Power System Transient Stability Design using Reachability based Stability-Region Computation

Transcription

1 1 Power System Transient Stability Design using Reachability based Stability-Region Computation Licheng Jin, student member, IEEE, Haifeng Liu, student member, IEEE, Ratnesh Kumar, Senior member, IEEE, James D.McCalley, Fellow, IEEE, Nicola Elia, Member, IEEE, Venkataramana Ajjarapu, Senior member, IEEE Abstract This paper presents a reachability based method to compute the stability region of a stable equilibrium point and uses it to design controls for transient stability of power systems. First, a Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) is obtained for the propagation of the backward reachability of a nonlinear system. This computation when used to obtain the backward reachable of a stable equilibrium point yields its stability region. Using the stability regions of various discrete controls (also called modes) transient stability design is performed. For example the effectiveness of a control can be verified by checking whether a post-fault initial state is in the stability region of the system with that control switched on. We illustrate our method by applying it to a single-machine infinite-bus system equipped with series and shunt capacitive compensation. Index Terms stability region, backward reachable, level method, transient stability design. I. INTRODUCTION Power system transient stability is related to the ability to maintain synchronism when subjected to a severe disturbance, such as a short circuit on the bus. The resulting system response involves large excursions of generator rotor angles and is influenced by the nonlinear power-angle relationship. Transient stability assessment essentially determines whether the post-fault operating state can reach an acceptable steady-state operating point or not. The conventional method to determine transient stability is to integrate the system equations to obtain a time domain solution of the system variables, for given system operating points and contingencies. The alternative method is to determine stability directly [1]. We can determine the stability of a post-fault power system by checking whether the fault-on trajectory just after the fault is cleared lies inside the stability region of a desired stable equilibrium point of the post-fault system. For a general nonlinear autonomous system, the stability region is defined as the of all points from which the trajectories start and eventually converge to the stable equilibrium point (SEP) as time approaches infinity [2]. In the last three decades numerous efforts have been undertaken to determine the stability region with the goal of power system This work was supported by funding from the National Science Foundation and from the Office of Naval Research under the Electric Power Networks Efficiency and Security (EPNES) program, award ECS and partly supported by National Science Foundation under the grants NSF- ECS-21827, NSF-ECS , NSF-EPNES , and NSF-ECS , and a DoD-EPSCoR grant through the Office of Naval Research under the grant N The authors are with the Iowa state University, Ames, IA 1,USA ( lcjin@iastate.edu,hfliu@iastate.edu,rkumar@iastate.edu, jdm@iastate.edu,nelia@iastate.edu,vajjarap@iastate.edu). transient stability analysis. The studies of [3] [6] provided the theoretical foundations for the geometric structure of the stability region. The authors in [] proved that the stability boundary of a SEP is the union of the stable manifolds of the type one unstable equilibrium points and proposed a numerical algorithm to determine the stability region. Recently, some algorithms have been developed to approximate the stable manifold of an UEP. For example, in [7], [8] the Taylor expansion is used to get a quadratic approximation and in [9], [1], the stable manifolds around an UEP are approximated by the normal form technique and the energy function methods [11]. A well-known alternative method called the closest unstable equilibrium point method [1] finds a sub of the true stability region and thereby need not obtain the stable manifold of an UEP. It is shown in [12] that the stability region estimated by the closest UEP method is optimal in the sense that it is the largest region within the stability region, which can be characterized by the corresponding energy function. However, the closest UEP method can give very conservative results for stability region approximation. In [13], the authors apply the singular perturbation theory to decompose a particular power system into slow and fast subsystems based on the assumption that a power system can be perfectly separated in time-scale. Then the stability region of a SEP is obtained by numerical simulation. Control is used to improve the transient stability as well as the damping of a power system. The control of a power system can be both continuous (generator exciter and governor control,etc) and discrete (line impedance control through series/shunt capacitors switch, and tripping of generators/loads). Thus, power system control is hybrid, making a controlled power system a hybrid system. The application of hybrid systems based modeling and control techniques for power systems is not new, but is relatively active [14] [18]. Also the application of switching control to stability of power networks has been reported in articles such as [19] [26]. A commonly used method to choose a control strategy for transient stability analysis is time domain simulation. For each initial fault on state, we need to perform simulation for each control in order to find an effective one. However, the stability region based analysis can form an effective control strategy for all possible post-fault initial state. In this paper, we first obtain a Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) for the propagation of the backward reachability of a nonlinear system. This computation when used to obtain the backward reachable of a stable equilibrium point yields its stability region. Using

2 2 the stability regions of various discrete controls (also called modes) transient stability design is performed. For example the effectiveness of a control can be verified by checking whether a post-fault initial state is in the stability region of the system with that control switched on. We illustrate our method by applying it to a single-machine infinite-bus system equipped with series and shunt capacitive compensation. The paper is organized as follows. Some fundamental concepts of the reachable analysis are introduced in Section II. In Section III a new algorithm to determine the stability region of a SEP is proposed. In Section IV an algorithm is developed for the transient stability design. Section V presents an example to illustrate the algorithm provided in this paper. Finally Section VI provides some discussion and conclusion. II. REACHABLE SET AND ITS CHARACTERIZATION Reachability analysis mainly focuses on finding the reachable s. Reachable s are a way of capturing the behavior of entire groups of trajectories at once. There are two basic types of reachable s, depending on whether an initial or a final condition is specified. The forward reachable is defined as the of all states that can be reached along trajectories that start in a specified initial. On the other hand, the backward reachable is the of states where trajectories can reach the specified target. The backward and forward reachable s are shown in Fig. 1. In Section III, we make use of backward reachable s to compute the stability region of a stable equilibrium point of a nonlinear system. In Seciton IV transient stability design is developed based on reachability analysis. Fig. 1. backward reachable target initial forward reachable Illustration of backward and forward reachable s Reachable is a sub of state space. One way of describing a sub of states is via an implicit surface function representation. Consider a close S R n. An implicit surface representation of S would define a function φ: R n R such that φ(x) if x S and φ(x) > if x / S. In [27], the author formulates the backward reachable in terms of a Hamilton-Jacobi-Isaacs (HJI) PDE, and proves that the viscosity solution of this PDE is an implicit surface representation of the backward reachable. This HJI PDE can be solved with the very accurate numerical methods drawn form the level literature [28]. Consider an autonomous system described by ordinary differential equations: = f(x) (1) where x R n is the state vector and f(x) is the vector field. Suppose φ(x, t) is the level function to describe the backward reachability at time t. φ(x, t) = is a surface in (n + 1) dimensional space. The surface φ(x, ) = is the boundary of target, whereas the surface φ(x, t) = is the boundary of the of all states x R n where the target can be reached in time t or less. Consider the surface φ(x, t) = in the (n + 1) dimensional space. For every (x, t) on this surface, the φ value is zero. So if a small variation is made along this surface, i.e., move (x, t) to a neighboring point (x +, t + ), that point still lies on the surface, then the variation in φ is zero: dφ = φ(x +, t + ) φ(x, t) = dφ = φ x φ x n n + φ t From this it follows that, φ T x Substituting (1) to (2), we can get + φ t = (2) φ T x f(x, t) + φ t = (3) Now suppose the system is given by a of Differential algebraic equations (DAE) as, = f(x, y, t); g(x, y, t) = (4) where y R m is a of auxiliary variables. In this ting, we will represent the boundary of the backward reachability at time t by the surface φ(x, y, t) = in (n + m + 1) dimensional space. Then as before, dφ = φ(x +, y + dy, t + ) φ(x, y, t) = φ T x + φ T y dy + φ t = From above it follows that, φ T x + dy φt y + φ t = () we have that = f(x, y, t). Now to compute dy. We consider the algebraic constraint equation, g(x, y, t) =. Then using argument similar to above, dg = g(x +, y + dy, t + ) g(x, y, t) = g T x + g T y dy + g t From here it follows that, g T x + dy gt y + g t = (6) Multiplying in both sides of ( 6) by (g y g T y ) 1 g y, yields, which yields: (g y g T y ) 1 g y g T x + dy + (g yg T y ) 1 g y g t = dy = (g yg T y ) 1 g y [g T x + g t] So ( ) can be written as: φ T x f + φ t = φ T y (g y g T y ) 1 g y [g T x f + g t ] (7)

3 3 Thus we obtain the desired PDE. (3) describes the propagation of the backward reachability boundary as a function of time of system (1). While (7) describes the propagation of the backward reachable boundary with algebraic constraints. Level methods are a collection of numerical algorithms for computing the dynamics of moving curves and surfaces. Given a target defined by an implicit surface function φ(x, t ), we use level methods to solve the HJI PDE and thus compute the backward reachable. III. COMPUTATION OF STABILITY REGION Section II introduces the concepts of reachable s and their computation based on the derived PDE. Here we apply the reachability analysis for the computation of the stability region for power system transient stability assessment. Given a postfault stable operating point, there exists an open neighborhood of it that is contained in the stability region. We pick a sufficiently small ε ball around the SEP as the target. The following algorithm summarizes the procedure to determine the stability region of a post-fault power system. 1) Form the state space equations of the post-fault power system, = f(x). 2) Find the stable equilibrium point of this autonomous nonlinear system, by solving f(x) = and let x R n be a SEP. 3) Specify a ε ball centered at the stable equilibrium point with sufficiently small radius ε. Define an implicit surface function at time t = as φ(x, ) = x x ε (8) Then the target is the zero sublevel of the function φ(x, ), i.e, it is given by {x R n φ(x, ) } (9) Therefore, a point x is inside the target if φ(x, ) is negative, outside the target if φ(x, ) is positive, and on the boundary of the target if φ(x, ) =. 4) Propagate in time the boundary of the backward reachable of the target by solving the following HJI PDE: φ T x f(x, t) + φ t = (1) with terminal conditions (8). The zero sublevel of the viscosity solution φ(x, t) to (8),(1) is the backward reachable at time t is {x R n φ(x, t) } (11) ) The backward reachable of the ε ball around the stable equilibrium point is computed using a software tool from [29]. It is always contained in the stability region of the stable equilibrium point. As t goes to infinity, the backward reachable approaches the true stability region. If the stability region is bounded, the level based numerical computation of the backward reachability eventually converges to the stability region within a finite computation time. Example 1: For the nonlinear system, 1 = x 2 2 = x 1 + (x 2 1 1)x 2 (,) is the only equilibrium point that is stable. So in this case we can not employ the manifold-based method to find the stability region of the equilibrium point. Our backward reachability based method is always applicable and using it we compute the stability region as shown in Fig. 2. Fig x Phase portrait Stability region computed by level method x 1 Stability region In Fig. 2, the region inside the dashed line is the stability region of the stable equilibrium point. We verify our result by drawing the phase portrait from which we can see that our computation is precise. IV. APPLICATION IN TRANSIENT STABILITY DESIGN Section III presents an algorithm to compute the stability region of a given power system. Based on this algorithm we develop a control strategy to maintain system transient stability, as follows. 1) Compute the stability region of a stable equilibrium point x 1 of the normal system, referred as the stability region of mode 1. 2) Check if the post-fault initial state x is inside the stability region of mode 1, if yes, stop. The system is stable without any controls on. Otherwise, the system is unstable, go to next step. 3) Compute the stability region of a stable equilibrium point x i of the system with control i on, referred to as mode i. 4) Check if the post-fault initial state x is inside the stability region of control i. If yes, control i can be switched on, in which case, the system will eventually reach the equilibrium point. Otherwise, control i can not prevent the instability of the system. ) In case control i can stabilize the system, but the SEP x i is not a desired operating point, the system can be steered to another SEP x j, whose stability region has a nonempty intersection with stability region of x i. To achieve this control j is switched on after switching on

4 4 the control i and waiting till the system trajectory enters the intersection of two stability region. 6) Step may be repeated to steer the system further to yet an SEP if desired. V. EXAMPLE In this section an example is presented to show how the stability region is computed and the transient stability design is developed. A. A Single-Machine-Infinite-Bus Model The classical single-machine-infinite-bus model of a power system is shown in Fig. 3. The system model is given as follows: dδ = d = P m P M e sin δ D M (12) Here, δ is the machine rotor angle and is the relative angular velocity of the rotor. Suppose the inertial constant M = T J =.26 s 2 /rad, the damping coefficient D =.12, the mechanical power P m = 1. per unit, and the maximum electrical power transferred is Pe M = 1.34 per unit, where E is the voltage of the generator, U is the voltage of the infinite bus, and X is the reactance of the transmission line. = EU X jx U Fig. 4. Fig.. ( rad / s ) δ ( rad) Phase portrait Computed stability region boundary Stability region and phase portrait for D =.12 s/rad ) ( rad / s Phase portrait δ (rad) Computed stability region boundary Stability region and phase portrait for D =.1 s/rad Fig. 3. E δ A single-machine-infinite-bus model 1) The computation of stability region of normal system: From the system equation (12) and the chosen parameter values, the point (.8324, ) is identified to be a stable equilibrium point of this system. We define the target as δ.8324) The stability region computed using our algorithm lies inside the solid line drawn in Fig. 4. From this figure we conclude that if the post-fault initial condition of the state variables is inside the stability region, the trajectories converge to the stable operating point. If the initial condition is outside the stability region, the system is unstable. We validate our result by drawing the corresponding phase portrait using time domain simulation of some sample trajectories, from which we can see that our method can precisely compute the stability region. In addition, when the damping coefficient D is increased, the stable equilibrium point remains the same as (.8324, ). For D =.1, we compute the stability region as shown in Fig.. The figure clearly shows that when D is increased, the size of the stability region also increases. The observation is validated by time domain simulations. Again, Fig. 6 and Fig. 7 show the time domain responses of the rotor angle and velocity for an initial condition (δ, ) = (, 1) when D is.12 s/rad and.1 s/rad, respectively. Fig. 7 shows that the trajectories eventually tle down to the post-fault stable operating point. However, when D is.12 s/rad, the system loses stability for this initial condition as shown in Fig. 6. This is not unexpected since a larger D implies a larger stability region. Fig t (second) Time domain simulation when D =.12 s/rad 2) Transient stability design: Fig. 8 shows the model of a single-machine-infinite-bus system with shunt and series controls. The system model is given in ( 12). Define the system with no controls on as mode 1, with series control δ

5 1 δ 2 t = ) ( rad / s t (second) -1 Fig. 7. Time domain simulation when D =.1 s/rad -1 jx1 jx 2 jx series U -2 - δ (rad) Mode 1 Mode 3 Mode 2 Mode 4 E δ jx shunt Fig. 9. Stability region of the 4 modes Fig. 8. System model with shunt and series control strategies on as mode 2, with shunt control on as mode 3, and with both series control and shunt control on as mode 4. As the mode is changed, the transmission line reactance changes causing the Pe M value as well as the equilibrium point to change. The reactance, the Pe M value, and the equilibrium point associated with each of the four modes is given in Table I. Mode Series Shunt X i Pe M Equilibrium Capacitor Capacitor value point 1 Off Off X 1 + X (.8342, ) 2 On Off X 1 + X 2 X series 2.2 (.463, ) 3 Off On X 1 + X 2 X 1 X 2 X shunt 1.43 (.784, ) 4 On On X 1 + (X 2 X series ) (.44, ) X 1 (X 2 X series ) X shunt TABLE I FOUR CONTROL MODES AND THEIR CERTAIN PARAMETERS The stability regions of all the four modes are shown in Fig. 9. The stability region of mode 1 is inside the dotted curve, that of mode 2 is inside the dashed-dot curve, that of mode 3 is inside the dashed curve, and that of mode 4 is inside the solid curve. Based on the stability regions, we can verify the effectiveness of different control strategies and provide an effective control strategy for a given post-fault initial state. When the post-fault state is inside the stability region of mode 1, no control is needed because the state will finally reach the stable equilibrium point. When the post-fault state is outside the stability region of mode 1, we need to switch on some controls to stabilize the post-fault state. For example, if the initial postfault state is inside the stability region of mode 2 and outside the stability region of mode 3, we have two choices to stabilize the system: switch on the series capacitor or switch on both the series and shunt capacitors. The system will then converge to the stable equilibrium point of mode 2 or mode 4 respectively. In general, if the post-fault initial state is inside the union of such stability regions, the transient stability can be achieved by switching one or more controls. In Fig. 9, we can also see that the equilibrium points of different modes are all inside the stability region of mode 1. That means whenever a control is switched on, the system will finally reach the equilibrium point of that mode. After that, we can switch off the specific control so that the system finally reach the stable equilibrium point of mode 1. It follows that if a transient-fault causes the system to deviate, then as long as this state lies in the union of 4 regions of stability, it is possible to switch the series/shunt capacitors on and later off to return the system to the equilibrium of normal configuration. VI. CONCLUSION A novel method for computing the stability region of a nonlinear system, such as a power system, is presented in this paper. We also apply this method in transient stability design. The proposed method has the following advantages: 1) It computes the stability region accurately. For large systems, the computation may be stopped after a certain amount of time to get a sub region to save time. 2) It is easy to implement. We only need to form the mathematic model of the post-fault power system and identify the stable equilibrium point. After that, we can use level methods to compute the stability region as a backward reachable. The limitation of our method is that the complexity grows exponentially as a function of the system dimension. This is because the computation of the propagation of implicit function bases on a gridding the state space. As part of future research we plan to explore faster and/or approximate techniques for reachability computation. This includes possibility of parallelization, of hierarchical computation, Sum of Squares (SOS) based approach, etc.

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