VOLTAGE COLLAPSE AVOIDANCE IN POWER SYSTEMS: A RECEDING HORIZON APPROACH

Transcription

1 Intelligent Automation and Soft Comuting, Vol. 12, No. X,. 1-14, 26 Coyright 26, SI Press Printed in the USA. All rights reserved VOLAGE COLLAPSE AVOIDANCE IN POWER SYSEMS: A RECEDING HORIZON APPROACH S. A. AIA, M. ALAMIR AND C. CANUDAS DE WI Laboratoire d'automatique de Grenoble Domaine Universitaire BP Saint Martin d'hères France attia@ieee.org, Mazen.Alamir@ing.fr, carlos.canudas-de-wit@lag.ensieg.ing.fr 1. INRODUCION ABSRAC In this aer, a receding horizon control aroach is develoed for voltage stabilization in ower systems. he rimary focus is on a benchmark system roosed in the context of a Euroean roject. he system is termed hybrid due to the finite set of control inuts, state automata descrition of the voltage rimary controller and their interactions with the load nonlinear continuous dynamics. Moreover, the network toology imoses hard constraints on the states. All these make the control roblem quite challenging. For different realistic scenarios, the roosed control algorithm is shown to stabilize the bus voltages at accetable levels. he comutational demand fits the real time allowed slot, making the scheme attractive for large scale alications as an area central controller. Key Words: Hybrid systems, redictive control, voltage stability, benchmark roblem Besides the challenging control roblems osed by the large scale and distributed character of ower systems, an imortant hybrid asect is ut forward by the discrete continuous interactions both at the hysical and control levels. Consequently, ower systems are considered as aradigms of hybrid systems, see e.g., [1]. he theoretical foundations for hybrid systems has reached a moderate degree of maturity [2]- [4] this include both modelling and stability frameworks, but to date no general controller synthesis methodologies are available, it seems that this is system class and alication's related. In the last decade, the ower community has shown a great interest in systematic design methods for assessing stability in ower networks. Voltage instability is undeniably one of the most inconstant and costly failures, for instance, the 14 August 23 blackout in North America cost between US $4 billion and $6 billion [5]. Basic material on voltage stability hilosohical concets may be found in the recent surveys [6]-[7]. he interested reader may also consult [8]-[9] for general definitions and basic roblems. Much of the work reorted on voltage stability assessment relies on heuristics. Where basically, knowledge of the system and exerience gained by the designers is ut forward in schemes such those develoed in [1] where heuristics based load shedding strategies are comared with branch and bound based ones. he multi ste design consist in time domain simulation for generating training scenarios. An otimization soundstage is then carried out to minimize the amount of load shedding with resect to the value found in the training set. Another family of methods rely on comuting a security margin or distance to voltage collase, see e.g., [11] for a survey. Based on these measures, sensitivities with resect to the different control actions are then comuted. hese rovide directions towards the control must be udated, this is in general followed by a constrained multile otimization stage for disatching the different control actions, see e.g., [12] where a coordinated controller is synthesized by taking as a margin the minimum distance between the oerating oint and bifurcation boundary as measured in the arameter sace. In the same sirit, [13] use a quasi steady state simulation aroach. A simle formula is then roosed to udate the different control actions based on off line bus ranking, see also [14] and [15] for other variants. In [16] a clear distinction is made between the different control actions. And a scheme is derived where corrective 1

2 2 Intelligent Automation and Soft Comuting and reventive control strategies interact, the first is used in extreme contingencies (loss of system solvability) while the latter is for enhancing the system stability margin. he method develoed in this aer fall into the category where knowledge of the system is reduced at its basic. Only a simulation model is indeed needed. his model should incororate at least the dominant features i.e., the load dynamics since voltage collase is driven by these dynamics. hese aroaches include mainly methods based on redicting and analyzing sensitivities of the system trajectories see e.g., [17] where this is discussed in the more general framework of hybrid systems. See also [18] where a redictive control aroach is investigated. Based on evolutionary rogramming techniques, an otimization stage is carried out over the sace of all control inuts. he continuous set oint inuts as well as the discrete ones i.e., the caacitor banks and load shedding mechanisms. he redictive control aroach is also exloited in the work [19] where an imroved deth first search method from the Artificial Intelligence community is used as an alternative to further reduce the combinatorics. Material on redictive control can be found in [2]-[23]. he aim of the resent aer is to exlore a articular controller design methodology for voltage collase avoidance in ower systems with emhasis on a articular benchmark roblem [24]. his is a ste towards a fully coordinated decentralized solution in the sirit of [25], where the focus is on area central controller. We will not deal with higher level controllers used to coordinate areas controllers, indeed we believe that at a higher level an exert system with advanced heuristics shall be used, the interested reader may consult the multi agent aroaches e.g., [26]- [27]. he controller develoed is based on an efficient oen loo arametrization that drastically reduces the combinatorics associated to a rediction based methodology. hen a terminal inequality constraint is imosed on the bus voltages, discrimination of "good admissible controls" from the "bad ones" thus follows. Based on that, an otimization stage is carried out to extract the otimal oen loo control sequence. he first comonent of which is alied and the rocess is reeated in a receding horizon fashion [2]. he controller comutational burden is shown to be comatible with the available comutation time for a large number of rediction horizons. he aer is organized as follows: section 2 resents a brief analysis the benchmark ower system under consideration. Section 3 is devoted to the redictive control aroach. In section 4 some validating simulations and algorithm comutational times are reorted. Finally, some conclusions and future orientations are given in section POWER SYSEM ANALYSIS he ower system under study is roosed as a benchmark by ABB [24] in the framework of the Euroean Project Control & Comutation. his in order to illustrate control strategies on hybrid systems. he ower system consists in a four node transmission system as shown in figure 1, see also [24] for a comlete descrition Figure 1. he ABB transmission test case Figure 2. he OLC finite state machine model

3 Voltage Collase Avoidance in Power Systems 3 he system comonents are as follow An infinite bus which reresents the rest of the network. he system under consideration can not influence areciably the rest of the ower system network. his bus is also used as a slack in the modelling hase. A generator G 1 which is modelled as a steady version of the three axis model. his is a valid model since the dynamics of interest are well beyond (slower) the angular frequency. hree transmission lines modelled as ure reactance. And refereed as X ij where the subscrits i and j refer resectively to the dearture and arrival buses. It is clear that X ij = X ji since the ath is unique. Next, it is assumed that only the variation of X 13 is allowed due to a fault occurrence. his variation is considered as a measurable disturbance. A reactive ower source reresented by the caacitor banc B. his is actually a control inut to the system. A transformer equied by an On Load a Changer (OLC). Which can regarded as a transformer with a controlled turn ratio n. he OLC is modelled as a finite state machine, see figure. he oeration of which is as follows, the system remains in the state wait as long as the voltage deviation ( v r v ) is less than a secified dead zone. When the limit is exceeded, a transition to state count occurs. Uon entering count, a timer is started and is ket running until either it reaches the delay time d causing a transition to the state action or the voltage deviation becomes less than the dead zone causing a transition to the state wait and reset of the timer. When entering the state action, an udate of the turn ratio is oerated n + dn vr v > n < nmax + n = n dn vr v < n > nmin (1) where the suerscrit + and - reresent resectively the instants just before and after the instant of udate. n min and n max is the maximal and minimal allowed values. Once the ta oeration comleted, the control system receives a ready signal and the control system returns to state wait. It is imortant here to recall that the control inut to the OLC is the voltage reference v r ( ). A nonlinear load with recovery dynamics described by the following first order differential equations [24] x x q = = x x q q + P ( v + Q ( v and the following outut equations reresenting resectively the absorbed active and reactive ower α s β s v v α t β t ) ) (2) P Q = = x (1 k) x (1 k) q q α t + P v + Q v β t (3) where x = [ x x q ] reresent the load internal states and P and Q the active and reactive ower steady state values. he others arameters can be found in [24]. he variable k reresent the load shedding ercentage and is the third control inut. he model of the ower system can be written as a DAE

4 4 Intelligent Automation and Soft Comuting x = = f ( x, y, ) g( x, y, u, ) (4) where the vector differential equation reresents the load dynamics, x are the internal states, y the algebraic variables (bus voltages), for the examle under study y regrous the bus voltages, are ower system arameters for instance, the line reactances X ij, u reresents the control vector and regrous devices such caacitor bank, load shedding, ta changer control and AVR steoints. By using a coordinated action of the ta changer, the caacitor bank and the load shedding. he designed controller should be able to stabilize and restore accetable equilibria for the ower system. In figure 3 are lotted the bus voltages against the line reactance. As shown, the case where X n 13 =.5. u corresonds to the nominal steady state solution. he solutions dro when the value of X 13 exceeds 1..u and above X13 b = 1.2. u no steady state solutions are feasible, so that for this value and beyond the ower system can not be solved (loss of the DAE system's solvability). his case corresonds in fact to a singularity induced bifurcation [28]. his is dealt with in this case study. Recall that when the ower system is in fault, the X 13 reactance asses from.5. u its nominal value to X f 13 = 1.5. u which is well beyond X 13 b = 1.2. u. It is a well known fact, that the steady state ower demand ( P in equations (2)-(3)) is a slowly but time varying arameter. Figure 4 shows the evolution of the steady state values of the voltages as a function of P where no fault is assumed. As one can see, beyond P b = u no solutions do exist for the voltages. his can be seen as an active ower limit that can be drawn from the ower system. his is to be connected with the P-V curves [29] where only the stable region is lotted. he same quantity is lotted in figure 5 with resect to variations in the steady state reactive ower demand Q in equations (2)-(3). he system loses its solutions when Q b =.89. u and above which is a limit reactive ower that can be absorbed by the load without inducing a voltage collase. In figure 6 is lotted the steady state voltage solution versus the caacitor bank and load shedding ( 1 k in equations (3)). his lot shows that in order to restore an equilibria about.3. u of the caacitor must be activated. Figure 3. Bus voltages steady state values as a function of the line reactance. Figure 4. Bus voltages steady state values as a function of the active ower.

5 Voltage Collase Avoidance in Power Systems 5 Figure 5. Bus voltages steady state values as a function of the reactive ower. Figure 6. Load voltage for variable caacitor and load shedding. As we shall see later, one can not rely only on static analysis to derive control olicies. his is the otimal ower flow based methods (urely static setting) as known in literature see e.g., [3]. his methodology can be satisfactory, when the static analysis's derived controls are alied as soon as the fault aears, or the case where some advanced heuristics are introduced leading to efficient timing of the different actions. he voltage collase as seen in this contribution is a dynamic rocess needing efficient timing of the actions and thus a closed loo control strategy. More exressively, the control objectives are to fulfill the following requirements Stabilize all voltages at values above.9.u following the fault. Minimize the amount of load shedding alied. Kee the voltage at bus 4 close to 1.u while minimizing the amount of caacitor control to do so. Before we roceed further, let us denote the vector of bus voltages [ v 2 v3 v4 ] as y. We have the following constraints on the control inuts. he load shedding ercentage k belongs to the following discrete set with k =.15 max k K := {.,.5, } (5) he caacitor banc belongs to the following discrete set of admissible caacitors with b =.3 max b B = {.,.1, b } (6) It is worth noting that because of the ta changer dynamics, the transformer ratio n cannot be rigourously considered as a control variable. he true control variable must be v r that is used in the ta changer definition (see figure 2). However, it is clear that if the following three conditions are resected he oen-loo control inut n (.) is iece-wise constant with a samling eriod d (where d is the time delay of the OLC model) he iece-wise constant oen-loo control inut n (.) is such that for all j ℵ, one has max max n( j + 1) n( = d n (=.2) where d n is the ste size modulus alied to n by the OLC model when entering the " action" mode. he iece-wise constant oen-loo control inut n (.) is such that for all j ℵ, one has (.8 =) nmin n( nmax (= 1.2)

6 6 Intelligent Automation and Soft Comuting then it is ossible to use n as a constrained control variable and to obtain v r by inverting the dynamic of n. his is done by taking for all t j, ( j + 1) [ v r [ d d ( t) = v( t) Γ sgn( n( j + 1) n( ) (7) where Γ is any constant greater than 1 and sgn ( ) is the usual sign function. Definition 1. A sequence { n ( } j ℵ that meets the above three requirements is said to be ta-changer comatible. 3. HE NONLINEAR PREDICIVE CONROLLER Nonlinear redictive control [23] is now widely recognized to be a feedback strategy roviding a relatively easy handling of both nonlinearities, constraints and otimality concerns. Recall that redictive control schemes amount to comute at each samling time an otimal oen-loo control sequence (in the sense of some given cost function), to aly the first art of the resulting otimal oen-loo control sequence until the next samling instant. At the next samling instant, the whole roblem is re-considered on a moving-horizon and the rocedure is reeated indefinitely resulting in a state feedback law. For nonlinear systems and for long rediction horizons, this may lead to oen-loo otimal control roblems with a high dimension of the decision variable. his together with the non convex nature of the roblem may render the on-line comutations necessary to imlement the resulting strategy unfeasible. hat is the reason why a key feature in nonlinear redictive control design lies in the choice of control arametrization that leads to reasonably reduced comlexity. he aim of the following subsection is to clearly define a reduced dimensional arametrization of oen-loo controls that are used afterward in the redictive control imlementation. Recall that a key feature in receding-horizon control is that the resulting closed-loo control is much more rich than the underlying oen-loo arametrization. he consequence of this is that in many cases, aarently over-simlified oen-loo arameterizations results in a sufficiently rich closedloo control behavior. 3.1 he oen-loo control arameterization Recall that the control is imlemented in a samling scheme with a samling eriod τ = d and that based on this assumtion, the control inut in our roblem is given at each samling instant j τ ( τ = d ) by u( := ( n( b( ( ) { n( j 1) d n, n( j 1),( j 1) + d n } B K (8) with the constraint nmin n nmax resectively. Remark 1. It is needless to say that the samling eriod τ may be different for the three control comonents. his enables k and b to be more reactive than n restricted by the ta changer dynamics. In this study, we referred to adot a unified samling scheme for simlicity. Like any redictive control scheme, one must first define a rediction horizon, that is, the future time horizon over which some cost function is to be minimized. Given some samling time, this rediction horizon may be defined as an integer number N of samling eriods. o define the control arametrization, one has to clearly define at each samling instant j τ the structure of the control u ( ) over the time horizon [ j τ,( j + N ) τ ]. Before, let us use the following notations to denote the oen-loo control rofiles at instant j τ over the rediction horizon [ j τ,( j + N ) τ ] (. Recall that the sets B and K have been defined in (6) and (5)

7 Voltage Collase Avoidance in Power Systems 7 u ( n ( b ( k ( := := := := ( u( jτ ),, u(( j + N ( n( jτ ),, n(( j + N ( b( jτ ),, b(( j + N ( k( jτ ),, k(( j + N 1) τ )) 1) τ )) 1) τ )) 1) τ )) he oen loo control arametrization is defined by adoting the following choices over the rediction horizon he control rofiles k ( j ) and b ( are constant. he control rofiles n ( j ) is monotonic starting from n ( j 1) he control rofiles n ( j ) is constant on [( j + N c ) τ,( j + N ) τ ] where N c is the control horizon [22]. he last two oints define 2 N c + 1 ossible choices for n ( j ). N c increasing rofiles, N c decreasing rofiles and 1 constant rofiles. As a result, the resulting arametrization leads to a decision variable of dimension (2N c + 1) card(b) card(k) Note also that all candidate sequences n ( j ) defined by the above rules are ta changer-comatible in the sense of definition 1. Next and for simlicity the control horizon N c is taken equal to the rediction horizon N. he imact of such is not imortant since the control rofiles are taken constant for the two first control inuts. 3.2 Definitions and notations In order to roerly resent the otimal control comutation, some further definitions and notations are needed. Let U ( n( j 1)) be the set of admissible rofiles at instant j τ which were defined before. he fact that this set deends on the ast value of n results from (8). his will be shortly denoted by U j. Sometimes, the index j is omitted when no ambiguity follows. We shall define an equivalence relation on U j by (1) (2) (1) (1) (2) (2) { u u } {( b, k ) = ( b, k )} (9) he need for such equivalence relation comes from the fact that while constraints are imosed on the use of caacitor changes ( b ) or load shedding ( k ), no exlicit constraint is considered to state that such ta changer comatible sequence (1) n is better or worst that some other one, still comatible sequence (2) n. Denote by := {U the set of equivalence classes that artitions U eq j, k ) } eq ( b U j j ( b j, k ) B K according to the equivalence relation (9), namely, an equivalent class (an element of U ) is defined by the corresonding ( b, ) air where ones ( r, s) denotes an r s matrix with all its elements at 1, ) U ( b k := { = (,, ) U u n b k b = b ones(1, N ) and k = k ones(1, N )} (1) Let j j k

8 8 Intelligent Automation and Soft Comuting O : B K {1,, card(b) card(k)} eq be an ordering of U j that reflects the riority in the choice of the control action. Namely (1) (2) (1) (1) (2) (2) { k > k } {O( b, k ) > O( b, k )} (2) { k (1) = k and (1) > (2) (1) (1) (2) (2) b b } {O( b, k ) > O( b, k )} his order relation reflects the concern that changes in the caacitors have to be referred rather than load shedding if the control n is unable to lonely recover the voltage collase. We shall denote by X ( t; x, u ) the solution x ( ) of the system's equations at time t starting from the initial conditions (, x ) and under the control sequence u. Using this notation, the following two admissible control rofiles are defined (subscrit "nc" is used for no collase) U nc ( x X (, x ) := { u U, u ), k is defined over [, N τ ]} ) ( b nc nc nc nc j ( b, k ) = arg ( b, min O( b, k ) : k ) B K (11) in other words, U nc ( x ) is the set of oen loo control rofiles such that starting from x, no voltage collase occurs on the rediction time interval [, τ ]. Finally, the following set needs to be defined U f ( x y ( N i ) := { u U τ, x( N, k ) τ ), u ) y N ( b f f f f j ( b, k ) = arg min ( b, k ) B K i, i {1,2,3}} his is the set of controls for which the resulting near steady state voltage meets the viability requirement. he last inequalities are comonent wise constraints. 3.3 he imlicit feedback law Having at hand the notations of the receding section, the roosed nonlinear redictive control may be roerly defined. he "otimal oen-loo" control rofile at instant j τ is obtained by solving the following otimization roblem arg min J ( u, x( if U f u U f uˆ( x( = arg min ( (13) J u, x( if U = f u U nc where J ( u, x( := N τ ( y( τ, x( u ) y ref ) O( b Q( y( τ, x( u ) y and y ref is a reference signal and Q is a weighting matrix. he otimization leads to the following sequence of inuts uˆ ( x( = ( uˆ ( x( n( j 1)) uˆ 1 ( x( n( j 1))) N where only the first comonent is alied. hus leading to the following imlicit state feedback u ( t) = uˆ ( x( n( j 1)) ; t [ jτ,( j + 1) τ ) (14) in accordance with the redictive control rincile. 4. SIMULAION RESULS In what follows see figures 7 and 8 are shown the results of the simulations for the case where the rediction horizon is N = 1 (a one samling eriod). his induces a chattering in the control inuts, more, k ref ) : ) dτ (12)

9 Voltage Collase Avoidance in Power Systems 9 Figure 7. Bus voltages and load states for N = 1 Figure 8. Control inuts for N = 1 recisely in the load shedding control inut. From a ractical viewoint the chattering of the different control inuts is not accetable since it drastically reduces the life san of the different control comonent. his chattering can be alleviated by using a longer rediction horizon. Indeed, a rediction horizon of N = 1 ( 3 sec ) does not essentially catures the dominant dynamics of the system since the load time constants are = q = 6 sec. For a rediction horizon of N = 5 ( 15 sec ) the chattering is suressed. he voltage resonses are shown together with the control inuts in resectively figures 9and 1. he results obtained here (for N = 5 ) are exactly the same as those obtained in [31]. he authors in [31] use the Mixed Logical Dynamical Framework [32] to aroximate and convert the ower system dynamics into a iecewise affine system. As ointed, this led to sensitivities both in the modelling and control hases, moreover the aroach roosed is not fully systematic in the sense that a heuristic value of the caacitor is used. he roblem is thus simlified by using as control only the transformer turn ratio. In figures are deicted the voltage, states and control inut corresonding to one of the most difficult scenarios. Indeed this has been identified as difficult based on an analysis carried out on a simle two nodes system, see [33]. his analysis revealed that for the case of low initial values of the transformer turn ratio, the steady state voltages corresond to the nearest equilibria to the singular surface (the surface described by the algebraic load flow equations). As may seen from these figures, chattering is also resent for the case where N = 1. In figures the same scenario is tested for the case where N = 5. he chattering is thus avoided but the caacitor bank is activated before the occurrence of the fault ( t fault = 1 sec ). his is due to the fact that the horizon of N = 5 remains shorter and the controller will not be aware that the sole action of the transformer turn ratio is sufficient to stabilize the voltages at a higher level. Load shedding is also used in the transient hase just after the fault detection. In order to avoid such a useless activation of the caacitor, a longer rediction horizon should be used. In figures are deicted the system resonses for N = 1. he caacitor is used only after fault detection, a load shedding of 5% is introduced for aroximately more than 2 min. In figure 17 is lotted the comutation time versus the rediction horizon. Desite the facts that the controller is imlemented on one of the highest level rogramming languages (Matlab) and that it sends most of its time getting simulations from the ower system model (converted from Dymola to Simulink). With all these in mind a rediction horizon u to N = 3 can be imlemented. In real time alications and for larger scale systems by using more advanced simulations techniques, one can hoe to imlement this strategy with comarable rediction horizons. Although, for the case study and beyond N = 5 no erformance imrovement is achieved. he

10 1 Intelligent Automation and Soft Comuting Figure 9. Bus voltages and load states for N = 5 Figure 1. he control inuts for N = 5 Figure 11. Bus voltages and load states for N = 1, Figure 12. he control inuts for N = 1, n ( ) =.8 and n ( ) =. 8 Figure 13. Bus voltages and load states for N = 5, Figure 14. he control inuts for N = 5, n ( ) =. 8 and n ( ) =. 8

11 Voltage Collase Avoidance in Power Systems 11 Figure 15. Bus voltages and load states for N = 5 Figure 16. he control inuts for N = 5 and a delay of and a delay of 11 sec 11 sec Figure 17. he comutation time versus N Figure 18. Performance versus N Figure 19. Performance versus the delay Figure 2. Bus voltages and load states for N = 1, Figure 21. he control inuts for N = 1, n ( ) =.8 and n ( ) =. 8

12 12 Intelligent Automation and Soft Comuting erformance is worsened, we conclude that the otimum rediction horizon for the system under nominal conditions ( n () = 1.2 and a delay fault-action τ d of 2 sec ) is 5, see figure 18. In order to show the imortance of a dynamic analysis. he delay τ d between the fault occurrence and its detection is varied, see figure 19. For a delay of about 14 sec an accetable voltage level can not restored and voltage collase is inevitable under the actual constraints on the different control inuts. his is closely related to critical time aroaches, [34]. For instance, in figure 2-21 are lotted the different variables when the delay is 11 sec. Although, the steady state inuts are the same as in the receding cases (static analysis see figure 6) load shedding is used in the transient hase, this is in accordance with the fact that delaying too much the action leads one to shed more load (if reaction is made instantaneously the steady states values are sufficient) see e.g., [1] where the same conclusions are drawn. Recall this tye of control behavior can not be inferred from the static analysis. 5. CONCLUSIONS In this aer a redictive control aroach is develoed for voltage stabilization of ower systems. Desite its tuning easiness, the aroach is shown to be comutationally efficient which makes it attractive for real time alications. Moreover, numerous simulation scenarios testify the viability of such an aroach. By using subotimal search methods, extension to large scale systems can be made ossible within the comutational time restriction. his is currently an issue under investigation, and some romising results are available. REFERENCES 1. W. H. Esselman, et al., Hybrid discrete and continuous control for ower systems, Discrete Event Dynamic Systems, 9, , P. J. Antsaklis and A. Nerode, Secial issue on hybrid systems, IEEE rans. On Aut. Control, Vol. 43, N 4, S. Morse, et al., Secial issue on hybrid systems, Automatica, vol. 35, N 3, P. J. Antsaklis, Secial issue on hybrid systems, Proc. of the IEEE, vol. 88, N 7, P. Fairley, he unruly ower grid, IEEE Sectrum, Van Cutsem, Voltage instability: Phenomena, countermeasures and analysis methods, Proc. of the IEEE, vol. 88, N 2, , K.. Vu, et al., Voltage instability: Mechanisms and control strategies, Proc. of the IEEE, vol. 83, N 11, , IEEE/CIGRE Joint ask Force on Stability erms and Definitions, Definition and classification of ower system stability, echnical reort, I. A. Hiskens, Analysis tools for ower systems- Contending with nonlinearities, Proc. of the IEEE, Vol. 83, N 11, , Van Cutsem, et al., Design of Load shedding schemes against voltage instability using combinatorial otimization, IEEE PESW Meeting, C.A. Canizares, et al., Comarison of erformance indices for detection of roximity to voltage collase, IEEE rans. on Power Systems, vol. 11, N 3, , Q. Wu, et al., Voltage security enhancement via coordinated control, IEEE rans. on Power Systems, vol. 16, N 1, , Van Cutsem, et al., A comrehensive analysis of mid-term voltage stability, IEEE rans. on Power Systems, vol. 1, N 3, , M. La Scala, et al., On line dynamic reventive control: an algorithm for transient security disatch, IEEE rans. on Power Systems, vol. 13, N 2, , A. akehara, et al., Voltage stability reventive and emergency-reventive control using VIPI sensitivity, Electrical Engineering in Jaan, vol. 143, N 4,. 22-3, Z. Feng, et al., A Comrehensive aroach for reventive and corrective control to mitigate voltage collase, IEEE rans. on Power Systems, vol. 15, N 2, , I. Hiskens and M. A. Pai, rajectory Sensitivity Analysis of Hybrid Systems, IEEE rans. on Cir. and Sys., vol.47, N 2,.24-22, 2.

13 Voltage Collase Avoidance in Power Systems J. Y. Wen, et al., Otimal Coordinated Voltage Control for Power System Voltage Stability, IEEE rans. on Power Systems, vol. 19, N 2, , M. Larsson and D. Karlsson, Coordinated System Protection Scheme Against Voltage Collase Using Heuristic Search and Predictive Control, IEEE rans. on Power Systems, vol. 18, N 3, , D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE rans. on Aut. Cont., vol.35, N 7, , W. H. Chen, et al., Model redictive control of nonlinear systems: comutational burden and stability, IEE Proc. On Cont. he. and Al., vol.147, N 4, , J. B. Rawlings, utorial overview of model redictive control, IEEE Cont. Sys. Magazine, vol. 2, N 3, , D. Q. Mayne, et al., Constrained model redictive control: Stability and otimality, Automatica, vol.36, , M. Larsson and P. Korba, Benchmark Model for Hybrid Systems Design based on a small-scale Power System Model, his issue. 25. B. Fardanesh, Future trends in ower system control, IEEE Com. Al. in Power, , Nagata and H. Sasaki, A multi-agent aroach to ower system restoration, IEEE rans. on Power Systems, vol. 17, N 2, , H. Ni, et al., Power system stability agents using robust wide area control, IEEE rans. on Power Systems, vol. 17, N 4, , V. Venkatasubramanian, et al., A axonomy of the Dynamics of Large Constrained Nonlinear Systems, Proc. of the IEEE, vol.83, N 11, , J. Zaborsky and M.D. Ilic, Dynamics and Control of Large Electric Power Systems, Wiley Interscience, J. A. Momoh, et al., Challenges to otimal ower flow, IEEE rans. on Power Systems, vol. 12, N 1, , Geyer, et al., Hybrid Emergency voltage control in ower systems, Proceedings of the Euroean Control Conference Oxford, UK, A. Bemorad and M. Morari, Control of Systems Integrating Logic, Dynamics, and Constraints, Automatica, vol.35, , M. Larsson, A Simle est System Illustrating Load Voltage Dynamics in Power Systems, echnical reort ABB, L. Varga and C. Canizares, ime deendence of controls to avoid voltage collase, IEEE rans. on Power Systems, vol. 15, N 4, , 2. ABOU HE AUHORS S. A. Attia received an engineer diloma in Electrical engineering (ower comonents) from the university USO, Oran in After sending two years as an associate researcher at the same university, He joined the LAG (Laboratoire d'automatique de Grenoble), Polytechnic of Grenoble, where He obtained the Msc in Automatic control. He is now a Phd candidate at the same deartment. His research interests are in the field of nonlinear otimal control with emhasis on the comutational asect and alications.

14 14 Intelligent Automation and Soft Comuting M. ALAMIR graduated in Mechanical Engineering from the Polytechnic of Grenoble (INPG), France (199) and in Aeronautics from the Ecole Nationale SuXerieure des IngXenieurs en Construction Aéronautiques (ENSICA), France (1992). He received a Ph.D. from INPG in Nonlinear Control (1995). Since 1996, he has been a research associate CNRS in the Laboratoire d Automatique de Grenoble in Saint Martin d Hères, France. His main research interests are nonlinear receding horizon control, nonlinear receding horizon observers, robust nonlinear and otimal control as well as industrial control alications. C. CANUDAS DE WI received his B.Sc. degree in electronics and communications from the echnological Institute of Monterrey, Mexico in 198. In 1984 he received his M.Sc. in the Deartment of Automatic Control, Grenoble, France. He was visitor researcher in 1985 at Lund Institute of echnology, Sweden. In 1987 he received his Ph.D. in Automatic Control from the Polytechnic of Grenoble (Deartment of Automatic Control), France. Since then he has been working at the same deartment as "Directeur de recherche at the CNRS", where He teaches and conducts research in the area of nonlinear control of mechanical systems, and on networked controlled systems. His research toics includes: vehicle control, adative control, identification, control of walking robots, systems with friction, AC and CD drives, and networked controlled systems. He has established several industrial collaboration rojects with major French comanies (FRAMAOME, EDF, CEA, IFREMER, RENAUL, SCHNEIDER, ILL). He has written a book in 1988 on adative control of artially known systems: theory and alications (Elservier Publisher). He has also edited a book in 1991 on Advanced Robot Control (Sringer-Verlag), a joint 12 author book "heory of robot control", on Sringer Control and Communication Series, in (1997). He has been associate editor of the IEEE-ransaction on Automatic Control, from Jan 1992, to Dec. 1997, and of AUOMAICA, from 1999, to 22. His research ublications includes: more than 12 International conference aers and more than 47 ublished aers in international journals. He has suervised 2 Ph. D. students.