A Model-predictive Approach to Emergency Voltage Control in Electrical Power Systems

Transcription

1 A Model-predictive Approach to Emergency Voltage Control in Electrical Power Systems Mats Larsson Abstract Recent blackouts have shown that traditional local protection against power system collapse is not always sufficient to arrest instabilities. This paper shows that centralized systems can perform better than traditional local protection, as well as outlining how the decision logic of such centralized systems can be implemented. We present a coordinated system protection scheme (SPS) against voltage collapse based on model predictive control and heuristic tree search. It coordinates dissimilar and discrete controls such as generator, tap changer and load shedding controls in presence of soft and hard constraints on controls as well as voltages and currents in the network. The response with the coordinated SPS is compared to an SPS based on local measurements using simulation of the Nordic 32 test system. In terms of the amount of load shedding required to restore voltage stability, the simulations indicate that load shedding based on local criteria is near optimal in the system studied. However, when also generator controls are considered as emergency controls, the coordinated scheme reduces the amount of required load shedding by 35% compared to the local scheme. I. INTRODUCTION Voltage instability mitigation has been discussed for some time now, and some utilities have protection systems against voltage collapse in operation [], [2]. Most of these systems use rather simple rules, such as low voltage, and quite rough actions, such as load shedding. At the same time phasor measurements units are becoming an increasingly accepted and available technology. Using these phasor measurements it is possible to monitor the power system using a time resolution down to milliseconds. This gives the possibility of a wide range of stability monitoring and control applications. Therefore, time has now come to introduce more smooth emergency control systems, where system-wide voltage levels, reactive power flows, etc., are used in a wide-area emergency control system. Such a system, to be activated when the power system is in transition towards instability, must include different voltage levels and act on transformer tap changers, AVRs on generators as well as reactive power compensation devices available. To save a power system exposed to a disturbance, where the system survived the initial disturbance, but the system dynamics have been triggered and a transition towards instability has started, powerful and synchronized actions are required as the harm to the customers has to be minimized. In such a situation, i.e., a system exposed This work was partly funded by a grant from the Bundesamt für Bildung und Wissenschaft, Bern, Schweiz in the Control and Computation project (IST ). M. Larsson is with Corporate Research, ABB Switzerland, 5405 Baden, Switzerland. (mats.larsson@ch.abb.com) to a long-term voltage instability, there is some time, tens of seconds to some minutes, available to counteract the transition. Basic rules are to lower the voltage on the load level as much as possible to achieve a temporary load relief. The transmission, subtransmission and distribution system voltages should, however, be kept as high as possible to reduce the losses and maximize the line and cable reactive power generation. Furthermore, most control variables have an inherently discrete nature; for example, capacitor banks and tap changers must be switched in fixed steps and while most utilities lack direct load control schemes, load shedding must still be carried out by disconnecting whole feeders. Recent blackouts have shown that traditional local protection against power system collapse are not always sufficient to arrest instabilities. This paper shows how centralized systems like can perform better than traditional local protection, as well as outlining how the decision logic of such centralized system can be implemented. The scheme presented uses a network model and wide-area measurements to account for the actual power flow in the system, load models to account for load recovery dynamics and constraints are imposed to ensure that voltage and generator current limits are not violated. The scheme presented is suitable for implementation on a wide-area monitoring and control platform such as that described in the paper. II. SYSTEM SETUP The wide area platform for dynamic monitoring of transmission systems comprises of hardware: Phasor Measurement Units (PMU) Communication Links Central Unit (Personal Computer) and software: Data preprocessing package Basic services Specific individual applications Graphical user interface (GUI) Package containing model/data of the supervised power system and coordination with other software packages PMUs are placed in the substations to allow observation of a critical part of the supervised power system under any operation conditions (network islanding, outages of lines, generators etc.), taking into account a certain degree of redundancy to provide sufficient results also in a case of unavailability of some data (PMU outage, communication

2 SPC PMU Data Preprocessing Basic Services GUI PMU 2 Application Application 2 Power System Actions Fig. 2. Functional architecture of wide area measurement platform based on synchronized phasor measurements. actual trajectory Fig.. Setup of wide area platform based on synchronized phasor measurements. failure etc.). Measured data are sent via dedicated communication channels/links to a central unit, which is a central computational unit where the collected measurements are synchronized and sorted, yielding the snapshot of the power system state. This setup is shown in Fig.. The snapshot is then processed by the Basic Services package (BS), which is part of the central unit. Basic Services denote the set of algorithms included in all installations of the wide area platform for different applications and they are comprising the following capabilities: ability to provide needed data for any application fast execution - leaving sufficient time for running applications within the sampling interval robustness - resistance against poor quality of some input data (unavailability, out of range, synchronization problems etc.) Applications, which are attached to the output of BS, address various phenomena occurring in power systems, such as frequency instability, voltage instability etc. They predict the state of the power system and trigger appropriate actions if an incipient instability is detected. Their output as well as the output of BS are displayed to the power system operator by an ergonomic GUI. The functional structure described above is illustrated in Fig. 2. The voltage stability control method described in Sec. III is one example of a method that can be applied in such an algorithm. Other examples have been previously published [3 5]. III. COORDINATED SYSTEM PROTECTION SCHEME The coordinated system protection scheme (CSPS) is based on the model predictive method described in [6]. The principle is illustrated in Fig. 3. A system model, including the load dynamics, is used to predict the output trajectories (dotted lines) based on the current state and for several different candidate input sequences. A cost function reference trajectory Fig. 3. sample time prediction interval t* t*+tp Principle of Model Predictive Control. predicted trajectories candidate control sequences time is defined based on the deviation of each predicted trajectory from a desired trajectory (dashed line). The optimal control, in the sense that it minimizes the defined cost function, is then obtained by solving an optimization problem on-line, and applied to the system. The interval between the current time t and the prediction horizon t +t p is referred to as the prediction interval, and is chosen based on the settling time of the slowest dynamics. Usually, the prediction interval is chosen as a multiple of the sample time. The problem of selecting controls at the instant t and operating point (x,u ) is formulated as the combinatorial optimization problem min J(x,u + ) subject to u + S(u ) where u + is the optimization variable corresponding to the new control state to be determined whereas u denotes the control state presently in use, in other words before the optimization starts at time t. S(u ) is the set of available control states. The scalar function J(x,u + ) is referred to as a cost function and should evaluate the benefit of using the new control state u + such that a smaller result indicates a more desirable state. The cost function is defined as J(u +,x ) = t +t p () t ỹ T Qỹ + ũ T Rũ + P dt (2)

3 where ỹ = (ŷ(t,x,u + ) y r ) is a prediction of the output deviations from the desired trajectories, provided the new control u + is applied during the prediction interval. The control state presently in use is denoted u, and thus ũ = (u + u ) is the applied change of controls. The reference trajectory is denoted y r and the weighting matrices for output errors and control variations Q and R, respectively. The scalar penalty term P = P(u +,x ) is introduced whenever a constraint violation is predicted to occur during the prediction interval. A. Modelling A power system network model is conveniently expressed in the differential-algebraic (DA) form [7] ẋ = f dae (x,w,u) (3) 0 = g dae (x,w,u,y) (4) The dynamic state variables x are variables that appear as derivatives in differential equations and cannot change instantaneously. On the other hand, the algebraic state variables w do not appear as derivatives, and can thus change instantaneously due to changes in x or z(k). Additionally, u and y are vectors of external control and output signals that may be arbitrarily chosen. The dynamic state variables relate to generator flux, control and load dynamics; algebraic state variables relate to network voltages and currents, and the discrete states z(k) typically arise from discrete control logic such as relay controls. The full system model can be obtained on the DA form, for instance using the modelling and simulation tool Dymola and the power system library ObjectStab [8]. Furthermore, assuming that the Jacobian of (4) is nonsingular, the continuous part (3) (4) of the DA form can be reduced to the ordinary differential equation form ẋ = f(t,x,u) (5) y = g(t, x, u) (6) This reduction eliminates the algebraic state variables. Some variables in w can be solved for symbolically and the remainder must be solved for by an iterative solver when the functions f and g are evaluated. B. Trajectory Prediction Introducing the coordinate changes x = x x, u = u + u, y = y y and ẋ = ẋ(t) ẋ and linearizing the system (5) (6) at the point (x,u ) yields ẋ = f f x + u = A x + B u (7) x u y = g g x + u = C x + D u (8) x u By computing the step responses from every input to every output of the linearized model, the output trajectories can be approximately computed using a linear combination of these step responses. The approximations are then inserted into (2) and the integral is computed numerically. The linearization and the corresponding step responses need to be computed only once, prior to the start of the optimization. IV. HEURISTIC TREE SEARCH When solving the combinatorial optimization problem () by tree search, the possible control states are organized in a tree structure. Using terminology from the literature on search, each control state is represented by a node or position in the tree and the transition from one node to another node is called a move. A move in this case corresponds to the switching of a single control. The maximum number of levels in the tree is called the depth of a search and thus corresponds to the number of controls that may be switched in a single search. A leaf is a special type of node where the search is not further continued. The width of the search is the number of successors that are expanded in each node, that is, the number of moves that are explored in any intermediate position. Furthermore, the Search space of the optimization problem contains all unique control states reachable from the current state. However, in most practical cases it is not computationally feasible to explore all of the search space. The Search tree is the part of the search space actually explored by a search algorithm. In [6], the combinatorial optimization problem () is solved using tree search using a standard depth-first search method as described by [9], which is an example of a socalled uninformed search method that explores all of the search-space. An informed search method makes use of search enhancements in order to reduce the size of the search tree and consequently the computational complexity. A survey of such enhancements are given, e.g., in [0]. A short description of the enhancements used in this application are given in the following sections. A. Transposition Table It can be observed that some nodes are visited more than once during a search. For example, if there are two on-off controls and the initial state is off off there are two ways of reaching the state on on, by switching either one of the two controls first. By storing previously evaluated states in a table along with some additional information about the status of the search, a transposition table may reduce the search tree considerably without sacrificing completeness of the search. A detailed study of transposition tables and their effect on search complexity can be found in []. B. Lower Bound The cost function (2) can be decomposed into three components f = f y + f u + f P (9) related to the output deviation, control and constraint violation costs respectively. All three components are implicitly determined by the control state in use. The control cost cannot decrease as progress is made deeper into the search tree, and the control cost can therefore be used as a lower

4 bound on the cost function. Without sacrificing completeness of the search, the expansion of nodes can be stopped in every node where the control cost alone is greater than the total cost in any node encountered so far. However, lower bound cutoffs cannot be expected to reduce the search tree significantly when there are constraint violations, since the constraint violation term is designed to dominate the cost function when such violations are present. C. Move Ordering The purpose of move ordering is to make sure that the search space is explored in a good order, meaning an order where the number of lower bound cutoffs is maximized. The following two measures have been used to order the moves: Firstly, the value of the cost function after the move and secondly, the change in the cost function divided by the cost of move. Thus, the first move considered in each node is the one with the best evaluation according to the first criteria, the second is the best one according to the second criteria and the third is the second best according to the first criteria and so on. D. Iterative Broadening In an iterative broadening search [2], a number of successive searches are made. Ranked according to the move ordering criteria, only the best move in each intermediate node is considered in the first iteration. In the second iteration also the second best move is considered and so on. The basic idea is to quickly find one reasonably good solution by venturing deep into the tree and then use remaining time available to improve this solution. Since the size of the search tree depends exponentially on the breadth, the effort spent on the search is dominated by the last iteration. In fact, the information collected in these intermediate searches can often enable additional lower bound cutoffs, and an iterative broadening search to a certain depth and width may therefore be computationally cheaper than a single search. Note, however, that the iterative broadening sacrifices completeness of the search. Because in most cases only a fraction of the game tree search space is explored, the search is likely to miss the optimal solution unless good move ordering criteria are used. There is a potential risk that the iterative broadening finds a local minimum or a plateau of the cost function [9]. No such behaviour has been detected in the work on this application, but can in such cases be circumvented by repeating the search using different initial control vectors. V. SPS BASED ON LOCAL MEASUREMENTS The simulations have also been carried out with a rulebased SPS using only local voltage measurements. Load shedding is carried out using the scheme described in [3] using an assumed lowest normal voltage of p.u., i.e., 5 % of the bus load is shed at 0.9 p.u. with 3.5 s time delay, another 5 % is shed at 0.92 p.u. with 3 s delay and yet another 5 % is shed at 0.92 p.u. with 8 s delay. N6 N406 N62 N4063 N4062 N2032 N4072 N407 N63 N02 N404 N4 N40 N203 N043 N04 N022 N402 N403 N042 N4044 N044 N045 N4022 Fig. 4. The Nordic test system from [5]. N405 N5 N4043 N43 N4032 N4045 N402 N0 N02 N4047 N47 North The tap changers are controlled by constant-time relays corresponding to model D in [4] with a delay time Td0 of 40 s. Tap changers are blocked if the primary side voltage stays below 0.93 p.u for 0 s or more. All loads and tap changers are equipped with these undervoltage and tap locking relays in the schemes LPSP and LSPS2. VI. THE NORDIC TEST SYSTEM The model predictive method will be demonstrated using the Nordic 32 test system originally described in [5], except for that generators connected to the same bus and parallel branches have been aggregated and generator saturation is neglected. All generators are equipped with field current limiters, and the thermal power generators also have armature current limiters. Both types of limiters are of integral-type and allow temporary overcurrents for the first 20 s. A detailed description of the limiters can be found in [5]. The armature current constraints are set at.05 p.u., whereas the field current constraints are set 5 % lower than in the report in order to compensate for neglecting saturation. The initial state is given by the load flow case at buses N042, N043, N4042, N4047, N405, N4062, N4063 N46 N42 N4046 N03 N04 N4042

5 TABLE I OVERVIEW OF THE SCENARIOS. Case # Tripped Component(s) Without SPS Generator 4062 Collapse at 50 s 2 Line Stable, no violation 3 Line Collapse at 340 s 4 Line Collapse at 45 s 5 Line , Generator 02 Collapse at 75 s 6 Line Collapse at 20 s 7 Generator 404 Stable, no violation 8 Line Collapse at 64 s called lf28 in the CIGRE report. The following components are considered as controls: Transformers with on-load tap changers (6 steps of.67 %) 22 Load shedding points (3 steps of 5 % of nominal load) 9 Generator voltage setpoints (8 steps of 2 %) Constraints are imposed on: Field currents of all generators, set to the same value as the internal generator field current limiter. Armature current of the thermal power generators, set to the same value as the internal generator field current limiter. All bus voltages should be kept above 0.92 p.u. Voltages below tap changing transformers are considered controlled, and should be kept close to p.u. unless transmission voltage constraints are violated. In cases where not all constraints can be satisfied, armature and field current constraints have priority over voltage constraints. VII. SCENARIOS AND SIMULATION RESULTS The pre-disturbance state lf28 has a higher load than would be allowed by normal reliability criteria. In fact, many single contingencies will lead to voltage collapse. Simulation of the scenarios listed in Tab. I is used to evaluate the different schemes. The table also shows the system response without an SPS. In most cases, voltage collapse occurs unless emergency controls are initiated. The simulations have been carried out with the local scheme in two versions; firstly, the LSPS where the generators control their respective terminal voltage and the LSPS2 where they use full line-drop compensation as described in [6], in order to make better use of their reactive capability. The coordinated scheme is used in two versions, the CSPS in which the generator controls have been disabled and the CSPS2 which has access to all controls. Thus, the results with CSPS should be compared to those with LSPS, and the results with CSPS2 to those with LSPS2. A. Case - Tripping of Generator at Bus N4062 In the pre-contingency situation, the generator at bus N4062 supplies 530 MW active and absorbs 8 Mvar reactive power. The tripping of the unit causes an active power deficiency in the south which is compensated mainly by the hydro generators in the north and consequently increases transfer through the central region. This leads to increased losses and low voltages in the south and central regions. In response, the remaining generators in the south almost instantaneously increase their reactive power output until after 20 s, the armature current limiters of generators 043 and 4042 are activated. Voltage support is then lost in the areas close to these buses. Load recovery and tap changer dynamics will then further depress voltages in the south until system collapse occurs, unless emergency actions are taken. Fig. 5 shows the terminal voltage and armature current of the generator at bus N4042. With both local schemes, the tap changing transformers in the southern region act to restore their controlled voltages after about 50 seconds. This depresses transmission voltages further and some tap changers are eventually blocked. Voltage decline then continues due to load dynamics and the first load shedding is ordered at about 00 s with the LSPS. Once enough undervoltage relays have been activated, the voltage instability phase ceases and voltages slowly recover close to their pre-disturbance values. Since the terminal voltages of generators with reactive capability to spare are kept higher with the LSPS2, reactive capability is better utilized, and therefore load shedding action is initiated. Also, less load shedding is required before the voltages become stable. While the voltage trajectories of the CSPS and CSPS2 look similar, the two schemes use different means of reaching a stable state. With the CSPS, load shedding at buses N43 and N044 combined with backstepping of transformer N4043 (which provides a temporary load relief at bus N43) relax the armature current constraints of the limited generators. The CSPS2 decreases the voltage setpoints of the generators at buses N043 and N4042, taking these out of their armature current limits. Generators at N2032, N402, N404 and N4047 have reactive capability to spare and are used to support the weak central-south region from the outside. Subsequently, the tap changers are used to restore the controlled voltages at nodes where this can be done without violating transmission voltage constraints. B. Summary of Simulation Results Tables II III list the system responses with the two versions of the local and coordinated SPS respectively. The local schemes stabilize the system in all cases except case 4, in which the collapse is due to instability of the shortterm dynamics, not modelled in the system model used by the coordinated schemes. Nevertheless, the coordinated schemes keep a more even voltage profile than the local schemes and thereby keep the system within the (shortterm) stability region even though the short-term dynamics are not explicitly modelled in the system model used by these schemes. Surprisingly, in case 3 the LSPS sheds less load than the CSPS. The voltage at bus N4042 is allowed to decrease below the voltage constraint of 0.92 p.u. with the local schemes since there is no undervoltage load shedding relay at this bus. On the other hand, the CSPS has the option of shedding load at neighbouring buses in

6 TABLE II SIMULATION RESULTS WITH THE LOCAL SPS. Case # LSPS LSPS2 Stable, 200+j69 MVA shed Stable, 20+j36 MVA shed 2 Stable, no load shed Stable, no load shed 3 Stable, 00+j33 MVA shed Stable, no load shed 4 Collapse at 44 s, 590+j90 MVA shed Collapse at 220 s, j68.7 MVA shed 5 Stable, 224+j68 MVA shed Stable, 208+j63 MVA shed 6 Stable, 60+j20 MVA shed Stable, 60+j20 MVA shed 7 Stable, no load shed Stable, no load shed 8 Stable, 424+j39 MVA shed Stable, 377+j22 MVA shed TABLE III SIMULATION RESULTS WITH THE COORDINATED SPS. Case # CSPS CSPS2 Stable, 60+j49 MVA shed Stable, no load shed 2 Stable, no load shed Stable, no load shed 3 Stable, 5+j37 MVA shed Stable, no load shed 4 Stable, 570+j85 MVA shed Stable, 500+j6 MVA shed 5 Stable, 60+j46 MVA shed Stable, 75+j22 MVA shed 6 Stable, no load shed Stable, no load shed 7 Stable, no load shed Stable, no load shed 8 Stable, 494+j55 MVA shed Stable, 270+j75 MVA shed Generator 4042 Armature Current (p.u.) LSPS LSPS2 CSPS CSPS Time (seconds).05 LSPS LSPS2 order to avoid this violation. Therefore, the amount of load shedding can be higher with the coordinated scheme. Case 6 is another interesting case, which illustrates misoperation of the local schemes both LSPS and LSPS2 sheds load. The simulation of the same scenario with CSPS and CSPS2 shows that load shedding is not necessary to avoid collapse. Fig. 6 shows a comparison of the amount of load shedding executed by the different schemes. The amount of load shedding by the LSPS2 compared to the LSPS is about 7% lower, since the generator reactive capabilities are better utilized. Although the simulations results are slightly biased in favour of LSPS because of less stringent handling of voltage constraints than with the CSPS, the benefits of coordinated control when only load shedding is allowed seems to be small. The required amount of load shedding is only 6 % smaller with the CSPS than with the LSPS. This reduction can be ascribed to better control of the tap changers and is presumably not due to the coordination of load shedding in different geographical locations. However, with the CSPS2, which has generator setpoint voltages included as control variables, remote generators can be used to support weak regions as an alternative to load shedding and the load shedding reduction is about 47 % compared to the LSPS and 35 % compared to the LSPS2. Also, because of better management of voltage constraints by both coordinated schemes, they are more likely to keep the trajectories of the system in a region where the shortterm dynamics are stable. VIII. DISCUSSION As the method in this paper aims at preserving longterm voltage stability related to the the dynamics present on the load side, a relatively long sample time of 30 s has been used. Using a Matlab implementation on a 900 MHz Pentium II processor, the search is successively Bus 4042 Voltage (p.u.) CSPS CSPS Time (seconds) Fig. 5. Voltage at bus N4042 and armature current of generator 4042 following tripping of the generator at bus N4062 with the different protection schemes. broadened allowing 30 s computation time in each control determination. The prediction horizon is 20 s and the maximum search depth is 5. The coordinated SPS has been tuned according to the tuning procedure in [6], aiming to ensure that load shedding is not used unless there will be a constraint violation during the prediction interval that cannot be relaxed by any of the other controls. The tuning procedure yields r i = for tap changer and voltage setpoint controls and r i = 000 for load shedding controls. The penalties for constraint violations and singularity-induced bifurcations in the penalty term P is 000 and 0000, respectively. The main task of the centralized scheme is then to steer the system trajectories away from operating points where the local scheme or other protection systems such as thermal line protection are activated. Since the centralized scheme offers better coordination, this can be achieved with less drastic emergency controls than would be executed by a standalone local scheme. To provide protection against short-term collapse or when communication links or the central processing unit malfunctions, some type of SPS based only on local measurements is necessary as a backup.

7 Load Shedding LSPS CSPS LSPS2 CSPS2 Pshed (MW) Qshed (Mvar) Fig. 6. Average amount of load shedding over the eight scenarios. Nominal precontingency load is 0940 MW and 3358 Mvar. Another issue to consider is the scalability of the method. The main factor that influences the computational complexity is the number of controls that may change in one search, which directly translates into the depth of the search tree. Also the total number of controls to choose from has some influence since it is normally related to the branching factor of the tree, however this relation is largely offset by the iterative broadening technique that is applied in this paper. Since efficient parallel implementations of search are available [8], a parallel computing architecture can provide the necessary computational capacity. The cost increase of the central processing unit will be small in comparison to the cost of the required communication and measurement equipment as controls in new substations are integrated. IX. CONCLUSIONS A coordinated system protection scheme for voltage control in the emergency state, capable of coordinating dissimilar and discrete controls such as generator, tap changer and load shedding controls using heuristic tree search, has been presented. Eight scenarios, some of which leads to voltage collapse unless emergency control measures are taken, has been used to evaluate system protection schemes based on local as well centralized measurements. Simulation results indicate that load shedding based on local measurements is near optimal when load shedding is the only emergency control considered. Another conclusion is that the reactive capability of generators is somewhat better utilized when generators use line-drop compensation. On the other hand, the results indicate that the protection scheme based on centralized measurements can use remote generators to support a weak region as an alternative to load shedding. The reduction in the amount of load shedding required is about 35 % compared to the best local strategy. Also, the centralized scheme maintains a more even voltage profile than the local schemes and may therefore avoid short-term voltage instability that occurs with the local scheme, even though the short-term dynamics are not explicitly modelled. REFERENCES [] Protection against voltage collapse, CIGRE Task Force 34.08, Tech. Rep., 998. [2] CIGRE, System protection schemes in power networks, CIGRE Task Force , Tech. Rep., [3] M. Larsson and C. Rehtanz, Predictive frequency stability control based on wide-area phasor measurements, in Power Engineering Society Summer Meeting, vol.. IEEE, 200, pp [4] C. Rehtanz and J. Bertsch, A new wide area protection system, in Bulk Power System Dynamics and Control V, August 26-3, 200, Onomichi, Japan, 200. [5] M. Larsson, C. Rehtanz, and J. Bertsch, Real-time voltage stability assessment of transmission corridors, in IFAC Power Plants & Power Systems Control, [6] M. Larsson, D. J. Hill, and G. Olsson, Emergency voltage control using search and predictive control, International Journal of Power and Energy Systems., vol. 24, no. 2, pp. 2 30, [7] T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, ser. Power Electronics and Power Systems Series. Kluwer Academic Publishers, 998. [8] M. Larsson, Objectstab an educational tool for power system stability studies, IEEE Transactions on Power Systems, vol. 9, no., pp , [9] S. J. Russell and P. Norvig, Artificial Intelligence - A Modern Approach. Prentice-Hall, 995. [0] A. Junghanns, Pushing the limits: New developments in singleagent search, Ph.D. dissertation, Department of Computing Science, University of Alberta, Canada, 999. [] D. Breuker, Memory versus search in games, Ph.D. dissertation, Department of Computer Science, Maastricht University, The Netherlands, 998. [2] M. L. Ginsberg and W. D. Harvey, Iterative broadening, Artificial Intelligence, vol. 55, no. 2 3, pp , June 992. [3] K. T. Vu, C.-C. Liu, C. W. Taylor, and K. M. Jimma, Voltage instability: mechanisms and control strategies, Proceedings of the IEEE, vol. 83, no., pp , 995. [4] P. W. Sauer and M. A. Pai, A comparison of discrete vs. continuous dynamic models of tap-changing-under-load transformers, in Proceedings Bulk Power System Voltage Phenomena - III : Voltage Stability, Security and Control, Davos, Switzerland, 994. [5] Long term dynamics phase II, CIGRE Task Force , Tech. Rep., 995. [6] C. W. Taylor, Line drop compensation, high side voltage control, secondary voltage control- why not control a generator like a static var compensator, in Power Engineering Society Summer Meeting, vol.. IEEE, 2000, pp [7] J. D. Glover and M. Sarma, Power System Analysis and Design. PWS Publishing Company, 994. [8] F.-H. Hsu, IBM s Deep Blue chess grandmaster chips, IEEE Micro, vol. 9, no. 2, pp. 70 8, March/April 999.