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IEEE TRANSACTIONS O W (IRCCITS .\\D 5> STFSIS. V o l . 37. NO. 6. J l ’ N r 1990 787 A Probabilistic Approach To Dynamic Power System Security Abstract --In this paper, the problem of dynamic power system secu- critical system state variables, i.e., generator frequency, rity is investigated. A probabilistic approach is chosen ayd an appropri- line currents, bus voltages, etc., which exceed the safe ate power system model is developed for analyzing the dynamic behavior of the different operational modes of the system. The model includes the operating limits of the system and ultimately cause inter- transients induced by primary and secondary disturbance events. Pri- ruptions in supply and loss of load. Cascading disturbance mary events are defined to be state-independent disturbances such as events may persist until the system completely separates line and unit faults and load changes and are modeled by random (a black out) [l]. There is a need for evaluating the switches in the power system model. Secondary events are state-depen- dynamic security of the system and using this information dent variations in the structure such a s forced line and unit outages and load shedding actions. They are modeled by switches in the model whose to develop timely control strategies to circumvent a possi- states depend on the instantaneous balues of certain system variables. ble system collapse. In this context, security is an instanta- The power system model developed characterizes the interaction between neous time varying measure of the robustness of the the dynamic state of the system and the network topology a s defined by system relative to imminent disturbances [2]. the status of the various protective relays in the system. A dynamic In a normal operating state the frequency and bus security region is defined a s a subset of state-structure space of the power system such that at any moment in time a n operating point within voltage are kept at prescribed values. The frequency and the region satisfies all the constraints required for secure power system voltage constancy results from a carefully maintained bal- operation. ance. Given a n initial probability distribution of the system state and of the Dy Liacco first introduced the concept of (steady-state) system structure, a fixed control policy, and a statistical characteriza- security in [3]as related to a multilevel decomposition of tion of the primary events, the probabilistic evolution of the state and structure is computed a s the solution of a system of linear partial the power system. Security was defined in terms of satisfy- differential equations with matching conditions at the switching bound- ing a set of equality and inequality constraints over a aries. A dynamic security measure is defined a s the integral of the joint subset of the possible disturbances called the “next con- a probability density of the state and structure of the system over the tingency set”. Dy Liacco’s approach [3] and a probabilistic dynamic security region for all possible structural bariations. This formulation of steady-state security by Patton [4] are computation yields the probability that the power system remains yecure for the range of primary events considered. Computer simulations for an pointwise approaches to the problem; security is assessed example power system are presented to illustrate the utility of the at specific sets of operating points. The work discussed in approach developed. [5]-[7] reduces the number of contingencies to be consid- ered using a performance index which measures the im- I. INTRODUCTION pact of a contingency on the system. The contingencies are rank ordered according to the performance index. T H E PRIMARY function of an electric power system is the reliable generation, transmission, and distribu- tion of electric power to meet a randomly variable de- An alternative methodology, the so-called regionwise approach, evaluates security of a particular operating point by verifying membership in an appropriate set. mand. These objectives must be accomplished at the Regionwise approaches are discussed in [8]-[12]. lowest possible cost over a wide range of operating condi- The concept of a security corridor was introduced in tions and random disturbances. A disturbance may be the 1131. A security corridor is composed of a number of result of environmental effects, equipment malfunctions. overlapping security sets covering a predicted daily trajec- or human operator error. Such disturbances, regardless of tory. As long as the actual trajectory stays inside a secu- their origin, can initiate transients and fluctuations in rity corridor, security is guaranteed and no additional computations are required. Manuscript received August 29, 1988. This work was supported by the Galiana [ 141, [15], investigated the limitations imposed U S . Department of Energy under Contract DE-ACOl-79-ET 29363. by the structure of a power network on the values of the This paper was recommended by Associate Editor M. Ilic. state variables of the system. The results led to the K. A. Loparo is with the Department of Systems Engineering, Case Western University. Cleveland. OH 44106. determination of a feasible region of bus injections for the F. Abdel-Malek is with the Department of Electrical Engineering. system. Wu [16] refined the definition of a steady-state Trenton State College. Trenton. NJ 08650. IEEE Log Number 8931168. security region in terms of a set of inequality constraints 0098-4094/9O/O600-0787$01 .OO 01990 IEEE 788 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 6, JUNE 1990 in the space of power injections. These constraints char- include continuous and jump process random distur- acterized the values of the state variables of the system bances which model the probabilistic aspects of the sys- for which a real load flow solution exists. tem. In the papers by DeMarco and Bergen [36] and [37], The regionwise approach has the following adcantages nonlinear wideband noise models are proposed for the over the pointwise approach. study of power systems security. The basic assumption is 1) It requires fewer computations compared to the that small signal instabilities can result from the “shrink- pointwise approach. ing” of the region of attraction of a particular equilibrium 2) The location of a specific operating point inside the point; for example, the small-signal instabilities can be security region can be determined, hence a security mea- caused by voltage instabilities or voltage collapse. In [38], sure for an operating point can be computed. we proposed an alternate model for small signal instabil- 3) It can identify which constraint has the most effect ity which was based on a linearized model of the power on the security at each operating point. This information system and parametric random fluctuations, possibly re- can be used in security enhancement to select the control sulting from perturbations in the load. which reduces the likelihood of the violation of that The work described in this paper is an outgrowth of constraint for a given set of contingencies. some of our earlier work [381, and other previous research All the methods discussed above relate only to steady efforts including [17], [MI, and [34]-1371. The problem state assessment. The limitations of steady-state analysis formulation presented here focuses on modeling the dis- are as follows. turbances as finite state jump processes. Continuous pro- t 1) Only initial disturbances are studied. The probabil- cesses can be included without any conceptional diffi- ity that other disturbance events may result from an initial culty; refer to [24] for more details. event, for example, network topological changes resulting Patton in [4] proposed another form of the security from protective relay actions, are ignored. function which takes into account both steady state and 2) It is assumed that the steady state (equilibrium) transient breaches of security. operating point determined by solving the post-fault flow The use of pattern recognition techniques [19], [201, for equations is reachable. Transient stability analysis ad- off-line computation of security functions is an effort to dresses this point. characterize the boundary of the set of steady state or In a recent set of papers, Wu and his coworkers [17], transient secure operating points. This method is in fact [181, have developed a procedure for constructing dy- an automated extension of the conventional techniques namic security regions and a probabilistic framework for used in power utilities to separate stable and unstable addressing the problem of dynamic security assessment. conditions. The pattern recognition techniques were not In this paper, power system dynamic security refers to the initially well received because of the excessive computa- following situation. Given that the power system is in a tional requirements involved in finding an adequate train- particular operating state at the time of a disturbance, the ing set for large scale power systems. Recent improve- system is dynamically secure if the initial operating point ments of these methods [21], coupled with the rapid is in the domain of attraction of a “secure” post-fault progress of computers however, have renewed optimism equilibrium state. In this context, the concepts of dynamic in their use in power system security analysis. security and asymptotic stability are considered to be A review of the works in both steady state and transient equivalent. Affine approximations are used to derive ap- security assessment of power systems are provided in [22]. proximate boundaries of the prefault angles which guar- These include methods for the construction of mathemat- antee transient stability. This region of dynamic security is ical models and dynamic equivalents and on line transient used along with a probabilistic model of the power system security analysis based on digital simulation, hybrid com- dynamics to estimate the evolution of the probability to puter simulation, and Lyapunov methods. insecurity. The model incorporates a finite state continu- Stability is a narrower condition of security and a more ous time (FSCT) Markov process which models the ran- general analysis of the dynamic security problem has been dom changes (state-independent) in system structure and proposed by Loparo et al. [23]-[251. The uncertainty in a smooth system of differential equations which describe the power system security problem is characterized by the evolution of the continuous state variables between considering the network structure itself to be uncertain. structural changes. This framework is used to develop Using the classical model of a multimachine power sys- lower and upper bounds on the distribution of the time to tem, the coupling between machines that is represented insecurity. by the transmission network, is considered to be random In [33], Blankenship and Fink presented a probabilistic in nature. However, the processes under consideration formulation of the dynamic security assessment problem are switching processes, representing both the primary motivating much of the work which followed in this area. disturbance events, faults, etc., and the operator or state- For example. in [34] and [35] probabilistic security assess- controlled secondary disturbance events, breaker opera- ment was studied in the context of transient stability. tion, etc. The effect of disturbances on the generators are Here, exit times or exit time probabilities from stability also modeled in a similar manner by modifying the power regions are computed. or estimated as measures of the output of each machine. Therefore, the state of the security of the power s!stem. The power system models system is dependent upon the evolving structure of the L O P 4 R O A%UD ABDEI UALEK: APPROACH TO DYNAMIC POWER SYSTEM SECURITY 789 network which is in turn, dependent to a degree on the state of the system. In this paper, for all possible structural variations of a power system, a dynamic security region is defined in terms of the system state variables. A system of quasi-lin- ear partial differential equations which describes the evo- trix. Then the elements Y , ( f )have the form: lution of the joint probability distribution of the state and / nh the system structure is presented. A probabilistic security measure is introduced as the probability that the system is in the dynamic security region. If the computed security i#j measure is found to be below a required security thresh- old then it is necessary to apply control actions to improve where Y, are the network parameters, S,,(t> is an inter- the situation. connection function such that: A detailed simulation example is presented for a sam- ple power system model to illustrate the results. if nodes i and j are connected at time t 11. A POWERSYSTEM MODEL S,,(t) = (il otherwise The main features of a probabilistic power system model for i, j = 1,2; . . ,nh and YL,(t)is the time-varying admit- are: tance of the load connected to the ith bus. YLf(t) be can (1) A detailed generator model for studying post-dis- written as turbance transient behavior of the machine. (2) Load variations and faults are modeled as stochas- YL, t ) = ( t ) + Y,, ( t ) , ( i = 1 , 2 , 3 , . . . ,nb tic processes capturing the random nature of primary where y ( t ) models the deterministic component of the events. load based on the projected nominal load for standard ( 3 ) Secondary events are modeled through an aggre- voltage and frequency conditions and Y,,(t) is the ran- gate representation of the protection system. dom component of the load modeled as a continuous time (4) A set of admissible controls is defined to include finite state Markov jump process with known transition all essential preventive and corrective security control probabilities and initial probability distribution. actions. The most general form of a mathematical model which A. Disturbance Models includes the above characteristics is The primary events are defined as those disturbances = F(x(t),u(t),A(t)) dt (1) which occur randomly independent of the system operat- ing conditions. Other than random changes in the load, where x ( t ) denotes the state of the system, u ( t ) is the the set of primary events includes generating unit outages input of the system, and A ( t ) , which takes values in a and line outages. These events may occur due to environ- finite set, encodes the various structures that the system mental effects, equipment failure or human error. The can have as a result of primary and secondary disturbance primary events are used to model the availability or events. For the power system model we can specialize (1) unavailability of power elements resulting from uncontrol- to the following form: lable events. These are modeled by random switches at each generator or transmission line (Fig. 1). It is assumed that the switches are normally closed (availability of the element) and that Vgl and 1/1,, take either the value 0 or 1 according to Here the state vector x ( t ) includes generator rotor angles and velocities measured relative to a reference angle and 1, i th generator available synchronous speed, the direct axis flux, quadrature axis I/gr = ( 0 , otherwise i = 1,2; . . ,ng flux, and field flux. The input vector u ( t ) includes me- chanical power input to each generator and the field 1 line ij available i , j = 1,2;. . ,n b , i # j circuit voltage, $ ( x ( t ) ) is a nonlinear term which includes Vrrr = (0: otherwise. the flux linkages for each machine, and f ( x ( t ) , A ( t ) ) is a nonlinear function of the system state variables, in partic- VRf and V I ,are random processes and historical data on ular the rotor angles of the machines and the network the time and duration of transmission line and generator parameters. i.e., the admittances and susceptances; A ( t ) outages is assumed to be available and can be updated at then accounts for the changes in the network structure as any time if the external factors (i.e., expected weather a result of primary and secondary events. This requires conditions, scheduled maintenance circumstances, etc.) more elaboration. Let n R and n b denote the number of indicate a change in the likelihood of occurrence of these generators and busses in the overall system, and let Y ( t ) events. The probability of the opening and closing time = { y , ( t ) ]i ; j = 1.7: . . . n b denote the bus admittance ma- , for these switches is derived from this data. 790 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 6, J U N E 1990 rated values, i.e., if I, denotes the current in generator i , U::: then 1 I 1, G 1 1 max > i=l,... 7 nK and if e,, is the angle between machines i and j , then i j @ ~ ' " G ~ @ , , ) Q ~ , ~=, l ; . - , n , ; i # j . Fig. 2. Protective relays in the network. The magnitude of the bus voltages should not exceed certain values in order that the power transmission capac- ity of the lines is not exceeded during the operation of the system (i.e., to prevent overloading any of the transmis- sion lines of the system). This defines an upper limit for the bus voltage magnitudes. Also, the magnitude of the bus voltages should not be below a certain limit during faults such as short circuits; this is a simplified model for Fig. 3. Controlled switches in the network. the maintenance of transient stability in the system in the following sense. A short circuit is generally accompanied by an instantaneous collapse of the bus voltages and consequently a sudden reduction of the generator power B. Protection System output. Since the power input to the generator can be A n aggregate model of the protection system is shown assumed to be constant during the first instants before in Fig. 2. The operation of the protection system induces the mechanical turbine controllers respond, each faulty structural changes in the power system network referred generator will be subject to a surplus accelerating torque. to as secondary events. These secondary events include Therefore, to dissipate the mechanical oscillations of the the triggering of generator (Rg,), line ( R f l , ) , and load synchronous machine rotors that result from a surplus (R,,) relays. The detailed operational characteristics of accelerating torque and maintain transient stability, the the relays are defined in terms of switching surfaces in the bus voltage levels should not be below a lower limit. Even state space of the power system. The determination of the though transient stability limits are in reality much more relay operating policies, in terms of the switching surfaces complicated, this formulation will suffice to illustrate our and possibly other time dependent phenomena, is part of approach and we will refer to these voltage limits as the the determination of the operating policy for the system. transient stability limit of the system. Therefore, the bus voltage magnitudes throughout the system must be within C. Modeling of Operator Initiated Control Actions certain limits for secure operation, i.e., if E, denotes the voltage at bus i then Operator initiated controls refer to those switching control activities which energize or deenergize power or E,""" G E, Q E,"'ax, i = 1,2; . . ,nh. load elements [26]. Fig. 3 illustrates how these types of These constraints on the operating variables of the power switching actions are accounted for in the power system model. Here U,,, Ufrl and U,, are operator controlled system can be written as switching devices which can be used to alter the generator ,Fin(x ,A j ) G ck(X , A j ) Q CY( x, A,), k = 1 2 , . . . ,K mix, the distribution of load on the transmission lines, and the load being served by the network, respectively. where ck(.,l) is a map from the state-structure space of In summary, the availability of a generating unit or a the power system to the real numbers. Here, x denotes transmission line is determined by the state of an equiva- the system state and A, is the structure descriptor for the lent switching element which accounts for the primary system, j = 1,2,3; . ., NP, where NP is the number of event status, the secondary event status, and the operator possible system structures. If X denotes the state space of initiated control action. The generator (or transmission the system and A denotes the set of possible system line) is in service if and only if the switches V (V,,,), structures then the constraints on the operating variables g! Rgl(R,,,)and U,,(U,,,) are all closed. The connection of define a subset, W , of the state-structure space ( X X A). the load to the network is determined by R,, and U,,, a It is assumed in our model that keeping the bus voltage load is connected if and only if the switches R,, and U,, levels in the system at their prescribed values will satisfy are closed. the reactive power balance requirements. Remark: The most realistic way of incorporating tran- sient stability constraints in the dynamic power system 111. DYNAMIC SECURITY REGION model is to associate a domain attraction (DOA) with During secure normal operation of a power system, the each system structure; see [17] for some results along operating variables have to be within certain limits. The these lines. In this way, the set W , is the set of ordered currents in the senerating units should not exceed their pairs ( x , A ) E ( X x A), the state/structure space, such LOPARO A N D ABDEL MALEK: APPROACH TO DYNAMIC POWER SYSTEM S E C U R l n 7Y 1 that if the system has structure A and if x is an element of iii) A , , ( x , t ) < 0 i =j; the DOA associated with the structure, the system is iv> C,N_,A,,(x,t)=0; transiently stable. This is consistent with our computation v> A , , ( x , t ) is continuous and uniformly bounded on of the security index as will become evident in Section V. x x[t,,m). The frequency is closely related to the stability of the overall network. Under normal and secure operating con- The process s(t> is intended to model the so-called ditions there is a balance in the system; generating units primary disturbance events such as generator outages, are supplying the system loads in addition to the real transmission line faults, etc. The matrix { A , , ( x ,t>)charac- transmission losses. Should this balance be disrupted, the terizes the rate at which the process s ( t ) evolves on S . difference would increase or decrease the kinetic energy Even though s(t> denotes a primary disturbance event, of the system. Since kinetic energy depends upon genera- the transition rates A , , ( x , t ) depend upon the state x ( t ) . tor speed, an imbalance will thus translate into a speed This is because the system structure is also random and if and frequency deviation. Therefore, keeping the fre- a generator is already out of service we do not want to quency within certain limits during the operation will allow any other primary disturbance events for this gener- guarantee such a balance condition for the system. This ator to occur. Also, the time dependence allows the inequality constraint is written as follows: modeling of nonstationary random behavior such as storms, equipment in need of maintenance, equipment 0;'" < 0,,< by"", i, j = 1; . . , n g . close to operating limits, etc. Let r ( t ) be a random process taking values in the finite Denote by W, the collection of all pairs ( x , A) in ( X X A ) set R = { r I , r 2 ;. ., rM).r ( t ) models the structural changes at which the active power balance of the system is satis- in the power system network which are state dependent. fied at all times. Partition the state space X into a set of open subsets of Let X denoted by X I , i = 1,2; . ., M such that: W, = W , n W , i) X = ( U E I X l , ) ' here ( >' denotes closure of the then W, defines a dynamic security region of the power set ( ); system as a subset of the state-structure space of the ii) X , n X , = @ for i , j = 1 , 2 ; . . , M , i z j . system. The system is said to be in the normal operating state if The evolution of r ( t ) on R is as follows: If the state the operating point remains within the dynamic secure x ( t ) E X I then r ( t > r,; i = 1,2; . ., M. Let b,, denote the = region defined above. The violation of the operating boundary separating the sets X I and X I ; i.e., b,, = XfC n constraints, i.e., being outside of the set W,, is referred to X f , i # j , i,j = 1,2; . ., M. If x ( t >E X I and crosses b,, at as an emergency operating state. The violation of frequency some time T > t and enters the set X I , then r(t>jumps a constraints, i.e., being outside of W,, is referred to as a from r, to r,. The boundary b,, can thus be interpreted as partial load operating state. Transitions between these op- a switching surface.for the process r(t>.The evolution of erating states are possible. the two processes s ( t ) and r ( t >are combined to define the process A ( t ) on its state space A. To be more precise, IV. PROBABILISTIC EVOLUTIONSYSTEMS OF for each subset X I of X we define a subset A, of A which WITH RANDOM STRUCTURE is the range of s ( t ) on S , i.e., the set of values in S that A probabilistic model of the power system with fixed the process s(t>can attain when x ( t >E X , . Then, when operating policy subject to primary and secondary distur- x ( ~ ) X I , s ( t ) evolves on A , as a finite state continuous E bances can be written in the general form: time Markov process. Let X , and X I be neighboring regions, i.e., Xf n X f is not empty and is equal to b,,. If &(t) =F(x(t),A(t)) dt A , and A, denote the subsets of A associated with the range of s ( t ) when x ( t ) is restricted to X I and X , respec- where x ( t > EX and A ( ~ ) E A, t > r o . A ( t ) models the tively, we partition A, and A, into an equal number of random changes in the system structure as a result of disjoint subsets and A , , k , for k E K,,, an index set. disturbances to the system. The primary disturbance Before defining the partitions, some additional clarifica- events are modeled by a continuous time finite state tion is necessary. When r ( t ) crosses the boundary b,, process s ( t ) . Let S = {sI; . .,s N ) denote the state space of from X , to X , because there has been a change in system d f ) ; the probabilistic evolution of s ( t )on S is determined structure, it is necessary to properly initialize the struc- by a continuous, bounded infinitesimal generator [30]. In tural evolution on A,. That is, if for example the crossing matrix form, the infinitesimal generator is an N x N of a switching surface b,, causes a transmission line in the matrix whose elements are the transition rates of the network to go out of service, then all primary events process s ( t ) . We assume that the transition rates are time which effect this transmission line are no longer admissi- and state dependent and that they satisfy the following ble in the current structure. We must account for this in properties for all x E X , t 2 to: defining the set A, and the probabilistic evolution of s ( t ) i) A ( x , t ) = { A , , ( x , t ) ) , i, ~ = 1 , 2 ; . . , 1 % ; on A, when x ( t ) is restricted to X I . If we define a ii) A , , ( x , t ) > 0 i zj ; boundary transition map p,, which defines how the struc- 1 792 IEEE TRANSACTIONS ON CIRCUITL A N D SYSTEMS, VOL. 3 7 , NO. 6, J U N E 1990 + 6 F , m B L XI -60' i-.166HZ -75O Ba +.166HZ 113 Her; 1- Fig. 4. The example power s y s t e m . Fig. 5. Security measure p versus time. TABLE IV CODING PRIMARY OF EVENTS Generalor , .Y' .YT H Rating So ' ip.u) (pu.) (MW.S./YVA) (!dVA) 1 028 008 5 50 2 1 0 2 5 I I007 I I I 4 1 1 120 5 Line No. 3 4 5 network model and the computer analysis which were X,, to 100 MVA bare I 0008 1 0003 I 0.13 used to obtain the dynamic security region illustrated in Fig. 5 can be found in [29]. TABLE I 1 1 The system model is in the general mathematical form LOADFI ON D \T.X of a system with random structure. The primary and Bru I Voltage I Load I Generator secondary events are modeled in terms of the switches So. mag.(p.u.) angle M.W. S1.V.A.R. M.W. M.V.A.R. 1 1.030 0 0 0 30.0 23.1 S1-S9 as indicated in Tables IV and V, respectively. 2 1.020 -0.5 80.0 09 4. 100.0 37.8 3 1.018. -1.0 50.0 20.0 0.0 0.0 There are seven primary events indexed by "i". The state of the six switches S1-S6 are modeled as a continuous time two-state jump process taking values in tural evolution is modified when x ( f ) crosses b,, from X , the set (0,l) as follows: to X,, then formally: 1 switch closed, no fault in T.L. 5 s1=( PI,: A , + -1, 0 switch opened, fault occurs, T.L. 5 is tripped b P,,: A, -2, 1 switch closed, no fault in T.L. 4. + s2={ 0 switch opened, fault occurs, T.L. 4 is tripped and the partitions and A,,k have the following prop- erties: 1 switch closed, no fault in T.L. 3 s3-( 0 switch opened, fault occurs, T.L. 3 is tripped i, PI,: ' 1 . k '1.k 0 switch opened, no fault in Generator 1 ii) P,r: ' , , k 'i,k s4={ iii) no other proper subsets of A , and A,.k satisfy i) 1 switch closed, fault occurs, G1 is tripped and ii). 0 switch opened, no fault in Generator 2 s5={ Example 1: To illustrate the development of a power 1 switch closed, fault occurs, G2 is tripped system model in the appropriate mathematical form, con- 0 switch opened, normal value of load 3. sider the two machine power system illustrated in Fig. 4; S6={ 1 switch closed, jump change in load 3. Tables 1-111, respectively, give the specific generator, transmission line, and load flow data that are used in the Define the open, disjoint subsets X , of R 2 as follows: example. Generator dynamics are modeled using the clas- X,, { x I , x z : lxll < 0.166 and -75"< x2 < 45") = sical swing equations with uniform damping. We recog- XI = { x I , x 2 :x , > 0.166 and -75"< x 2 < 45") nize the limits of such a generator model for predicting long term system behavior, but our only interest at this X , = { x , , x 2 : lxll < 0.166 and x 2 > 45") point is to illustrate the basic concepts of our approach to X , = { x I , x 2 :x I < -0.166 and -7S0< x 2 < 45") dynamic security evaluation. X , = { x I , x 2 :lxll < 0.166 and x2 < -75") Let 6, denote the rotor angle of the ith machine measured relative to a shaft rotating at synchronous speed X,= { x l , x 2 :x 1> 0.166 and x 2 > 45") and let d,,, denote a stable equilibrium rotor angle for the X 6 = { x I , x 2 :x l < - 0 . 1 6 6 a n d x , < - 7 5 " } ith machine. A choice for state variables of the example X,=(x,,x,: xl>0.166and x,<-7So) system. which is convenient for our purposes, is x 1 = ( 8 , - 8:) and sI= ( 6 , - 62)-(610- d2(,). Details of the X, = { x 1,x 2 : xI < - 0.166 and x 2 > 45"). LOPARO A N D ABDEL MALEK: APPROACH T O DYNAMIC POWER SYSTEM SECURITY 193 TABLE V TABLE VI SECONDARY EVENTSCODING FAILURE RATESOF THE SYSTEM 4 0 SFR Then R 2 = ( U ~ = , X k > c we define the boundaries b;, and a7 by the formula: bii = X; n X;, i, J = 0, * * * ,8, i f J associate a subset A iof A with each region X i , i = 0; - *,8. Then, For example, the boundary b,, = { x , , x , : X , = 0.166) and the remaining bij are computed in a similar way. A, = the set of primary events { 1,2,. . . ,7] which are The partitioning of the state space R 2 as defined above defined according to Table IV. is used to describe the evolution of the secondary distur- A , = A, bance events. The secondary events are modeled by the A, = (1,2,3,4,6,7), G 2 separated from the network and state dependent operation of the switches S7, S8, S9. The the primary event corresponding to S5 is not rele- status of the switches is determined by the following vant once G 2 is separated. , rules: R1) R2) x E X, -S 7 = S 8 = S9 = 1 x ~ X , u X , - S 7 = S 8 = 1 and S 9 = 0 A 3 = A,, A, = (1,2,3,5,6,7}, G 1 is separated from the network and the primaq event S4 is not relevant once G1 is separated. A, = A , , As = A,, A, = A,, A8 = A,. R3) S x ~ X ~ - S 7 = O a n d 8=S9=1 R4) x ~ X ~ - S 7 = S 9 = 1 S8=0. and The probabilistic properties of the primary disturbance These rules are summarized in Table V. events are defined according to a 7 x 7 stochastic matrix. The operation of the three state dependent switches For this example each switch is modeled by a two-state S7-S9 is as follows: Markov jump process with independent exponentially dis- i-' tributed switch times. Let ai denote the rate of event i, i switch closed, over-current relay at G1 does ' i.e., a,- is the mean time between jumps of the event i. not operate Then, ' switch opened, over-current relay at G1 oper- a , = failure rate of T.L. 5 ates, separating G1 a , = failure rate of T.L. 4 (, switch closed, over-current relay at G 2 does cy3 = failure rate of T.L. 3 not operate a , = failure rate of generator 1 switch opened, over-current relay at G 2 oper- a, = failure rate of generator 2 s8 = ates, separating G2 a6 = equivalent failure rate of load 13 switch closed, under-frequency relay at 13 does a , = failure rate between T.L. 5 and T.L. 4. not operate The numerical values used in the simulation are shown 0 13 operates shedding part of the load. in Table VI. The infinitesimal generator of the seven-state primary From Table IV there are seven primary events and from disturbance process is given by Table V there are four (state dependent) secondary events. SFR a, a 2 ag a4 a5 a6 Let A denote the set of possible system structures. In 0 -a, a, 0 0 0 0 principle, A would contain 29 elements even in this simple example, but most of these can be eliminated by practical a, -a, 0 0 0 0 considerations. For example, security assessment requires 0 0 0 : . : . : . : . that the current operating mode is normal, i.e., the oper- ating state of the system x ( t ) E X,. If we are interested in evaluating the security of the system given a single pri- 0 0 . . . o mary disturbance as an initiating event, then the cardinal- Here a , models a multiple contingency where T.L. 4 and ity of A is twenty one. Given A, the set of system T.L. 5 both go out of service and SFR = Cp=,ai. structures to define the structure evolution on A , we need The evolution of the primary disturbance process on to determine the subsets A , of A. In this example, there the subsets X , is determined by eliminating the appropri- are a total of nine regions in R 2 which are of interest. We ate rows and columns from the matrix A and renormaliz- 794 ILtL. rKAN5A<TIONS O N C'IKCUITS A N D SYSTEMS, VOL. 37, N O . 6, JUNE 1990 The boundary transition map po2 ( p I 2 )simply models the fact that in X,, the primary event 6 can not occur. The - remaining partitions Ai,, are determined in a similar way. - l% a2 a? 0, a(> We will return to this example later for our numerical 0 -a7 a, 0 0 0 studies. Next, we continue with our analysis of systems A,= 0 a7 -a, 0 0 0 with random structure. The structural evolution is characterized by the bound- 0 ... .. . ... ... ary transition maps p,,, the subsets A , , and the partitions, - 0 The problem is to analyze the probabilistic evolution of the joint Markov process ( x ( t ) , A ( t ) ) in the state-struc- ture space X x A , t to. Here ( x ( t ) , A ( t ) ) satisfies the differential equation dx ( t ) = F ( x ( t ) ,A( t ) ) dt , t 2 t,, and the problem is complicated because of the discontin- uous behavior of the vector (or velocity) field F on 61 X x '4.Our approach to the problem is to consider the restriction of the dynamics to the set A i , or more conve- where p4= SFR - a,. niently. the subsets A,,, of A , where the discontinuities in < A, = A ? ;A , = A,; A , = A,; A, = A2. the velocity field result only from the primary disturbance events. Let A,, E C A , . For each hi, fixed we have The boundary transition maps p,, define how the pri- mary disturbance process evolution is affected by state dx ( t ) = F ( x( t ) , A,, ) dt , x( t ) E X i dependent events, i.e.. when the state process crosses a where we assume that F ( . , A , , ) is a differentiable (C') boundary b,,. In this example we have mapping. Define the probability density pol = p l o is the identity map on A,, Plk(x, t ) dx = Probability { x G x ( t ) G x -tdx, A( t ) = Ai,}. Po, =PI:: - 0+ - 1 2 1 ' : Then P l k ( x t, ) has A as its support. Let Pi(x,t ) denote (1.3,3,4} -+ (1,2,3,4} the vector of probability density functions with compo- {5,6.7) -{5,7} nents P , k ( x ,t ) . From Morrison [27], P,(x, t ) satisfies the linear partial differential equation ~ 2 =~ 2 1 : 0 (1,2.3.4} (1,2,3,4} (5.7) (5,6,7) - + - - + .lo a -P,(x,t) at = L,P,(x,t)+AiP,(x,t), i=1,2;..,M S on the sets XI where A , is the restriction of the forward po3 = pill is the identity map on A,, . generator of A to X I , and L,=diagonal ( L l k } . L,, is The remaining p,, are defined in a similar way. The defined by partitions ,41,k.i = 0 . 1.2.. . . .8, k . suitably restricted, are 1 defined by the boundary transition maps p , , . For exam- -a I ple, consider the maps p,,? and p_.o.The boundary transi- L,kg(x) = , a( F ( x ,A r k ) g ( x ) ) ! tion map po2 (pl,) defines how the primary event struc- where g ( x ) is a smooth real-valued function defined tural evolution is affected b- crossing the boundary h,,2 on X I . ( b , 2 ) . If the state trajectory crosses h,,? from X,, into X z , Remark: The partial differential equation for the vector I then the status of the state dependent switches change P , ( x , t ) is a generalization of the well-known result for 1 from S7= S8 = S9 = 1 (in X , , )to S7 = S9 = 1, S8 = 0 in smooth dynamical systems with random initial data. Con- (X2); is rule 4 in Table V. In this situation the sider the system this over-current relay has actuated and generator G2 is sepa- j rated from the network. An examination of the primary disturbance events, Table I V indicates that S 5 (primary dx(t)=f(x(t))dt, t20, where f : R " + R" is a C' vector field and x(0) is a event 6) models the tripping of (72 as a result of a random variable with density p o ( x ) .Let cp,(.) denote the i primary event. However, if the state of the shstem r ( t ) E flow associated with the vector field f . Then, cp,(x) is the X,,then G2 is already separated and the priman e\'ent solution of the above differential equation initial from the corresponding to the tripping of G I i h not relevant and point x at t = 0. Let p , ( x ) d x = probability { x G x ( t ) < x must be eliminated from , Thus . partition the set .Il,nu), t 0 with x ( t > the solution of d x ( t ) = f ( x ( t ) ) , t z 0 I 2we i into two subsets, and A0,2 and .I2 into trio corre- given by x ( t ) = cp,(x(O)). It follows by an elementary calcu- sponding subsets and A?,? respectively. huch that: lation that the density p,(x) has the following representa- tion: = { 1.2,3,4}, = {5.6.'j .12,1=(l.2.3,4}, LOPARO A N D ABDEL MALEK: APPROACH TO DYNAMIC POWER SYSTEM SECURITY 795 where det Dq,l,-,~,,is the determinant of the Jacobian of a positive constant. The boundary b,, is the subset of R 2 the flow qr evaluated at q-,(x). Here, D,, is a linear map defined by from R“ + R“ and the det Dq,l,-,(,) can be calculated b,, ={x E R 2 :( X , n ) = C ) . directly: Even though Liouville’s equation is not valid on the whole det DcpJ,-,(,,= e x p l trace D f , , d ~ . of R 2 , at all points where x E R 2 such that the initial density p&’p - l ( x ) ) is continuous we have the formula: Note the trace of Ofl, = div. fix, the divergence of the vector field f . It follows that PI(X) = P”(’p-,(x))[det D&-,(X)] -’ PI ( x ) = Po (cp - I ( x ) ) exp 1 div fix 0 - dT. We will assume that p o ( . )is continuous on all of R 2 . It follows that + at on XI From the formula given for p,(x), p , ( x > satisfies the linear first order partial differential equation known clas- sically as Liouville’s equation: cPl(X> = ( x x + bt on X , and similarly that a -PAX) at = -div ( p , ( x ) f ( x ) ) cp-,(x) = i x - ut on XI x - bt on x,. To avoid a trival solution, we assume that the vectors a , b E R 2 satisfy If p,(x) is evaluated along the flow of i I= f ( x f ) , initial from x, then it follows that i> ( a , n ) < c ii) ( b , n ) > c which guarantees that the boundary b,, is not an invari- ant set for the flow cp,(x) on R 2 and that neither XI or or X , is an invariant set for the restriction of ‘ p l ( x ) to XI and X,, respectively. Let p l ( x ) and q,(t) denote the solutions of the Liouville equation restricted to the open sets XI and X,. The a time variant linear ordinary differential equation. Note “matching condition” at the boundary b,, is determined the solutions of 1,= f(x,) are simply the characteristic curves associated with Liouville’s equation. Y b The fact that the vector field f is C’ was used implic- lim P l ( Y >det D’pllq,(y) lim 4 1 ( 2 ) det Dcpllq-,(z). = y+x+ 2-x- itly in the derivation of Liouville’s equation. If the vector field f is C ’ , except along surfaces in R”, then Liouville’s Here, x + and x - denote the limit point X E b,, of equation is valid only in the regions where f is C ’ . The sequences converging to x from the interior of the sets important question is then: Given that Liouville’s equa- X , and X,, respectively. tion holds in certain regions of the state space, how are The matching condition requires that solutions of the individual equations “matched” at the boundaries of the regions? The mathematical details are P A X + 1 - det D’ptlv-,(x--) ~- discussed in [23]-[25]; we will explain the results here in q,(x- 1 det Dcpllvp,(x+) . terms of an example. A recent paper by MalhamC and For this problem it is possible to calculate the ratio on Chong [28] treats a similar problem. the right-hand side of the above equation. It follows that Example 2: Consider a nonlinear system evolving on R2 defined by [23] det D’pl,qp,(x-)( b , n ) - dw(t)=f(x(t))dt, x(t)ER2;t>0 det DcpI’P-,(X+)u , n >. ( where the vector field f on R2 is given by This leads to the matching condition: x, f(X) = ( f: x x E E x,. pt(x+)(a,n) q,(x-)(b,n). = The terms ( a , n ) and ( b ,n ) are the magnitudes of the Let n be a vector in R 2 , and define velocity field on the boundary in the direction n for the systems defined on XI and X , respectively, i.e., the XI = { x E R 2 :( x , n ) > c) matching condition can be interpreted in the context of a and conservation law. The “flow densities” p , ( x > aand q,(x)b, associated with evolution in the sets XI and X,, is X, = {x E R 2: ( X ,n ) <C} conserved along directions normal to the switching sur- where (., .) is the standard inner product on R 2 and c is face. In this context we recognize that Liouville’s equation 796 ILEE TKAhSAC'TIONS O N C'IRC'LJITS A N D SYSTEMS, VOI , 37, N O . 6. J l J N E 1990 is essentially derived from the differential of a conserva- acterization of contingencies, faults, disturbances, etc., in tion law for probability mass (or volume) throughout the an interval [t,,,t,,+ TI; the probabilistic power system sate space. This completes the example. model is used to evaluate how the system will respond on The results presented in the example can be stated this interval. The security measure p ( t ) is calculated by formally as a theorem. Theorem: Consider the dynamical system & ( t ) =F(x(t),A(t))dt where the structure evolution is as described above. Let where the sum is over all indexes ( i ,k ) which encode the rTL(.v.t) denote the flow density associated with x E X , possible structural variations of the system and where D and A = A,, E -1,. . Then if P , ( x ,t ) is the vector of proba- k is the dynamic security region D. p ( t ) is the probability bility densities as defined previously: that the system is dynamically secure at time t. This i, requires the following steps. - P , ( s , t ) =diag{ -div rjk(X,t)} +AjP,(x,t). 1) Determine a dynamic security region D which is a at subset of the state-structure space as outlined in Sec- Subject to the matching conditions at the boundary b,,: tion 111. 2) Given the initial distributions of x(t,,), A(t,,), the m C( r i k ( x , t ) , n i j ( x ) ) = C ( r j , ( x , t > , n , , ( x ) ) probabilistic evolution of x(t), A ( t ) over [ t o , o + T ] is ob- i n 7 t k I tained by solving the system of partial differential equa- 0 iL 0 where the indexes k , l range over all possible values tions presented in Section IV. The equations are subject encoding the state dependent structure changes across b,, to the conservation constraints and the use of the mathe- and is the normal to the boundary b,, at the point matical model of the system described in Section 11. The x E blj. solution of the partial differential equations determines Here, i = 1,2; . ., M , where M is the number of sets in the evolution of the joint probability distribution of the the partition of the state space X . j ranges over the set of system for a fixed operating policy and for a given statisti- indices encoding the shared boundaries between the par- cal characterization of the primary events. tition X , and the sets X,, k = 1,2;. .. M , k # i. 3) A probabilistic measure of dynamic security p ( f ) is Remarks: 1) A similar mathematical problem formula- then calculated as the integral of the joint probability tion models the development of shock waves in fluids, see distribution calculated in step 2) over the dynamic secu- [311 or the propagation of electromagnetic shock waves, rity region ( D ) , defined in step 1). The system is said to i.e., the Eikonal equation, see [32]. 2 ) The conservation be dynamically secure if this measure is above a security law in the theorem states that even though the velocity threshold (determined a priori by the operator); otherwise c field of the system is discontinuous at the boundaries b,, it is said to be dynamically insecure and then security which induces impulsive behavior in the gradient of the enhancement procedures are required. velocity field at the boundaries, that the flow density is Example 1: (continued) The security measure p ( f ) is conserved along directions normal to the boundary sur- defined as the probability at a time t that the system state face bij. This of course, assumes there is no complete ( x , , x,) is in the interior of the dynamic security region, as reflection or absorption at the boundary. 3) The P , k ( x , t ) given in Fig. 5. Since only a simplified model of the are initialized by a density P , , ( x , t , , ) which accounts for generators is used, the results are only valid over a limited the initial uncertainty of x(t,,) on A . 4) For example 2, the time interval. As defined, the security measure is an matching conditions from the theorem requires that instantaneous time-varying measure of the vulnerability of p , ( x ) ( a ,n ) = q,(x)( b, n ) , for x E b , 2 .This is the same as the current system state and network topology to stochas- derived in the example by analytical methods. tic contingency events. Fig. 6 illustrates the results of the computation for the example system being studied. All V. SECURITY APPLICATION DYNAMIC TO ASSESSMENI possible changes in the system structure, as a result of The power system can be modeled by a system of primary and secondary events, were considered in the differential equations of the form determination of the security measure according to their probability of occurrence. From Fig. 5 , at t = 2 s, the dw ( t ) = F ( x ( t ) , A( t ) ) dt. security measure is 0.849. The sensitivity of the security A dynamic security region D c X x A is defined for the measure to changes in the probability of occurrence of system such that whenever the state and structure of the the primary events was also studied. This was accom- D system ( . v ( f ) . A ( t ) ) E , then the system is said to be plished by decreasing the mean time between failures for dvnamicallv secure. We introduce a security measure for each state independent failure event, one at a time, and the system. p(r 1 which is the probability that the system is then recomputing the security measure. For the same d!.namicall! secure at the time t . Our interpretation of system, the security measure at t = 2 s is equal t o Oh2 and the security assessment problem is as a dynamic forecast- 0.60 for the cases that generators 1 and 2, respectively, ing or prediction problem. That is. given that at the are out of service with a failure rate of 0.2 s- for each. current time. say r , ) . ( d r , , ) )E D. and a probabilistic char- For the case of transmission line losses, the security LOPARO A N D ABDEL MALEK: APPROACH TO DYNAMIC POWER SYSTEM SECURITY 797 During emergency conditions, this measure can be 0.9 h extremely important to identify the impact of cas- cading events and to further collapse of the system by providing an indicator the operator for timely preventive control actions which are necessary to 06 I maintain normal operation. I I O5 015 1’0 1’5 2’0 2’5 1 30 In the way of an example, a simple power system model TIME, sec which includes the interaction between the dynamic state Fig. 6. of the system and the network topology is presented. The model includes state independent (random) switches and state dependent (controlled) switches which characterize the primary and secondary disturbance events which are disrupting normal operations. measure at t = 2 s slightly decreased to 0.84, 0.82, and Although a simplistic model of the power system was 0.83 for the loss of transmission lines 5, 4, and 3 , respec- used in this example, the basic concepts of the approach tively; again a failure rate of 0.2 s - ’ is used for each. have been tested and the results are encouraging. Even VI. CONCLUSIONS through a transient analysis was carried out in the exam- ple, the real emphasis of the work is directed toward The problem of dynamic security assessment of power long-term dynamic security assessment. In this context, systems is addressed in this paper in a rather unique and the importance of cascading events and the interaction of general form. In conclusion, the contributions of this the dynamic state of the system with the network configu- work may be outlined as follows. ration is important. Also, for long-term analysis the pre- Develop a power system model adequate for ana- diction horizon will be sufficiently far into the future so lyzing the dynamic behavior of the system. The that the approach may be computationally feasible. The power system is modeled as a dynamical system a priori information used in the method, e.g., the distribu- with state-dependent and state-independent varia- tion of primary events, can be updated as necessary to tion of its structure; i.e., all the different possible reflect changing environmental, weather, or operating changes in the structure of the system are included conditions. This was briefly addressed in the sensitivity in the model. The effect of primary (state-indepen- studies performed in the power system example. dent) events on the system is modeled by random switches. The state of these switches is governed by a continuous time finite state jump process, with state-dependent transition rates. The effect of sec- REFERENCES ondary (state-dependent) events on the system is [ l ] J. L. Convin and’ W. T. Miles, “Impact assessment of the 1977 New York City blackout,” Systems Engineering for Power, Div. of modeled by controlled switches. The state of these Electrical Energy Systems, U,B. Dep. Energy, July 1978. switches is governed by a random process whose [2] L. H. Fink and K. Carlsen, Operating under stress and strain,” IEEE Spectrum, vol. 15, Mar. 1978. change in time corresponds to the crossing of a [3] T. E. D; Liacco, “Control of power systems via the multi-level switching surface by the state process. These conce t Case Western Reserve Univ., Systems Research Center, Rep. [RC-68-19, June 1968. switching surfaces, defined in the state space, cor- [4] A. D. Patton, “A probability method for bulk power system respond to the setting of the protective relays. This security assessment, I-Basic concepts,” IEEE Trans. Power App. Syst., vol. PAS-91, pp. 54-61, Jan./Feb. 1972. model of the system is an appropriate and conve- [5] G. C. Ejebe and B. F. Wollenberg, “Automatic contingency selec- nient way of representing a power system for evalu- tion for on-line security analysis-real time tests,” IEEE Trans. Power App. Syst., vol. PAS-98, Sept./Oct. 1979. ating dynamic security. [6] G. Irisarri, A. M. Sasson, and D. Levner, “Automatic contingency A dynamic security region was defined as a subset selection,” IEEE Trans. Power A p p . Syst., vol. PAS-98, Jan./Feb. 1979. of the state-structure space such that at any mo- [7] G. Irisarri and A. M. Sasson, “An automatic contingency selection method for on-line security analysis,” IEEE Trans. Power App. ment, an operating point within the region satisfies Systs.. vol. 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Fischl, “The identification of feasible strategies in the presence of uncertainties for planning and operation of power systems,” probability density of the state-structure pair over U.S. Department of Energy Rep. DOE/ET/29115-3, Sept. 1981. the dynamic security region. This yields an instan- [12] F. D. Galiana and M. Banakar, “Approximation formula for dependent load flow variables,” IEEE Trans. Power App. Syst., vol. taneous measure of the vulnerability of the system. PAS-100, Mar. 1981. 798 IEEE TRANSACTIONS O N CIRCUITS AND SYSTEMS, VOL. 37, NO. 6, J U N E 1990 M. H. Banakar and F. D. Galiana, “Power system security corri- [34] J. Baillieul et al., “Stochastic methods of dynamic security assess- dors-Concept and computation,” I€€€ Trans. Power App. Syst., ment for electric energy systems,” U.S. Dep. of Energy Rep. vol. PAS-100, NCJY. 1981. DOR/ET/29361, Dec. 1984. F. D. 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Circuits Syst., vol. CAS-29, Sept. 1982. [38] K. A. Loparo and G. L. Blankenship, “A probabilistic mechanism F. F. Wu and Y. K. Tsai, “Probabilistic dynamic security assess- for small disturbance instabilities in electric power systems.” IEEE ment of power systems: Part I-Basic model.“ IEEE Trans. Cir- Trans. Circuits Syst., vol. CAS-32, pp. 177-184, Feb. 1985. cuits Syst.. vol. CAS-30, Mar. 1983. C. K. Pang et al., “Security evaluation in power system using pattern recognition,” IEEE Trans. Pobver App. Syst., vol. PAS-93, May/June 1974. Y. H. Pao, “Feasibility of using associative memories for static security assessment of power system overloads,” EPRI Rep. EL- 2343. Ap. 1982. D. Crevier and M. A. Nourmoussavi, ..Steady-State and transient stabilitv domains of power systems.” presented at the IEEE Cana- dian Communications and Power Conf., Montreal, P.Q., Oct. 1978. A. S. Debs and A. R. Benson. “Security assessment of power Kenneth A. Loparo (S’75-M’77-SM’89) re- systems,” Systems Engineering for Power: Status and Prospects, in ceived the Ph.D. degree in systems and control Proc. ERDA Conf., 760867, Aug. 1Y75. engineering from Case Western Reserve Uni- A. S. Khadr and K. A. Loparo. “Variable structure stochastic versity, in 1977. systems, conservation laws. and the method of characteristics,” in He was an Assistant Professor in the Mechan- Proc. of Berkeley-Ames Conf. on Non-Linear Problems in Control ical Engineering Department at Cleveland State and Fluid Dynamics, June 1983. University from 1977 to 1978 where he received K. A. Loparo et al., “Probabilistic methods for dynamic power system security evaluation.“ U.S. Dep. of Energy Rep., the distinguished faculty award for contributions DOE/ET/29363-T1, Dec. 1985. to teaching and research. From 1978 to the F. Maghsoodlou, “Dynamic security assessment in electric power present time he has been on the faculty of Case systems,” Ph.D. dissertation, Dep. of Systems Engineering, Case Western Reserve University where he is cur- Western Reserve Univ., Aug. 1983. rently an Associate Professor of Systems Engineering and Mechanical E. W. Kimbark, “Improvements of power system stability by and Aerospace Engineering and Associate Director of the Center for changes in the network.“ I€€€ Trans. Power App. Syst., vol. Automation and Intelligent Systems. He has also received wards for PAS-88, May 19$9. distinguished graduate and undergraduate teaching at Case Western J. A. Morrison, Moment and correlation functions of solutions of Reserve University. His research interests are in the areas of nonlinear some stochastic matrix differential equations,” J . Math. Phys., vol. 13, 3, pp. 299-306, Mar. 1972. and stochastic stability, nonlinear filtering and control of stochastic R. Malhamt and C. Y. Chong. “Electric load model synthesis by systems with applications to large-scale electric power systems. diffusion approximation of a high-order hybrid-state stochastic Dr. Loparo is a member of SIAM and AMS and is a past Associate system,” IEEE Trans. Automat. Corirr. A C , vol. AC-30, pp. Editor for the IEEE TRANSACTIONS AUTOMATIC ON CONTROL. e 854-860, Sept. 1985. F. Abdel-Malek, “Dynamic power system security,” Ph.D. disser- tation, Dep. Electrical Engineering. Case Western Reserve Univ., Cleveland, OH, 1985. E. Wong, Stochastic Processes in Information and Dynamical Sys- tems. Huntington, NY:Krieger. 1979. P. D. Lax, “The formation and decay of shock waves,” Amer. Math. Mthly, vol. 79, 3, p 227-241, Mar. 1972. Y. Choquet-Bruhat et a r , Analysis. Manrfolds, and Physics. Am- sterdam, The Netherlands: North-Holland. 1977, p. 455. G. L. Blankenship and L. H. Fink, “Statistical characterizations of power system stability and security,” in Proc. Second Lawrence Fayez Abdel-Malek (M’86), photograph and biography not available at Symp. System and Decision Science. Berkeley, CA, 1978. time of publication.

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