).: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1395 as well-posed if it maps square integrable truncated bility. corresponds to closed curves 2) Select an appropriate stability definition. or as functions in the vast majority of cases of practical interest today. see ). A large size of power systems and the need to consider event-type problem of some analytical interest is determining the perturbations that will inevitably lead to nonlinear models. In that case. librium set (which may be distinct from ). hybrid as. We use the notion of an equilibrium set to may be found in ..g. is its derivative. for nonlinear systems is very challenging. Restrictions apply.g. that our main tools in response remains stable. transmission network followed by a line disconnection in- efficient. system setting. of time. disturbances may also be initiated by human operators. on the other hand. cluding the duration of the event—“fault clearing time”). In a power of systems (including infinite dimensional ones). limit cycles imply periodic behavior. 1) The system is initially operating in a pre-disturbance Our focus is on time horizons of the order of seconds equilibrium set (e. A 3) Analyze and/or simulate to determine stability. and the particular nature 3) After an event-type disturbance. that mechanical inputs to all generators We consider the system are constant. a related problem of whether all trajec. is suf- by their size e. ficiently differentiable and its domain includes the origin.g. However. strange necessary. in phase space. or that they vary according to the known response of turbine regulators.. The we will return to this issue shortly. or strange) attractor corresponds using a scenario of events. and we want to characterize the system motion with Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. p. differentiable and its domain includes the origin. Note also • stable if. when forcing (control input) term is not included i. 2. • uniformly stable if. In stability analysis in the Lyapunov framework that tion converges to the equilibrium set. (a two-dimensional system in the space of real vari- ables). is not a limitation. such that: The definition of stability related to linear systems finds wide use in small-signal stability analysis of power systems. voltage levels) are that: inherently soft i. however. In the case of norm-type disturbances. NO.g. C.. it is said that (and sometimes as well) are stable. the uncertainty structure is different—the per. after the disturbance. ping or a partial load shedding) may lead to a new stability study for a new (reduced) system with new starting and cept of partial stability is useful in the classification of power viable sets. sociated system whose states are deviations of the states of the itly. is its derivative. 19. input. Note that we assume that our model is accurate within Note that in (2) any norm can be used due to topological in the sense that there are no further system changes (e. assuming differentiability of functions in- it is assumed that the analyst has determined all that are rele. very often we have . to stability and asymptotic stability are the ones most applicable • asymptotically stable if it is stable and in addition there is to power system nonlinear behavior under large disturbances. causing line trip. The con- Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. Illustration of the definition of stability .” For technical reasons. . rium at the origin could be a translation of a nonzero equilibrium For example.e. we do not write the initial state is in the given starting set. perturbations. equivalence of all norms.e...2010 at 13:39:06 EST from IEEE Xplore. and operating con. Note. and not necessarily in the most general context. entire trajectory. We vant for a given and the disturbance. for each . that norm-type perturba. In Figure 2. 132] as it implies that there exist paths (in graph-theoretic terms) The equilibrium point of (1) is: from any given bus to all other buses in the network. By choosing the initial conditions in a sufficiently small C. Stability Definitions From System Theory spherical neighborhood of radius .e. output stability. we present definitions for cases that are typical for power fied deterministically (the same may apply to as well). 2. the Lyapunov stability definitions related • unstable if not stable. like periodic orbits. If the system response turns out to be stable (a precise definition will follow shortly). turbation is characterized by size (in the case of the so-called can often be reduced to the study of equilibrium points of an as- small-disturbance or small-signal analysis this is done implic.. and system models (e. MAY 2004 respect to i. there is . system stability into different categories. volved). In the case of The definitions collected here mostly follow the presentation in event-type disturbances. a study of more intricate equilibrium sets. tions (or several such studies. the power system analyst may be interested (2) in stability characterization with and without these constraints. the system mo. (1) We propose next a formulation of power system stability that will allow us to explore salient features of general stability con.g. if the system trajectory will remain in- side the technically viable set (which includes ). after a suitable coordinate transformation [54. the perturbations of interest are speci. if a system gets partitioned into several disconnected parts). ). and partial stability. An equilib- type. Downloaded on March 04. we will assume that the origin is an equi- The operating constraints are of inequality (and equality) librium point (meaning that .g. for each . Fig. since all control inputs are assumed to be straints are satisfied for all relevant variables along the known functions of time and/or known functions of states . In the case of norm-type will concentrate on the study of stability of equilibrium points. Note that the • “An equilibrium set of a power system is stable if. system connectedness is a collective feature.g. so that linearized analysis remains valid). stability of linear systems. VOL. . Restrictions apply. . where is the state vector.. another possibility is to equilibrium set typically characterizes the system before and study periodic orbits via sampled states and Poincare maps .. and pertain to individual variables and their collections. Of these various types. 1396 IEEE TRANSACTIONS ON POWER SYSTEMS. we can force the trajectory of the system for all time to lie entirely in a given cylinder In this section. and the same original system from the periodic orbit. independent of .1 Lyapunov Stability: local.. we provide detailed analytical definitions of of radius . we depict the behavior relay-initiated line tripping) until the trajectory crosses the of trajectories in the vicinity of a stable equilibrium for the case boundary . there is such that some of the operating constraints (e. The stability analysis of power systems is in general non. A detected insta- bility (during which system motion crosses the boundary of the technically viable set —e. is sufficiently cepts from system theory. such that (2) is satisfied. as various equilibrium sets may get involved. We again consider the nonautonomous system: tions could in principle be used in large-signal analyses as well. several types of stability including Lyapunov stability. and possibly with different modeling assump. Downloaded on March 04. A related notion of practical • exponentially stable if there are . and for sufficiently large time . is • globally uniformly asymptotically stable if it is uniformly bounded by a fixed constant. tial stability –. as is • globally exponentially stable if the exponential stability the case with total stability. Illustration of the definition of exponential stability . namely the set of points • uniformly asymptotically stable if it is uniformly stable in the state space with the property that all trajectories initiated and there is . and can be most naturally checked arbitrarily small by reducing the model perturbation term. that is. the so called converse Lyapunov theorems estab- the equilibrium. By choosing the initial operating points in nominal system has a stable equilibrium at the origin. tive (along trajectories of (1)) is negative definite. In the case of large scale power systems. we can not be the case for the actual perturbed system. This is a stricter requirement of the system be. Illustration of the definition of asymptotic stability . and of a Lyapunov function for the nominal (non-perturbed) system. and we cannot make that motion proach to system stability. 5. and. Quali. bility. Roughly speaking. tions (some leads for the case of simple power systems are given bility combines the aspect of stability as well as attractivity of in .2010 at 13:39:06 EST from IEEE Xplore. smooth positive definite “energy” function whose time deriva- Fig. for a specific system via so called Lyapunov functions. i. Char- there is such that: acterization of stability in this case requires knowledge of the size of the perturbation term. Even if the stability pictorially. which is not en- force the trajectory of the solution to lie inside a given cylinder tirely known to the analyst. in a ball of a fixed radius. While unfor- tunately there is no systematic method to generate such func- It is important to note that the definition of asymptotic sta. 4. the solution of the perturbed system will approach the origin.KUNDUR et al. we are to construct a The basic idea is to relax the condition for stability from one that Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO.: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1397 Fig. If the equi- . . tonomous characteristics of the system. for practical sta- condition is satisfied for any initial state. as . One does not require a more narrow In Figure 5 the behavior of a solution in the vicinity of an interpretation that the deviation from the origin of can be exponentially stable equilibrium point is shown. the region of attraction of a given equilibrium set .e. Another concept of interest in power systems is that of par- tatively speaking. we depict the property of uniform asymptotic alizations that are necessary because of system size. made arbitrarily small by a suitable choice of the constants. there is shown that the region of attraction has nice analytical properties such that (it is an open and connected set. even for small perturbations.e. in case the initial values and the perturbation are bounded by suitable constants” . and its boundary is formed by system trajectories).. introduced already by Lyapunov himself. This concept certain sense. Illustration of the definition of uniform asymptotic stability . such that for all at the points will converge to the equilibrium set . disregarding subtleties due to the nonau. In power systems we are interested in. but could if the solution is ultimately bounded i. we allow that the system will move away from the origin These definitions form the foundation of the Lyapunov ap. is pictorially presented in Figure 3. then it can be . uniformly in and librium set is a point that is asymptotically stable. 3. We cannot necessarily expect that for all . . independent of . given that the initial condition is stable. such stability is motivated by the idea that a system may be “con- that: sidered stable if the deviations of motions from the equilibrium remain within certain bounds determined by the physical situa- tion. for each pair of positive numbers and . Fig. for each . lish the existence of such functions if the system is stable in a havior to eventually return to the equilibrium point. this may a sufficiently small spherical neighborhood at . Restrictions apply.. ). we are naturally interested in the effects of approximations and ide- In Figure 4. It is said to be input-to-state stable if the local input-to-state property holds for the entire input and output spaces.function defined on . and some of the main tools for establishing input-to- Note that if is the space of uniformly bounded signals. extended bounded or square integrable sig- there exists a class. This property is typically established by This approach considers the system description of the form: Lyapunov-type arguments . Input-to-output stability results are Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. this is allowed in the notion of small-signal erties of the norm used in (2) and elsewhere) to one that requires stability. ever. in some adaptive systems). Next. input . Restrictions apply. with respect to and as . for all . for each fixed . A system (6) is said to be locally input-to-state stable if there chical (two-level) approach to stability determination . where the norm of the input signals is constrained. and we may be interested in a hierar. function . and the first step. where it equals the maximal singular value It is finite-gain stable if there exist non-negative constants (the supremum of the two-norm of the transfer function evalu- and such that: ated along the imaginary axis. such behavior from only some of the variables. this results in the study of (7) composite Lyapunov functions. • Definition: A continuous function It is said to be input-to-output stable if the local input-to-state is said to belong to class if.g. The above definitions exclude systems a system. the maximal amplification from (4) a given input to a given output is denoted as the gain of the system. One of main goals of control design is then to mini- (5) mize this gain. we analyze the stability of each subsystem sepa. if the input represents a disturbance. only for all and . function . In the case tive constant such that: of square-integrable signals. or the norm) of the transfer function. and leads to substan- tial simplifications in others (e. In a Lyapunov framework. we combine the results of the first step with information . bility inputs are not assumed to be known functions of time. meaning that all truncations of such signals (set to zero positive constants and such that for any initial state for ) are bounded or square integrable (this allows inclu- with and any input with sion of “growing” signals like ramps etc. stability. output stability come from the Lyapunov approach. we consider the system (6) with the output determined (3) from: where is an operator (nonlinear in general) that specifies the (8) -dimensional output vector in terms of the -dimensional where is again assumed smooth. An important qualitative result is that if the isolated subsystems are sufficiently stable. Downloaded on March 04. for each fixed . Note. MAY 2004 requires stable behavior from all variables (because of the prop.function . so in its standard form.2010 at 13:39:06 EST from IEEE Xplore. while the function in the second term bounds class.. and inequality the mapping belongs to class with respect to (9) is satisfied for any initial state and any bounded input and.function . VOL. a class. the solution exists and satisfies: about the interconnections to analyze the stability of the overall system. property holds for entire input and output spaces. The first term describes the (decreasing) effects of the ini- • A mapping : is stable if there exists a tial condition. At exists a class. how- then this definition yields the familiar notion of bounded-input. in general. for linear systems.e. 19. and nals. it is not suitable for study of for which inequalities (4) and (5) are defined only for a subset individual equilibrium sets.2 Input/Output Stability: bounded input . and a non-nega- the “amplification” of the input through the system. The input belongs to a normed linear space of vector A system (6) is said to be locally input-to-output stable if signals —e. that are of interest in . This gain can be easily calculated. of the input space. • Definition: A continuous function (9) is said to belong to class if it is strictly increasing and .g. com- pared to the strength of the interconnections. 1398 IEEE TRANSACTIONS ON POWER SYSTEMS. while ignoring the interconnections). It (6) has been used in the context of power system stability as well Note the shift in analytical framework. as in input/output sta- . In the second with and any input with step. the mapping is decreasing . lower-order subsystems.. then the overall for all . . that input-to-output stability describes global properties of bounded-output stability. a class. the solution exists and the output satisfies: stability analysis). and inequality (7) is satisfied for any initial state and any C. system is uniformly asymptotically stable at the origin. 2. but A power system is often modeled as an interconnection of assumed to be in a known class typically described by a norm. There exist a number of theorems relating Lyapunov and input-to-output for all and . NO. This formulation Let us consider a nonautonomous system with input: is natural in some engineered systems .. positive constants and such that for any initial state rately (i. Note in the suitable sense. Advances in this direction come from improved computer tech- tual value. see . the stability of power sys- tional value. particularly accounting for load models. Stability Definitions and Power Systems C. the input-output properties are indexed by the operating equilib- rium. In the case of autonomous systems (i. The essence of the approach is that if the lin. the starting set is a point. that standard definitions like to the properties of a linearized model at a certain operating (2) are not directly applicable. every eigenvalue of having method and related ideas. we focus on a subset of variables describing the (10) power system. they are of significant concep. whereas Lyapunov stability protects against a single functions are known to exist for simplified models with spe. polynomial of . but again not for many realistic models. 209-211]. norm-type bound is not very suitable. lier. Lyapunov bances. it also connects input-to-output stability using (9) for nonlinear systems. The precise technical conditions re. that conditions like (9) are difficult Authorized licensed use limited to: UNIVERSIDADE FEDERAL DO RIO DE JANEIRO. For the case of power systems. The autonomous system case when some also that partial stability may be very suitable for some system eigenvalues have zero real parts. as it is impossible to derive an analytical expression tems to large disturbances is typically explored in simulations. positive definite matrix solution. p. case). especially if the operating point is ju. for a finite list of dif- in [62.e. An attempt to use such bounds diciously selected. as it would likely include or if all eigenvalues have negative real parts (in the autonomous many other disturbances to which the system may not be stable. impulse-like disturbance” [55. responds to a quadratic Lyapunov function that establishes sta- phisticated use of the input-to-output stability concept..3 Stability of Linear Systems: D. we consider a system of the form: damental importance in voltage and angle stability studies. they are still of the technically viable set are difficult to characterize with the great practical interest. Because of these two reasons. Note. p. General stability conditions for the nonautonomous case the ignored variables do not appear. however. some persistent disturbances acting on power systems (e. like generator angles. the starting set will be a collection of earized system is uniformly asymptotically stable (in the nonau. where it is equivalent to exponential stability). we are typically parts. is covered by the center manifold theory. 103]. for in the general case.. Downloaded on March 04. While such results are necessarily local. it is pos- origin is stable if and only if all eigenvalues of have nonpos.1 Complementarity of Different Approaches: The direct ways to establish stability in terms of the preceding While Lyapunov and input/output approaches to defining definitions are constructive. 193-196] are of little computa.. Restrictions apply. load values of is to solve a linear Lyapunov Matrix Equation for a variations). it cor- punov stability results in a global sense . the long experience with Lyapunov system stability have different flavors. a detailed exposition with geometric a zero real part is a simple zero of the minimal characteristic and topological emphasis is presented in . The requirement that such be “covered” by a tonomous case. In the case of power if and only if all eigenvalues of have negative real parts. they serve complemen- stability offers guidelines for generating candidate Lyapunov tary roles in stability analysis of power systems. an alternative to calculating eigen. In practice. and others have negative real models . speaking. as both the starting set and point. if such solution exists.g. Intuitively functions for various classes of systems. The concept of partial stability is hence of fun- In this subsection. . see  for an not interested in some states. we tend to use simpler reduced models where origin. and we assume that the disregarded variables will not influence the outcome of the analysis in a significant which is the linearization of (1) around the equilibrium at the way. sible to approximate the viable set using the so-called BCU itive real parts. For a more so. then the original nonlinear system is also locally stable and which are not of interest to the power system analyst. The input/output framework is a natural choice for analysis of In the autonomous case. and as- in the power industry today. however. their differences. and in addition. ): The for a recent review of key issues in power system simulations origin of (10) is (globally) asymptotically (exponentially) stable that are related to stability analysis see . in which bility of the system. nology and from efficient power system models and algorithms. (as contrasted with simulation-based) software packages used For outage of a single element (e. mostly related to the topic of practice is then to try to relate stability of a nonlinear system voltage collapse. Note.2010 at 13:39:06 EST from IEEE Xplore. The ability to select specific equilibrium sets for analysis is Similarly. for example. The system models for which there exist energy functions. but no general sys. norm-type bounds used in (2). cial features . In a very straightforward example. In such studies. p. The other key difficulty in analysis stems from the fact that the construction of Lyapunov functions (11) for detailed power system models. but only in introduction.KUNDUR et al. as we commented ear- While such conditions [62. D. is still an open question. distinct points. system model. . but conceptually we effec- are given in terms of the state transition matrix : tively use partial stability. there are no general constructive methods to establish a major advantage of the Lyapunov approach. “input/output stability protects against noise distur- tematic procedures.: DEFINITION AND CLASSIFICATION OF POWER SYSTEM STABILITY 1399 thus sometimes used in the stability analysis to establish Lya. This is the method of choice for analytical would produce results that are too conservative for practical use. suming a known post-equilibrium set and an autonomous quired from the linearization procedure are given. ferent outages of this type. transmission line).g. naturally with studies of bifurcations  that have been of One approach of utmost importance in power engineering great interest in power systems. 2000. I. 347. and control theory. 1978. turns out to be unstable. . 1948. Feb. pp. stability analyses  T. and P. 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