Power System Dynamic Security Region and Its Approximations

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < preventive control actions to take. After defining dynamic security region (DSR) in Section II, we show in Section III that the expression characterizing of the boundary of the stability region is the solution of a partial differential equation. We then extend the idea to define the boundary of DSR in Section IV and derive methods of calculating quadratic and linear approximations to the boundary of DSR. The approximations are tested using simple power systems in Section V. Potential applications of the proposed method are discussed in Section VI. A summary is provided in Section VII.

II. MATHEMATICAL MODEL A. Power System Consider a power system with generators and loads interconnected together by a transmission network. The system dynamics is assumed to be represented by a differential equation x = f ( x, u ) (1) where x ∈ R n are the state variables, and u ∈ R m are the control variables. Due to power flow equations of the transmission network, the electromechanical interaction of a power system for stability analysis is typically represented by a differential-algebraic equation instead. Additional assumption is therefore necessary for the model introduced in (1) to be valid for stability analysis. One is to assume that the loads are all constant impedance loads. Another one is to assume that the structural preserving model is used [11]. For example, if the classical model of a power system is used for transient stability analysis, then for a power system consisting of ng generators with the loads modeled as constant impedances, the dynamics of the k-th generator can be written with the usual notation as δ k = ω0ω k k = 1, , ng (2) 2 H k ωk = Pmk − Pek − Dk ω k where Dk and H k are damping ratio and inertia constant of ng

machine k. Pek = Ek2 Gkk + Ek (∑ E j (Gkj cos δ kj + Bkj sin δ kj )) is j≠k

the electrical power at machine k, δ kj = δ k − δ j , δ k is the rotor angle of machine k, Ek is the constant voltage behind direct axis transient reactance, Pmk is the mechanical power.

ω0 = 2πf B , and Y = (Yij ) ng ×ng = (Gij + jBij )ng ×ng is the reduced admittance matrix. If, furthermore, as usual, uniform damping is assumed, i.e. d 0 = Dk / 2 H k ( k = 1, , ng ), then using the ng-th machine as the reference, (5) can be transformed into the form of (1) as follows:

δ kn g = ω0ωkn g

k = 1,

ωkn g = −d0ωkng + ( Pmk − Pek ) / 2 H k

, ng − 1

2

(3)

− ( Pmn g − Pen g ) / 2 H n g

In such a case, the state x = (δ T , ω T )T = (δ1, ng , , δ ng −1, ng , ω1, ng , control variables could be u = ( Pm1 ,

variables , ωng −1, ng )T

will be and the

, Pmn g ) . More discussion

on control variables will be presented in Section II.C. B. Transient Stability Transient stability is the ability of the power system to maintain synchronism after a fault such as short circuit. The system can be thought of going through three stages in relation to a fault. Before the fault, or the pre-fault stage, the system is in steady-state. Immediately after the fault, before the protective relay system responds, the power system is under stress and this is the fault-on stage. After the fault is cleared by the protective relay system, the concern is, in this post-fault stage, whether the power system is able to settle to a stable operating condition. Transient stability can therefore be considered as the study of the stability of the post-fault system. The stability depends on the transmission loading of the pre-fault system, the clearing time of the protective relay, and the strength of the post-fault system. It is commonly conducted in transmission planning to design the protective system and for that purpose, clearing time has been used as an important measure in the study. Our focus, however, is on system operation. We, therefore, assume that the clearing time of the protective system is already set and all automatic control actions for transient stability enhancement are built in the dynamic system model. Furthermore, we assume, for simplicity, that the control variables u in the short period during the fault and immediately after the fault are constants. We now state the mathematical model of transient stability. At the pre-fault stage, the system is operated at a stable equilibrium point x0 (u ) of the pre-fault system x = F1 ( x, u )

t < 0 x ∈ Rn ,u ∈ Rm

(4)

In other words, x0 (u ) satisfies F1 ( x0 (u ), u ) . Here, we assume that other quantities of the operating condition are pre-determined and the setting of the control variables u completely determines the operating state x0 (u ) . At time t = 0 , the system undergoes a fault that results in a structural change in the system. Suppose the fault is cleared at time t = t F . Then during the fault, the system is governed by a fault-on dynamics described by y = F2 ( y, u ) , y (0) = x 0 , y (t ) = φ (t , x 0 , u ) , 0 ≤ t < t F (5) We use the notation φ (t , x0 , u ) for the solution of (5) starting at x0 , with the operating point u explicitly indicated. Once the fault is cleared, the system is henceforth governed by a post-fault dynamics described by the following differential equation (6). The initial condition of the post-fault system is the state of the fault-on system at fault clearing, φ (t F , x0 , u ) .

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < Notice that since the clearing time is given and x0 is a function of u, the system state at the time of clearing is really only a function of u, we therefore write φ (u ) = φ (t F , x0 , u ) . The post-fault dynamics is described by: z = f ( z , u ) z (t F ) = y (t F ) = φ (u ) , t ≥ t F (6) Next, assume the post-fault system has a (asymptotically) stable equilibrium point z s (u ) . The fundamental issue of transient stability analysis is the following: starting from the post fault initial state φ (u ) , will the system trajectory be able to settle down to the steady sate condition z s (u ) ? The problem can be expressed in terms of the concept of the stability region. The stability region (or the region of attraction A( z s (u )) ) of a stable equilibrium point, z s (u ) , is defined as the set of points from which the trajectories converge to z s (u ) . The transient stability analysis is to determine whether the initial point of the post-fault trajectory is located inside the stability region of the equilibrium point z s (u ) . Mathematically, it can be expressed as to check the condition: φ (u ) ∈ A( z s (u )) (7) Eq. (7) implies that transient stability is completely determined by the setting of the control variables u at the pre-fault operating condition. This is true because of the assumptions we have made so far. As a consequence, the pre-fault operating state, the fault-on dynamics and the post-fault dynamics are all specified, once the control variables u are specified. C. Dynamic Security Region We say that a power system is dynamically secure with respect to a given fault if the operating state x(u0) is transiently stable (belong to the stability region of post fault operation equilibrium point) for the given fault. As stated earlier, in the context of system operation, the setting of the control variables u completely determines the transient stability of the system. We therefore can define a region in the space of control variables u in which the system is transiently stable and called it the dynamic security region of the power system (with respect to a given fault). Mathematically, the dynamic security region (DSR) can be described by: Ω d = {u : φ (u ) ∈ A( zs (u ))} (8) Previously dynamic security region was defined in the space of power injections which include all generation and loads in the system [6-10]. We now modify it to be in the space of control variables u, which is the set of controls that the dispatcher can use when dynamic security, i.e., transient stability, of the power system is being threatened. The rationales for the modifications are twofold. While the old definition might be reasonable in the era of regulated monopolies when the dispatcher had the authority to control all generations and loads in the system, it is no long true in the deregulated environment. The dispatcher of an ISO (independent system operator) or RTO (regional transmission organization) now has only control over some generators and

3

loads. On the other hand, there may be other means, such as ancillary services, or FACTS devices, under his disposal to control power flows and they should be taken into account. The control variables are therefore those under the control of the dispatcher, such as dispatcher generation, interruptible load, curtalable trade, various controllable devices etc. that can be used to mitigate dynamic security problems. The dimension of the control variables u is typically not high in practical situations. III. STABILITY REGION AND ITS BOUNDARY A. Characterization of the Stability Region In this section, we present results characterizing stability regions of general dynamical systems described by the following differential equation: z = f (z ) (9) where f is twice differentiable, z ∈ R n . An equilibrium point is a solution of the equation 0 = f ( z ) . A hyperbolic equilibrium point is an equilibrium point z at which the Jacobian J ( z ) = ∂f / ∂z has no eigenvalue whose real part is zero. (The notation ∂f / ∂z for a vector-valued function f ( z ) stands for a matrix whose (i,j)-th element is equal to ∂f i / ∂z j ). A hyperbolic stable equilibrium point z s of the system (9) is an equilibrium point at which all the eigenvaules of J (z ) are having negative real parts. A type-k equilibrium point is a hyperbolic equilibrium point at which k eigenvalues of J (z ) are having positive real part. The flow of the system (9) is the solution of (9) at time t starting at z and is expressed as φ (t , z ) . The stable manifold of a hyperbolic equilibrium point zk is the set of all those points from which the flow will converge to zk as time approaches positive infinite, i.e. W s ( zk ) = {z : φ (t , z ) → zk , as t → +∞} (10) The unstable manifold of the hyperbolic equilibrium point zk is the set of all those points from which the flow will converge to zk as time approaches negative infinite, i.e.,

W u ( zk ) = {z : φ (t , z ) → zk , as t → −∞} (11) The stability region of an asymptotically stable equilibrium point z s is the stable manifold of z s , i.e.,

A( zs ) = {z : φ (t , z ) → zs , as t → +∞} (12) The following theorem from [5] completely characterizes the stability region of a dynamical system. Theorem 1 Suppose the system (9) satisfies the following assumptions: A1) All the unstable equilibrium points on the stability region boundary are hyperbolic. A2) The stable and unstable manifolds of equilibrium points on the stability boundary satisfy the transversality condition. A3) Every trajectory on the stability boundary approaches one of the equilibrium as t → ∞ .

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < Then the boundary of the stability region A( z s ) of a hyperbolic stable equilibrium point z s is composed of the stable manifolds of all unstable equilibrium point ze on the boundary of the stability region, i.e., (13) ∂A( z s ) = ∪ W s ( ze ) z e ∈∂A

B. Characterization of the Stable Manifold We characterize the stable manifold of an unstable equilibrium point (uep) in Theorem 2 below. We will show that the equation describing the stable manifold satisfies a partial differential equation. Similar result for type-1 uep was obtained by Cheng et.al. [12]. The result can be generalized to any type k uep as shown in Theorem 2. The proof is included in the Appendix because our proof, based on elementary mathematical analysis, is more accessible to the engineering community. Theorem 2. Consider system (9). Let Λ = diag (Λ1 , Λ 2 ) be the real Jordan canonical form of the Jacobian matrix

J at a

type-k unstable equilibrium point zek , where Λ1 is the unstable part of the real Jordan canonical form matrix. Then, the stable manifold of zek can be locally written as h( z ) = 0 , ∂h / ∂z is full rank and h( z ) is the solution of the following partial differential equation:

[∂h( z ) / ∂z ] f ( z ) = Λ1h( z ),

h( zek )

=0

(14)

In other words, the stable manifold of zek can be expressed as: ⎧⎪ z : h( z ) = 0, h( z ) ∈ R k , rank (∂h / ∂z ) = k ⎫⎪ W s ( zek ) = ⎨ ⎬ (15) ⎪⎩ such that [∂h / ∂z ] f = Λ1h, h( z ek ) = 0⎪⎭ Proof: See Appendix A. The special case of Theorem 2, when k=1, is stated below as a Corollary. Corollary. For the system (9), the stable manifold of a

type-1 unstable equilibrium point z1e can be described as

⎫⎪ ⎧⎪ z : h( z ) = 0, rank (∂h / ∂z ) = 1 W s ( z1e ) = ⎨ (16) ⎬ ⎪⎩ such that [∂h / ∂z ] f = µh, h( z1e ) = 0⎪⎭ where µ is the unstable eigen-value at the unstable equilibrium point z1e . C. Quadratic Approximation The above Corollary can be used to derive a quadratic approximation to the stable manifold of a type-1 equilibrium point [12]. Theorem 3. For the dynamic system (9), z = f (z ) (17) let us assume that ze is a type-1 unstable equilibrium point with unstable eigen-value µ , and the system has a series expansion at ze

4

f ( z ) = f ( ze ) + J ( ze )( z − ze ) ⎛ ( z − ze )T H1 ( ze )( z − ze ) ⎞ ⎟ 1⎜ ⎟+ + ⎜ 2⎜ ⎟ ⎜ ( z − z )T H ( z )( z − z ) ⎟ e n e e ⎠ ⎝

(18)

where J ( z ) = ∂f / ∂z is the Jacobian of f , H i ( z ) = ∂ 2 f i / ∂z 2 is the Hessian matrix of f i . Then a quadratic approximation of the stable manifold h(z ) at ze can be expressed as: hQ ( z ) = η T ( z − z e ) + ( z − z e )T Q ( z − z e ) / 2

(19)

where the coefficients η and Q of the quadratic expression can be determined as follows: (a) The coefficient vector η ∈ R n of the linear term is the left eigenvector associated with the eigenvalue µ of the Jacobian J at the equilibrium point ze , i.e., J Tη = µη , η Tη = 1

(20) (b) The coefficient matrix Q of the quadratic term is the solution of the Lyapunov equation:

CQ + QC T = H

(21)

where C = ( µI / 2 − J ) , I is the n × n identical matrix, and T

H = [∑ in=1ηi H i ] . Proof: Substituting (18) and (19) into the partial differential equation (16) characterizing the stable manifold of the unstable equilibrium point, with appropriate manipulations, we obtain (20) and (21). The solutions to (20) and (21), for eigenvectors and Lyapunov equations, are standard problems in matrix computation. There are several numerical methods for solving the Lyapunov matrix equation. Cheng [13] has shown that the solution Q to the Lyapunov equation, when Q is symmetric, can be obtained directly by the solution of the following linear equation: [C ⊗ I + I ⊗ C ]Vc (Q ) = Vc ( H ) (22) where Vc is the matrix column stacking mapping defined by:

Vc ( A) = (a11 , a21 ,

an1 , a12 ,

an 2 ,

, a1n

ann )T

(23)

for any n × n square matrix A = (aij ) n× n , and the operator ⊗ is the Kronecker tensor product, defined by: a1t B ⎞ ⎛ a11B ⎜ ⎟ A⊗ B = ⎜ (24) ⎟ ⎜a B ⎟ ast B ⎠ ⎝ s1 for any s × t matrix A = (aij ) s×t and k × l matrix B = (bij ) k ×l . The dimension of A ⊗ B is thus ( s × k ) × (t × l ) . As a solution method, obviously, it is very useful for small systems. Methods of obtaining expressions for the stable manifolds generally fall into two categories: power series and the normal form. The use of the first term of the power series expansion, or linear approximation by hyperplanes, has been suggested very

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < earlier [14, 15]. Recursive formulas for the coefficients of the power series expansion have been derived [16, 17]. The use of the normal form [18, 19] is based on a coordinate transformation, for which the laborious work of computation of all eigenvalues and eigenvectors of the Jacobian matrix is required.

IV. LOCAL APPROXIMATIONS OF THE BOUNDARY OF DSR

A. Boundary of Dynamic Security Region In power system operation, as argued previously, the pre-fault operation point, x0 (u ) , the fault-on trajectory at the time of fault clearance, φ (u ) , the unstable equilibrium point, ze (u ) , and the function describing the stable manifold of ze (u ) , h( z , u ) , are all determined by the control variables u.

We have defined the dynamic security region (DSR) with respect to a given fault as Ω d = {u : φ (u ) ∈ A( z s (u ))} . The

already calculated. Let us assume: (i) ∆u is small, (ii) the controlling uep does not go through sudden change, and (iii) x0 (u ) , φ (u ) , and ze (u ) are differentiable functions of u. We can then use sensitivities to approximate: ze (u ) ≈ ze (u0 ) + Z (u0 ) ⋅ ∆u (29)

η (u ) ≈ η (u0 ) + E (u0 ) ⋅ ∆u Q (u ) ≈ Q (u0 ) +

where

(30)

m

∑ Qi (u0 )∆ui

(31)

i =1

Z (u ) = ∂ze (u ) / ∂u

,

E (u ) = ∂η (u ) / ∂u

and

Qi (u ) = ∂Q (u ) / ∂ui . The fault-on trajectory will also be

approximated by sensitivities: φ (u ) ≈ φ(u 0 ) + Φ(u 0 ) ⋅ ∆u

(32)

where Φ(u ) = ∂φ (u ) / ∂u . Methods of calculating these sensitivities are discussed in Appendix B. Substituting the approximate expressions of ze (u ) , η (u ) , Q (u ) and φ (u ) using sensitivities in (29)-(32) into (19), we

boundary of DSR is reached when the fault-on trajectory φ (u ) at the time of clearing reaches the boundary of the stability region ∂A( zs (u )) . Suppose that ze (u ) is the controlling unstable equilibrium point of the system with respect to the fault-on trajectory [20]. The local boundary, by the Corollary of Theorem 2, is represented by {z : h( z , u ) = 0} . The boundary of dynamic security region that is of interest to the study of transient stablity can therefore be written locally as: ∂Ω d = {u : h(φ (u ), u ) = 0} (25)

obtain a quadratic approximation to h(φ (u ), u ) = 0

B. Approximations Let us consider the situation where at the present time the operating state is at u0 , we are interested in finding the local boundary of the dynamic security region or its approximations. We may use the variable ∆u from u0 to describe the local

where

boundary, i.e., ∆u has the property that u = u0 + ∆u is on the boundary. Our objective is not to find a point on the boundary; rather, we want to find an expression that can characterize the local boundary. In other words, we want to derive an equation in the terms of ∆u , and all the solutions ∆u of that equation are points on the boundary. As the operating condition u changes from u0 to

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hQQ (φ (u0 + ∆u ), u0 + ∆u ) = C0 + C1∆u + ∆uT C2 ∆u = 0

(33)

The coefficients of the above quadratic approximations are: (34) C0 = hQ (φ (u0 ), ze (u0 )) C1 = η (u0 )T D (u0 ) + d (u0 )T E (u0 ) + d (u0 )T Q(u0 ) D (u0 ) +[d (u0 )T Q1 (u0 )d (u0 ),

, d (u0 )T Qm (u0 )d (u0 )]

C2 = E (u0 )T D(u0 ) + D (u0 )T Q (u0 ) D(u0 ) / 2 + D (u0 )[Q1 (u0 )d (u0 ),

, Qm (u0 )d (u0 )]

(35) (36)

d (u0 ) = φ (u0 ) − ze (u0 ) is the difference between the

system state at the time of fault clearing φ (u0 ) and the uep ze (u0 ) , and D(u0 ) = Φ (u0 ) − Z (u0 ) is its sensitivity with

respect to u. Ignoring the second-order term, we get a approximation to the quadratic approximation (33): hQL (φ (u0 + ∆u ), u0 + ∆u ) = C0 + C1∆u = 0

linear (37)

equation of the local boundary of DSR h(φ (u ), u ) = 0 as well.

If we start directly from a linear approximation to the stability boundary: (38) hL ( z , u ) = [η (u )]T ( z − ze (u )) and substituting (29)-(32) into it, we get: hLL (φ (u0 + ∆u ), u0 + ∆u ) = L0 + L1∆u = 0 (39) where L0 = hL (φ (u0 ), u0 ) (40)

Let us assume we have calculated the relevant quantities at u0 ,

L1 = η (u0 )T D (u0 ) + d (u0 )T E (u0 )

u = u0 + ∆u , φ (u ) , ze (u ) , and h( z , u ) all change, hence the

i.e., ze (u0 ), φ (u0 ), and hQ (φ (u0 ), u0 ) , where hQ (φ (u0 ), u0 ) = η T (u0 )(φ (u0 ) − ze (u0 )) + (φ (u0 ) − ze (u0 ))T Q(u0 )(φ (u0 ) − ze (u0 )) / 2

J T (u0 )η (u0 ) = µ (u0 )η (u0 )

(26) (27)

C (u0 )Q(u0 ) + Q(u0 )C (u0 ) = H (u0 ) (28) The question is how we can characterize the local boundary of the dynamic security region (25) in terms of the quantities T

(41) We have thus obtained three approximations to the equation of the local boundary of dynamic security region h(φ (u ), u ) = 0 . Two of them start from the quadratic approximation (19). One starts with only the linear term in (19). These approximations will be tested with two simple examples in the next section. Major computational tasks involved in the proposed method include the following (in the order decreasing computational burden):

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < (a). Finding controlling uep of the given fault ze (u ) .

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For this example, it is possible to draw the stable manifold

(b). Calculate system state at fault clearing φ (u0 ) and the trajectory sensitivity Φ(u0 ) . (c). Calculate the quadratic coefficient Q (u0 ) and its sensitivity Qi (u ) . (d). Calculate the linear coefficient η (u0 ) . (e). Calculate sensitivity terms, Z (u0 ) and E (u0 ) . Step (a) and Step (b) are present in most of the Direct Methods for transient stability analysis. Steps (c)-(e) involve the solutions of Lyapunov equation, the solution of an eigenvalue problem and the solutions of linear equations. The computational cost of Steps (a) and (b) is the most serious. Steps (c)-(e) are standard problems in linear algebra. The computational feasibility for the applications of the proposed method to practical large-scale power systems, therefore, depends very much on the efficiency of the methods used for these steps, especially Steps (a) and (b), for which significant progress has been made recently with power system specific methods.

Fig. 2. Stability boundary and its approximations in state space.

h( z , u ) = 0 , Eq. (19), in the post-fault state space z = (θ , ω ) in

Fig. 2. Next we show the sensitivity of the approximations. Fig. 3

V. EXAMPLES

A. A single machine system

Fig. 1. Single machine system.

The approximation methods proposed in Section IV is first applied to a single machine infinite bus system (Fig. 1.) Complete data for the system can be found in [21] (p.844). Line 2 (XL2=0.93p.u.) experiences a three-phase fault at the sending terminal side. The fault is cleared at tF = 0.087s. In this case, u=Pm, i.e., the control variable u has dimension 1. Hence the boundary of DSR is just a point, i.e., the maximum Pm for secure operation. We wish to estimate this operating limit. Let the current operating point u0=0.85p.u.. Table 1 displays calculations based on the estimated boundary of DSR using three approximations, the LL method, the QL method the QQ method. The boundary obtained from the time-domain numerical integration method is used as a

Fig. 3. Effect of initial value.

TABLE I SIMULATION RESULTS FOR THE EXAMPLE SYSTEM AT PM=0.95.P.U. Methods Max Pm(p.u.)

LL Approx.

QL Approx.

QQ Approx.

Time Domain

1.0167

0.8937

0.8981

0.8996

benchmark for comparison. It is seen that the proposed QL and QQ approximations offer fairly accurate results of the DSR for the fault.

Fig. 4. Effect of clearing time.

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < shows the effect of the current operating point u0 on the approximations. In normal operation, the operating point changes as load demand and generation dispatch vary and can not be set arbitrarily. However, the operator has the authority to alter the operating point if the security of the power system is threatened. It is shown in Fig. 3 that as the operating point u0 gets closer to the boundary (i.e., the operating limit), the approximations get better. This result is not at all surprising as the accuracy of the sensitivities used in the approximations becomes better as the difference between u0 and the boundary gets closer. The practical implication of this result may be very significant. The proposed method has the very desirable property that if the current operating point is close to the operating limit, which is the case when the operator is most concerned, the more accurate the result becomes. In other words, the more critical is the situation when actions may be required, the more confident we may place on the method. Fig. 4 shows the effect of clearing time on the approximations. All these results indicate that the proposed QQ and QL approximations are fairly accuracy for this system (e.g. the maximum error of the QQ method is 6%) and superior to the LL approach.

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Fig. 6. Boundary of DSR and its approximations.

B. A 3-Generator 9-Bus System

Fig. 7. Projection on a plane.

Fig. 5. IEEE 3-generator 9-bus system.

The 3-generator and 9-bus IEEE test system [22](p.38), showed in Fig.5, is used to test the proposed methods. A three-phase fault occurs at bus 7 on the line between nodes 7 and 5. The fault is cleared at tF = 0.20s. Classical models of generators with uniform damping d0=0.1661 and network reduction procedure are used to derive the differential equation (1) of the model. We assume that the generator powers are the control variables. Therefore, the dimension of u is equal to three. A portion of the boundary of the three dimensional DSR, along with the three approximations, QQ, QL and LL methods, is shown in Fig. 6. A two-dimensional cross-sectional view of the boundary at Pm1=0.68p.u. is shown in Fig. 7. Again QQ and QL methods provide much better approximations than LL method. The error

of the QQ approximation is about 5%, except there are points (for example, at Pm1=0.74p.u., Pm2=1.635p.u.) where the quadratic equation (33) fails to have a real solution; The QL approximation has no problem in solutions, and its errors is around 10%. But the error of the LL method may exceed 50%.

VI. APPLICATIONS In this section, we discuss potential applications of the dynamic security regions in power system operation, namely, dynamic security assessment and preventive control.

A. Dynamic Security Assessment Let the current operating point be represented by u0 and we are concerned with the ability of the system to withstand transient stability for a specific contingency. Suppose that the system is transiently stable with respect to the current operating state, i.e., the current operating point is inside the dynamic security region, u0 ∈ Ω d = {u : φ (u ) ∈ A( zs (u ))} (42) or φ (u0 ) ∈ A( zs (u0 )) . The fact that the operating point u0 is in

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < DSR indicates that there is still room for movement of the operating condition where the system remains transiently stable. This range is provided by the boundary of the dynamic security region ∂Ω d |local = {u0 + ∆u : h(φ (u0 + ∆u ), u0 + ∆u ) = 0} . As long as the change ∆u does not exceed what is given by the equation h(φ (u0 + ∆u ), u0 + ∆u ) = 0 , the system will remain transiently stable. The three approximate expressions for the boundary of DSR, (33), (37) and (39), expressed directly in terms of ∆u from u0, that provide the limits of current operational movement. The example in Section 5.1 indicates that the closer the operating condition is to the limit, the more reliable the approximations (33) and (37) become. This is a very desirable property for dynamic security assessment.

B. Preventive Control Suppose, on the other hand, corresponding to the current operating point u0 the system will be unable to maintain transient stability for the given fault, i.e., u0 ∉ Ω d = {u : φ (u ) ∈ A( zs (u ))} (43) or φ (u0 ) ∉ A( zs (u0 )) . To ensure that the system can withstand the fault, we have to change the operating point with control variables ∆u . The boundary of the dynamic security region ∂Ω d |local = {u0 + ∆u : h(φ (u0 + ∆u ), u0 + ∆u ) = 0} gives how far the change ∆u needs to be taken in order for the system to be transiently stable. Any ∆u that is a solution of h(φ (u0 + ∆u ), u0 + ∆u ) = 0 will move the system to a point where transient stability is guaranteed. To choose which one of the solutions depends on the criterion used, which should be a measure that reflects the effectiveness of the control. The control strategy is then to pick the most effective means to move the pre-fault operating condition into the boundary of DSR. Of course this is done prior to the fault as a preventive measure. The preventive control problem can therefore be formulated as an optimal power flow problem with objective function L( ∆u ) : max L(∆u ) (44) s.t. h(φ (u0 + ∆u ), u0 + ∆u ) = 0

VII. CONCLUSION In this paper, we define dynamic security region (DSR) in the context of operation. Our focus is on variables that are under control of the dispatcher, such as dispatchable generation, interruptible load, and curtalable trade. A complete mathematical characterization of the boundary of the DSR is derived. Locally, if the direction to which unstable equilibrium point (the controlling uep) the trajectory of the faulted system is moving is known, a local quadratic approximation to the boundary of DSR can be derived. Methods to calculate a quadratic approximation, and two linear approximations, to the boundary of DSR, have been derived. Examples are provided to show the effectiveness of the approximations. The two approximations QQ and QL based on quadratic expressions of

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the boundary of DSR are shown to provide consistently better results than the one based on a linear expression. Potential applications of the proposed method based on the boundary of DSR, for dynamic security assessment and preventive control, are briefly sketched. The proposed method, as it stands, requires significant computational efforts. The computational issues have not been thoroughly examined and are beyond the scope of this paper. A recent implementation of the proposed method has been tested on the 10-machine 39-bus New England system [24] where the results indicate that the accuracy of the approximations and the computational complexity of the method are well within acceptable range. For applications to practical systems, efficient computational methods need further exploration. However, this paper provides a first step towards laying a theoretical foundation on which further improvements of the applications of the concept of dynamic security regions can be based.

APPENDIX A. Proof of Theorem 2 With shift of coordinate x = z − zek on (9), the system x = f ( x) has a type-k equilibrium point at x = 0 . There exists

a similarity transformation G , with which we have J ⋅ G = G ⋅ diag (Λ1 , B )

(A1)

where the k × k matrix Λ1 is the unstable part of the real Jordan canonical form. With G , we have the coordination transformation w = G −1 x, w = (u , v ), u ∈ R k , and

w = G −1 x = G −1 f ( x) = G −1 f (Gw) = f ( w) It follows from (9), (A1) and (A2),

(A2)

u = Λ1u + Ru (u , v) = f1 (u , v)

(A3)

v = Bv + Rv (u, v) = f 2 (u, v)

It can easily be verified that the partial differential equation (14) is invariant under coordinate transformation, i.e., h( x ) , where is of full ∂h / ∂x [∂h( x) / ∂x] f ( x) = Λ1h( x), h(0) = 0 if

rank, and

satisfies only if

h ( w) satisfies [∂h / ∂w( w)] f ( w) = Λ1h ( w), h (0) = 0 , where

h ( w) = h(Gx) = h( x) . The proof will be stated in the transformed coordinates w = (u, v) and proceed in two parts. We first show that (necessity) if h ( w) = 0 , with ∂h / ∂w full rank and h (0) = 0 , is the stable manifold W s (0) of the origin, then h ( w) satisfies

[∂h / ∂w( w)] f ( w) = Λ1h ( w), h (0) = 0

(A4)

By definition, if w is a point in the stable manifold

W s (0) = {w : h ( w) = 0, h (0) = 0} , then the solution trajectory

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < J T (u )η (u ) = µ (u )η (u )

−1

(B4)

η T (u )η (u ) = 1 Taking partial derivative with respect to ui we obtain ∂J T ∂η ∂η ∂µ η (u ) + J T (u ) = µ (u ) + η (u ) ∂ui ∂ui ∂ui ∂ui

η T (u )

∂η =0 ∂ui

It can be rewritten in a matrix form: ⎛ J T − µ I −η ⎞ ⎛ ∂η / ∂ui ⎞ ⎛ −[∂J T / ∂ui ]η ⎞ ⎟⎟ ⎜ ⎟⎜ ⎟=⎜ T 0 ⎠ ⎝ ∂µ / ∂ui ⎠ ⎜⎝ 0 ⎝ η ⎠

(B5)

(B6)

Taking partial derivative with respect to ui we obtain ∂C ∂Q ∂Q T ∂C T ∂H + = Q+C C +Q ∂ui ∂ui ∂ui ∂ui ∂ui

(B9)

∂Q is the solution of the ∂ui

following Lyapunov equation ∂Q ∂Q T ⎡ ∂H ∂C ∂C T ⎤ C + C =⎢ − Q −Q (B10) ⎥ ∂ui ∂ui ∂ui ⎦ ⎣ ∂ui ∂ui The partial derivatives terms on the right-hand side can easily be obtained by recalling that (B11) C (u ) = µ (u ) I / 2 − J T (u )

φ (t , x0 , u ) = F2 (φ (t , x0 , u ), u )

Interchanging the differentiations,

The solution of the matrix differential equation (B18) with the initial condition (B19) evaluated at t = t F is the sensitivity matrix that we are seeking ∂φ (t F , x0 , u ) ∂φ Φ(u ) = = ∂u ∂u t =tF

solution trajectory φ (u ) = φ (t , x0 , u ) of the fault-on system evaluated at t = tF , we will first find a more general problem of

[2]

[3] [4]

[7]

[8]

[9]

[10]

the sensitivity of the trajectory φ (t , x0 , u ) with respect to u. Then the sensitivity of φ (u ) is obtained by setting t = tF . But first let us look at the sensitivity of the initial point φ (0, x0 , u ) = y (0) = x0 (u ) , ∂y (0) / ∂u = ∂x0 (u ) / ∂u . Recall that x0 (u ) is the pre-fault equilibrium point, i.e., F1 ( x0 (u ), u ) = 0 .

Therefore, similar to (B3)

(B20)

REFERENCES [1]

[6]

n ⎛ ∂η ∂H ⎞ ∂H = ∑ ⎜ i H i (u ) + ηi (u ) i ⎟ (B14) ∂ui i =1 ⎝ ∂ui ∂ui ⎠ 4) Sensitivity of φ (u ) : Since φ (u ) = φ (t F , x0 , u ) is the

(B18)

The above matrix differential equation has an initial condition ∂φ ∂y (0) ∂x0 (u ) (B19) = = ∂u t = 0 ∂u ∂u

(B12)

(B13)

(B17)

∂ d φ d ∂φ , we obtain = ∂u dt dt ∂u

d ∂φ ∂F2 ∂φ ∂F2 = + ∂y ∂u ∂u dt ∂u

i =1

Therefore we have ∂C 1 ∂µ ∂J T = I− ∂ui 2 ∂ui ∂ui

(B16)

Taking partial derivative with respect to u, we obtain ∂φ ∂F2 ∂φ ∂F2 = + ∂u ∂y ∂u ∂u

[5]

n

H (u ) = ∑ηi (u ) H i (u )

∂x0 ⎛ ∂F ⎞ ∂F1 (B15) = −⎜ 1 ⎟ ∂u ⎝ ∂x ⎠ ∂u To find the sensitivity of the trajectory φ (t , x0 , u ) , let us

recall that φ (t , x0 , u ) is the solution of (5), i.e.,

The nonsingularity of the matrix in (B6) can easily be established. The solution ∂η / ∂ui of (B6) provides the i-th column of the sensitivity matrix ∂η (B7) E (u ) = ∂u 3) Sensitivity of Q(u): The Q(u) matrix is the solution of the Lyapunov equation (21) (B8) C (u )Q (u ) + Q (u )C T (u ) = H (u )

By re-arrangement, we can see that

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[11]

[12]

[13]

M. A. Pai, Energy Function Analysis for Power System Stability. Boston: Kluwer Academic Publishers, 1989. A. A. Fouad and V. Vittal, Power System Transient Stability Analysis Using the Transient Energy Function Method. Englewood Cliffs, N. J. Prentice Hall, 1992. M.Ribbens-Pavella and P. G. Murthy, Transient Stability of Power systems: Theory and Practice. New York: Wiley, 1994. H. D. Chiang, C. C. Chu, and G. Cauley, “Direct stability analysis of electric power systems using energy functions: theory, applications and perspective,” Proceedings of the IEEE, vol. 83, no.11, pp.1497-1529, Nov. 1995. H. D. Chiang, M. W. Hirsch, and F. F. Wu, “Stability regions of nonlinear autonomous dynamical systems,” IEEE Transactions on Automatic Control, vol. 33, no. 1, pp.16-27, Jan. 1988. R. J. Kaye, F. F. Wu, “Dynamic security regions of power systems,” IEEE Transactions on Circuits & Systems, vol. CAS-29, no.9, pp.612-623, 1982. Y. Zeng, J. C. Fan, Y. X. Yu, F Lu, and Y. G. Fang, “Practical dynamic security regions of bulk power systems,” Automation of Electric Power Systems, vol. 25, no.16, pp.6-10, 2001. (In Chinese) Y. X. Yu, Y. Zeng, and F. Feng, "Differential topological characteristics of the DSR on injection space of electrical power system," Science in China, Series E, vol. 45, no.6, pp.576-584, 2002. Y. X. Yu, “Security region of bulk power system,”in Proceeding of the 2002 International Conference on Power System Technology, Kunming, China, vol.1, pp.13-17, 2002. Y. Zeng, Y. X. Yu, “A practical direct method for determining dynamic security regions of electrical power systems,”in Proceeding of the 2002 International Conference on Power System Technology, Kunming, China, vol. 1, pp. 1270-1274, 2002. A. R. Bergen and D. J. Hill, “A structure preserving model for power system stability analysis,” IEEE Trans. Power Apparatus Systems, vol. PAS-100, no.1, pp. 25-35, Jan. 1981. D. Z. Cheng, J. Ma, “Calculation of stability region,” in Proceedings of 2003 (42nd) IEEE Conference on Decision and Control, vol. 6, pp. 5615-5620, 2003. D. Z. Cheng, Matrix and Polynomial Approach to Dynamic Control System. Beijing: Science Press, 2002.

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < [14] H. Yee and B. D. Spading, “Transient stability analysis of multi-machine systems by the Method of Hyperplances,” IEEE Transactions on PAS, vol. PAS-96, pp. 276-284, 1977. [15] P. A. Cook and A. M. Eskicioglu, "Transient stability analysis of electric power systems by the method of tangent hypersurface,” IEE Proceedings. Part C, vol. 130, pp. 183-193, July 1983. [16] S. Ushiki, “Analytic expression of the unstable manifolds,” Proc. Japan Acad. 56, Ser. A, pp. 239-244, 1980. [17] F. M. S. Faithi, A. Arapostathis, and P. P. Varaiya, “Analytic expressions for the unstable manifold at equilibrium in dynamical systems of differential equations,”in Proceedings of the 22nd IEEE Conference on Decision and Control, vol. 3, pp. 1389-1392, 1983. [18] S. Saha, A. A. Fouad, W. H. Kliemann, and V. Vittal, “Stability boundary approximation of a power system using the real normal form of vector fields,” IEEE Transactions on Power Systems, vol. 12, no. 2, pp.797-802, May 1997. [19] V. Venkatasubramanian, Weijun JI, “Numerical approximation of (n-1) dimension stable manifold in large systems such as power systems,” Automatica, vol. 33, no. 10, pp. 1877-83, 1997. [20] H. D. Chiang, F. F. Wu, and P. P. Varaiya, “Foundation of direct methods for power systems transient stability analysis,” IEEE Transactions on Circuits and systems, vol. CAS-34, no.2, pp.160-173, Feb. 1987. [21] P. Kundur, Power System Stability and Control. New York: Mc-Graw-Hill, 1992. [22] P. M. Anderson and A. A. Fouad. Power System Control and Stability. The Iowa State University Press, Ames, Iowa, U.S.A. 1977. (Revised Printing, IEEE Press, 1994. (p.375)) [23] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990. [24] A. Xue, F. F. Wu, Y. Ni, Q. Lu, S. Mei, “ Power system transient stability assessment based on the quadratic approximation for the stability region,”in Proc. IEEE PES Asian-Pacific T&D Conference, Dalian, China, June 2005.

Ancheng Xue was born in Jiangsu, China, in Apr. 1979. He received his B. Sc degree in applied mathematics from the Department of Mathematics, Tsinghua University in 2001 and Ph. D. Degree in electrical engineering from Department of Electrical Engineering, Tsinghua University in 2006. He is currently with the Institute of System Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include the nonlinear systems and their applications to the power systems. Felix F. Wu (SM’86– F’89) received his Ph. D from University of California at Berkeley (UCB). He is currently Philip Wong Wilson Wong Professor in Electrical Engineering, at the University of Hong Kong, and Professor Emeritus, Department of Electrical Engineering and Computer Sciences at University of California, Berkeley. Qiang Lu (SM’85– F’02) graduated from the Graduate School of Tsinghua University, China, in 1963 and joined the faculty of the same University. He was a visiting scholar and a visiting professor in Washington University, St. Louis and Colorado State University, Ft. Collins, respectively in 1984-1986 and a1993-1995. He is now a professor in Tsinghua University, and an academician of Chinese Academy of Science (1991). His research interest is in nonlinear control theory applications in power systems. Shengwei Mei (M’99) was born in Xinjiang, China, on Sept. 20, 1964. He received his B.S. degree in mathematics from Xinjiang University, M.S. degree in operations research from Tsinghua University, and Ph.D. degree in automatic control from Chinese Academy of Sciences, Beijing, in 1984, 1989, and 1996 respectively. He is now a professor of Tsinghua University. His current interest is in control theory applications in power systems.

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