A New Rigidity Method for Stabilizing Transient

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A New Rigidity Method for Stabilizing Transient Processes By Coordinated Control Actions Yuri V. Makarov, Senior Member, IEEE [email protected] Southern Company Services, Inc., Birmingham, Alabama, USA Abstract – This paper introduces a new concept of rigidity and its first example applications for stabilizing transient processes in power systems. By definition, a rigid system maintains a close-to-original state despite passing disturbances or smoothly transforms into a new state if the disturbance is permanent; it also responds to faults and disturbances with a smooth monotonous or minimum oscillatory behavior. Rigidity implies a stable behavior pattern of the controlled system for at least a specified group of finite disturbances over certain pre-set observation period. But stable systems are not necessarily rigid systems because they may be poorly damped and/or exhibit excessively oscillatory responses. In contrast to stability, rigidity has a simple and convenient mathematical description. It provides an objective function that can be optimized along the transient trajectory to obtain effective stabilizing control impacts. Rigidity can be pursued by both preventive and on-line control actions. Several different controls can be robustly selected in coordination for a variety of operating conditions, controlled variables, and uncertainties. The proposed rigidity criterion can be easily computed by using the output files of any standard transient simulator. Theoretical suggestions along with the first numerical tests are presented for the judgment of the power system community aiming further discussions, developments, and testing.

I. INTRODUCTION Power system transients are described by large systems of differential and algebraic equations (DAE). Nonlinearity, discreteness, uncertainty, and randomness are immanent features of these processes. They may include such nonlinear phenomena as saddle-node, Hopf, transcritical, and singularity induced bifurcations, limit cycles and chaos; various constraints such as limited reactive power support from generators and voltage control devices; discrete events due to disturbances, tap changer and protection system operations, uncertainties due to unknown a priori initial conditions and load characteristics, hardly predictable faults or incorrect control actions; complicated dynamic interactions, say, between tap changers and dynamic loads, etc. [1, 2]. The set of on-line stabilizing control actions usually includes high-speed fault clearing, series capacitor compensation and FACTS control, blocking of tap changers, reactor and shunt capacitor switching, SVC control, dynamic breaking, steam turbine fast-valving, generator and line tripping, system separation, load shedding, etc. [1] The on-line control impacts are usually selected in advance. This off-line procedure involves determining the

proper magnitudes, tunings, timing, sequence, selectivity, coordination, and places of application of stabilizing impacts. An example can be given here regarding the process of selecting the size and timing for the dynamic braking resistors or tuning the power system stabilizers gains. Besides the on-line time-domain control problem, there is a need in developing a more preventive type of adjustments used on the stages of system expansion planning (e.g. building new lines, placing of stabilizing devices, etc.) or planning of operations (e.g. stability-constrained active and reactive power dispatch, unit commitment, etc.) Another problem is to determine a criterion of power system performance (directly related to damping or attenuation) for these complicated nonlinear transient processes. Without this criterion it is difficult to compare variants and assess the effectiveness of various controls, set a control objective for various stability enhancement algorithms, for example, the ones based on the artificial neural networks [4]. The modern stability theory has not yet provided such a general criterion to judge whether such complicated and oversized systems are stable for a variety of initial conditions, disturbances and uncertainties, what are the optimal controls, when, in what extent, and in which order they must be applied. Despite some recent theoretical and practical successes (many new approaches have been first developed by Russia’s power analysts), for example, development of the energy function method, the main practical approaches are still being reduced to analyzing simplified system models or conducting multiple time domain simulations. Even so, it is often hard to find adequate control actions from these experiments, determine their sequence, and appropriate time of application. The energy function criterion could be a very effective comprehensive stability criterion because it is based on Lyapunov’s functions [5,6]. If a general form of the Lyapunov’s function had been found for real systems, it would have efficiently and conclusively solved numerous problems related to transient stability [6]. Unfortunately, only some partial forms of the energy functions have been found for power systems by the moment. They are based on simplified power system models and have many limitations that are not always acceptable in practical applications. For example, they cannot cover substantial time intervals, discrete events, on-line controls, and constraints

applied parameter limits yet. This means that the search for adequate stability criteria must be continued. The Lyapunov’s exponent [5,6], lim t →∞ t −1 ln f (t ) , which determines the growth of solutions of the linearized system ∆x& = f&x ∆x and is used in the Lyapunov’s stability criterion for the original nonlinear system x& = f ( x, t ) , would serve as a possible measure of the degree of instability. Nevertheless the computational techniques for assessing this criterion are far from being clear. In [7], a minimization technique is successfully used to minimize the system eigenvalues real parts along the integral trajectory, and by doing so to arrest voltage collapse. The technique was able to predict accurately the appropriate timing for transformers tap locking as well as the other discrete stabilizing actions. Unfortunately, this technique has not been completely explained theoretically, and the extent of its applicability is not clearly seen. Moreover, eigenvalues and their sensitivities used in [7] require significant computational efforts for real size systems. In [8] and other related works, a meaningful achievement has been made in developing a new stability criteria based on the sensitivity analysis of solutions x(t , p) of the differential system x& = f ( x, p, t ) with respect to system parameters p. The peak magnitudes of the sensitivity trajectories, S (t ) = x& p , are suggested to be used as a measure of system stability. Actually, the lesser these peaks are, the smoother the integral trajectory is, and, by our definition, the system is more rigid. If S (t ) ≡ 0 , the system is absolutely rigid, that is it does not feel the disturbances at all. In unstable systems, S (t ) tends to infinity. A beauty of this approach is that instead of the existing stability criteria with their complicated mathematical formulation, computational burden, and nevertheless limited applicability to the real power system problems, a refreshing idea is elaborated that directly links the stability measure with the type of behavior (the shape of the integral trajectories) that we expect from a “good “ (i.e. rigid) system. But there are also some serious difficulties associated with this approach. First, one have to analyze so many different state variables x, system parameters p, and time moments t, that makes these multiple peaks difficult to handle and hard to compare if any changes are inflicted to the system. Second, these peaks cannot be adequately determined for an unstable system, so that the measure or instability is not provided. In [9], a new rigidity-based technique has been first suggested. That paper was focused on the on-line type of the rigidity control. In [10], a similar technique was proposed for a voltage collapse prevention algorithm. This paper presents a generalization of the newly proposed rigidity concept for both on-line and preventive stabilization. New developments suggested in the paper also include handling multiple operating conditions and uncertainties.

II. THE IDEA BEHIND THE METHOD Suppose that x( p,0) is a vector of m state variables at the initial operating point t = 0 (before a disturbance occurs). Vector p denotes control parameters that can be varied to influence the transient process caused by a disturbance. If the system is asymptotically stable, the postdisturbance steady-state vector of state variables is referred to as x( p, ∞ ) = lim t →∞ x( p, t ) . For an asymptotically stable system, a process when the minimum integral deviation from the final steady-state point can be provided by using the following heuristic objective:

ϕ I ( p) = ∫ µ [x( p,τ ) − x( p, ∞)] dτ → min T

2

(1)

p (t )

0

where µ is a diagonal matrix of weight coefficients that normalizes state variables of different physical dimension and range of variation, and emphasizes the most important variables (say, rotor angles). [0, T ] is a sufficiently big interval. Obviously, optimization task (1) requires that the steady state point exists and is a priori known. Therefore, (1) cannot be used for unstable and non-asymptotically stable systems. It may happen that x( p, ∞) depends upon the integral trajectory itself, and is effected by controls or events in a way that is not seen in the beginning. Objective (1) forces the system to react the new state as fast as possible. That is it leads to effective decrease of the system inertia. For the electromechanical motion, a decrease in inertia leads to increasingly oscillatory behavior of the system. Leaving formulation (1) for future studies, let us consider the following objective:

ϕ II ( p) = ∫ µ [x( p,τ ) − x( p,0)] dτ → min T

2

(2)

p (t )

0

This task aims to preserve the original state x( p,0) . A similar result can be reached by pursuing the following objective: 2 ϕ III ( p) = ∫ µ [x& ( p,τ )] dτ → min T

0

(3)

p (t )

The last objective minimizes the integral derivative of state variables x. Objectives (2) and (3) lead to increasing system inertia. A compromise between the different objectives (preserving the initial state plus minimum oscillations) can be reached by minimizing the following composite function: ϕ ( p ) = Aϕ II ( p) + Bϕ III ( p ) → min (4) p (t )

Weight coefficients A and B are used to achieve better flexibility in finding a solution. Objectives (1) and (3) can be also combined in one function. Composite problem (4) can be solved locally or globally. In local problem, vector p includes the controls available locally, and vector of state variables x consists of a limited number of local parameters. In global problem, all states variables are included. III. PREVENTIVE RIGIDITY CONTROL The purpose of the preventive rigidity control task is to select a time-domain signal p(t ) that would stabilize the

system over a variety of operating conditions, disturbances, and uncertainties. For example, p(t ) can be the size and switching times for of a dynamic braking resistor, or FACTS control sequence. In more simplistic cases, p can be just a system change or constant tuning, for instance, improved PSS gains, power dispatch, or even a transmission system improvement. A) Preventive Control Rigidity Function The preventive rigidity objective function will have the following general form: n 2  T ϕ ( p, u ) = ∑ C j  A ∫ µ [ x j ( p, u, τ ) − x *j ] dτ + j =1  0 (5) T 2 + B ∫ µ x& j ( p, u, τ ) dτ → max min p (t ) u 0  In (5), a minimax problem is stated. Along with the minimization of ϕ by changing control parameters p, the same objective is maximized by varying uncertain parameters u. The last feature provides a robust solution by addressing for the worst combination of uncertain parameters. An additional summation over j is done for n different operating conditions/fault locations covered by the study. The weight coefficients C j reflect the relative importance of an operating point/fault location j. For instance, the probabilities of contingencies and/or faults can be used to determine C j .Values of x *j can be assigned as x j ( p, u ,0) or x j ( p, u, ∞) . B) Computing Rigidity Functions Results of transient simulations can be obtained by conducting computer or field experiments for different operating conditions and fault locations j. The data file consists of the time instants t jk with t j 0 = 0 , t jK = T , and the corresponding values of state variables x j ( p, u, t k ) . The rigidity objective function has the following simplified form: ϕ ( p, u ) = n 2  K −1 ∑ C j  A ∑ µ [ x j ( p, u, t k +1 ) − x j ( p, u, t k )] (t k +1 − t k ) j =1  k =0 K −1 2 1 +B∑ µ [ x j ( p, u, t k +1 ) − x j ( p, u, t k )]  k = 0 (t k +1 − t k )  (6)

C) Sensitivity of Rigidity Function Sensitivity of (6) with respect to p and u can be computed by increasing some parameters and repeating the simulation for the new values of p and u. The sensitivity expressions are as follow: ∂ϕ ≈ ∆p −1 [ϕ ( p1 , K pl −1 , pl + ∆p, pl +1 , K p L , u ) − ∂pl − ϕ ( p1 , K pl −1 , pl , p l +1 , K p L , u )] (7) ∂ϕ −1 ≈ ∆u [ϕ ( p, u1 , K u l −1 , u l + ∆u, u l +1 , K u M ) − ∂u l − ϕ ( p, u1 , K u l −1 , u l , u l +1 , K u M )] D) Optimization Procedure

The optimization procedure required to find out the optimal control may be based, for example, on the steepest descent method. It consists in subsequent variation of the control and uncertain parameters along the gradient vector t

 ∂F ∂F ∂F ∂F ∂F ∂F  G = − ,− ,L ,− , , ,L,  (8) ∂ p ∂ p ∂ p ∂ u ∂ u ∂ uM  1 2 L 1 2  so that  p k +1   p k  (9)  u k +1  =  u k  + ρ k Gk where ρ k is the step length parameter selected at kth iteration. The step length may be selected using different approaches that are not being discussed in this paper. IV. ONLINE RIGIDITY CONTROL The online rigidity control aims stabilizing the system by applying control impacts at certain points of a transient process. Unlike preventive rigidity control, the online rigidity is not pursuing a common control sequence for a variety of transient processes; on the contrary, its aim is to address a specific system transient condition at a specific instant of time. The model behind the online rigidity scheme includes sensors that measure instantaneous values of parameters, a computing device that determines the realtime control signals based on an imbedded power system model, and executive devices that apply controls in real time. A digital power system stabilizer (PSS) can serve as an example of such an arrangement. A) Definition of Online Rigidity In practical terms, the time axis can be partitioned into small control intervals ∆t k +1 = t k +1 − t k , where problem (1) can be reduced to a local rigidity problem, which is similar but not identical to the global one: ϕ k +1 ( p, u ) = n 2  ∑ C j  A µ[ x j ( p, u, t k +1 ) − x j ( p, u, t k )] (t k +1 − t k ) +  j =1 2 1 +B µ[ x j ( p, u, t k +1 ) − x j ( p, u, t k )]  (t k +1 − t k )  (10) The set of constraints associated with the above minimax problem can include restrictions posed on p and u, algebraic equations from DAE, and any other desirable options. The constraints as well as the weights µ can be changed over time. For instance, at final stages of a transient process, steady-state operating restrictions on voltage magnitudes, currents, etc. may be applied. The differential part of the power system DAE consists of the first and second order equations: &x&1 = f1 (x, y, z ) (11) x& 2 = f 2 ( x, y, z ) (12) As ∆t is small, a linear approximation of x can be used in Fj for the first order states x 2 : ∆x 2 (t j + ∆t , y, z ) = x& 2 ∆t = f 2 ( x, y, z )∆t (13)

T m, X d, D G

160

Xc

Xe

V i, 0

140

Fig. 1 A simple power system

For the second order states x1 (rotor angles), a quadratic approximation of ∆x must be considered: ∆x1 (t j + ∆t , y, z ) = x&1∆t + 12 &x&1∆t 2 = (14) = x1s ∆t + 12 f1 (t j , y, z )∆t 2 where x1s = x&1 represents the speeds of x1 - additional variables used in the normal form of DAE. They are known at the beginning of and are taken fixed within each interval j as they are not explicitly affected by y and z. Unlike ∆x2, the linear approximation of ∆x1 cannot be used in Fj because ∆x1 becomes here not sensitive to y and z. The quadratic approximation allows to get an objective function which is explicitly dependent on optimized variables y and z. V. POWER SYSTEM TESTS Numerical tests were conducted for a power system model with FACTS control used in [4] to illustrate a neural network technique for stabilizing power system transient processes. The model is presented in Fig.1. Differential equations for this model are the following:  Vt Vi 1  dω = sin δ  Tm − D(ω − 1) − dt M  Xd + Xe − Xc  (15) dδ = ω b (ω − 1) dt System (15) does not have an algebraic part and first order states x2. The state variables are δ (rotor angle or x1) and ω (rotor speed or x1s ). The constants are the mechanical input at generator Tm = 0.3383, damping coefficient D = 2.0, voltages Vt = 1.0 and Vi = 0.9, impedance Xd = 1.5 and Xe = 0.7, and ω b = 120π. Initial conditions are ω 0 = 1.01 and δ 0 = 75 deg. The problem to be solved at each interval ∆t = 0.05 sec has the following expression: 160

Delta, grad, and Xc in % of Xe

140 120 100 80 60 40 20 0 -20 0

0.5

1

1.5

2

Time, sec

Fig. 2 Solutions of (7) with and without FACTS control

2.5

Delta, grad, and Xc, % of Xe

V t, δ

120 100 80 60 40 20 0 -20 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time, sec

Fig. 3 Stabilizing the system around the unstable equilibrium point by means of persistent control actions

min(∆δ ∆t )2 XC

where

(∆δ

∆t ) 2 =

(δ& + 12 ω& ∆t )2 =

(16)

 Vt Vi sin δ   ∆t  ω b (ω − 1) + Tm − D(ω − 1) −  2  X d + X e − X c    subject to the constraint 0.35 X e ≤ X c ≤ 0.7 X e (17) Fig. 2 displays solutions of (15) obtained with and without FACTS control impacts determined by (16). The rotor angle δ is plotted against time. The dotted curve corresponds to the case without any control. The step line represents the control signal Xc - the reactance of a regulated series capacitor placed on the transmission line that models a FACS device. By applying a control signal selected using (16), the system performance becomes significantly better see the solid curve in Fig. 2. The initial growth of δ is explained by nonzero initial conditions for the rotor speed ω at t = 0. Fig. 3 presents a more impressive result. Negative damping D < 0 was artificially introduced into the system to make it unstable - see the dotted curve in Fig. 3. The proposed rigidity control makes system stable (solid curve) by applying persistent control impacts (step line in the Figure). Moreover, the control acts in such a way that the system is stabilized at a point where its state matrix has eigenvalues with positive real parts, that is at an unstable equilibrium point. Fig. 4 and 5 represent a case of coordinated control. All the system conditions correspond to Fig. 2 except that an additional control with varied Tm is introduced. The value of Tm, which is set to vary from 80% to 100% of its nominal value (the upper step line) is an idealized model of either dynamic braking or steam turbine fast-valving. It is seen that the control Xc (the lower step line) acts in a different mode providing (along with Tm) much more effective stabilization within 1.5 sec. Some excessive actions

are observed after the initial 1.5 sec control interval (Fig. 4). If control actions are prohibited beyond the interval, the system performance remains well enough despite some small oscillations (Fig. 5).

Delta, grad, Xc in % of Xe, and Tm, %

160 140 120 100 80 60 40 20 0 -20 0

0.5

1

1.5

2

2.5

Time, sec

Fig. 4 Stabilizing the system by means of continuous coordinated control actions

Delta, grad, Xc in % of Xe, and Tm, %

160 140 120 100 80 60 40 20 0 -20 0

0.5

1

1.5

2

BIBLIOGRAPHY [1] P. Kundur, Power System Stability and Control, McGrawHill, 1995. [2] D. J. Hill, Editor, Local Bifurcation in Power Systems: Theory, Computation and Application, Special Issue on Nonlinear Phenomena in Power Systems, Proceedings of the IEEE, Vol. 83, No.11, November 1995. [3] A. A. Fouad and V. Vittal, Power System Transient Analysis Using Transient Energy Function Method, Prentice Hall, 1991. [4] R. R. Zakrzewski, R. R. Mohler, and W. J. Kolodziej, “Hierarchical Intelligent Control with Flexible AC Transmission Systems Application”, Control Eng. Practice, Vol. 2, No. 6, pp. 979-987, 1994. [5] A.M. Lyapunov, A General Problem of Stability of Motion, Gostechizdat, Moscow-Leningrad, 1950 (in Russian), or A. M. Lyapunov, Stability of Motion, English translation, Academic Press, Inc., 1967. [6] I. G. Malkin, Theory of Stability of Motion, Gostehizdat, Moscow-Leningrad, 1952 (in Russian). [7] Y.V. Makarov, D. H. 3RSRYLüDQG'-+LOO ³6WDELOL]DWLRQ of Transient Processes in Power Systems by Eigenvalue Shift Approach”, IEEE Trans. on Power Systems, Vol. 13, No. 2, May 1998, pp. 382 – 388. [8] M. J. Laufenberg and M. A. Pai, “A New Approach to Dynamic Security Assessment Using Trajectory Sensitivities”, IEEE Trans. on Power Systems, Vol. 13, No. 3, August 1998, pp. 953 – 958. [9] Y. V. Makarov, “Rigidity concept for stabilizing transient processes in power systems”, Paper BPT99-343-12, Proc. IEEE Budapest Power Tech’99 Conference, August 29September 2, 1999. [10] E. De Tuglie, M. Dicorato, M. La Scala, and P. Scarpellini, “Online Dynamic Security Control in a Large-Scale Power System”, Paper BPT99-147-12, Proc. IEEE Budapest Power Tech’99 Conference, August 29- September 2, 1999.

2.5

Time, sec

Fig. 5 Stabilizing the system by means of coordinated control actions applied within the first 1.5 sec only

V. CONCLUSION In the paper, a new rigidity approach to stabilize transient processes in power systems is proposed. The rigidity concept gives a practical criterion to measure power system transient performance and stability level based on the shape of real transient processes. It helps to effectively improve them by optimal coordinated control impacts. Numerical tests show the effectiveness of the new rigidity concept. The system performance (damping) was improved drastically in all cases studied. By applying persistent control actions, the proposed approach was even able to stabilize a system that has no a stable equilibrium point. Further efforts are required in connection of the both theoretical and practical aspects of the method.

Yuri V. Makarov (M’97, SM’99) received his M. Sc. degree in Computers and Ph.D. degree in Electrical Engineering from the Leningrad Polytechnic Institute (now St. Petersburg State Technical University), Russia, in 1979 and 1984 respectively. From 1990 to 1997 he worked as an Associate Professor in the Department of Electrical Power Systems and Networks at the same University. From 1993 to 1998, he conducted his research at the University of Newcastle and University of Sydney in Australia, and at Howard University, Washington, D.C., USA. Now he works for the Transmission Planning Department, Southern Company Services, Inc., Birmingham, Alabama, USA, as a Senior Engineer. His current interests and activities include various theoretical and applied aspects of power system analysis, planning and control.