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Stability analysis for digital controls of power systems
Electric Power Systems Research 55 (2000) 79 – 86 www.elsevier.com/locate/epsr
Stability analysis for digital controls of power systems Luonan Chen a,*, Hideya Tanaka b, Kazuo Katou c, Yoshiyuki Nakamura c a
Department of Electrical Engineering and Electronics, Osaka Sangyo Uni6ersity, Nakagaito 3 -1 -1, Daito, Osaka 574, Japan b KCC Ltd., Fukagawa 2 -2 -18, Koto-Ku, Tokyo 135, Japan c Electric Power De6elopment Co. Ltd., Ginza 6 -15 -1, Chuo-Ku, Tokyo 104, Japan Received 29 December 1998; received in revised form 28 June 1999; accepted 16 August 1999
Abstract The digital control systems of power grids are typical hybrid dynamical models or sampled data models which include not only differential–difference equations but also algebraic equalities. Therefore, they can be formulated as differential – difference–algebraic equation systems, which are generally nonlinear. This paper aims at analyzing the digital controls of power systems in a nonlinear manner. We first define a power system with digital controllers as a hybrid dynamical system, and then give a general analyzing methodology for the steady-state stability of digital controls in power systems with a special emphasis on the digital PSS. Numerical simulations for small power systems have verified our theoretical results. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Digital control; Hybrid dynamical system; Asymptotical stability; Bifurcation; Singular perturbation; PSS
1. Introduction It has been recognized that digital control provides various advantages over the usual analog feedback controls [11], and several kinds of digital controlled devices, such as digital AVR and digital PSS, have been put into practical use in power systems for the last decade. Power systems with digital devices yield mathematical descriptions with not only ordinary differential – algebraic equations, but also difference equations. For instance, the dynamics of the generators as well as their continuous-time controllers and the load dynamics together define the ordinary differential equations, while algebraic equalities are defined by the power balance equations of the transmission network [5]. On the other side, the difference equations govern the dynamics of the digital devices. Therefore, the digital controlled system of power grids is a hybrid dynamical model or a sampled-data model, which can mathematically be formulated as differential – difference – algebraic equations (DDA) where all of the states are continuous. * Corresponding author. Tel.: +81-728-753001; fax: + 81-728708189. E-mail address: [email protected] (L. Chen)
So far, both continuous-time and discrete-time nonlinear systems have attracted considerable attention and a variety of techniques have been developed [1–5,7]. In contrast, however, the researches for the hybrid systems are mainly for the systems defined by both differential equations with continuous states and logical-discreteevent equations with discrete states [12–16]. On the other hand, the sampled-data models or differential– difference equations where all of the states are continuous are investigated mostly for the linear systems [17–19]. Less attention has been focused on the nonlinear analysis of the DDA or the hybrid dynamical systems where the differential and the difference equations not only have continuous states but also are constrained by algebraic equations. This paper aims at analyzing the asymptotical stability of the digital controls in power systems with a special emphasis on the digital PSS, in a nonlinear manner. In other words, we intend to treat power systems with digital controllers as nonlinear hybrid dynamical systems so that the power systems can be analyzed in a more exact way, compared to the conventional linear analysis for digital controls (e.g. operation point linearized model, or small signal analysis). We first address the notation and give definitions of the
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equilibrium for general hybrid dynamical systems, and then describe three theorems for asymptotical stability, singular perturbation, and Neimark – Sacker bifurcation [8]. Finally, the theoretical results are applied to the analysis of the digital PSS in a one-machine – infinitebus power system with four buses, and numerical simulations have verified that our theoretical results are effective.
x (t) is continuous due to Eq. (1) while z(t) is generally discontinuous at t= nt, n= 0, 1, 2, …, namex(t) and lim ly lim x(t)=x(nt)= xn = t lim t nt − 0 nt + 0 t nt − 0 z(t)" z(nt)=zn = t lim z(t) as far as hz " 0 and nt + 0 hz + hzn " 0. In this paper, sampling instants mean t=nt while intersampling instants imply ntB tB (n+ 1)t where n= 0, 1, 2, …. Therefore, we have the following definition.
2. Hybrid dynamical system
Definition 2.1. Vectors p¯ = (x¯, x¯, y¯n, z¯, z¯ ) are called a equilibrium of hybrid dynamical system Eqs. (1)–(3) if the following equations hold
The following notation is used throughout the paper [8 – 10]. Let
f(x¯, x¯n, y¯n, z¯, z¯n )= 0
x; (t) = f(x(t), xn, yn, z(t), zn )
(1)
yn + 1 = g(xn, yn, zn )
(2)
g(x¯n, y¯n, z¯n )= y¯n
h(x(t), xn, yn, z(t), zn ) =0
(3)
h(x¯, x¯n, y¯n, z¯, z¯n )= 0
(nt 5tB(n+1)t;
n = 0, 1, 2, …)
be the hybrid dynamical system under study, where vectors x, xn, yn, z and zn belong to the Euclidean spaces Rnx, Rnx, Rny, Rnz and Rnz, respectively. f:Rnx + nx + ny + nz + nz Rnx, g:Rnx + ny + nz Rny and h:Rnx + nx + ny + nz + nz Rnz are all assumed to be functions of class C 1, which means that their partial derivatives of the first order with respect to x, xn, yn, z, zn exist and are continuous. t is a sampling interval (nonnegative real number). Define xn =x(nt), zn =z(nt) and yn is the value at the instant nt, where xn, yn and zn are constantly held during nt 5t B(n + 1)t for Eqs. (1) and (3). Then Eqs. (1) – (3) are an ordinary differential– difference–algebraic equation model (DDA). Let fx = (f(x, xn, yn, z, zn )/(x and fx is the determinant of fx. eA is the exponential function of a matrix A. Although Eqs. (1) – (3) are an autonomous DDA model with both continuous and discrete time, all variables take continuous values, which are different from the conventional hybrid systems [12 – 16] where the states of discrete-time equations are discrete. In other words, the conventional hybrid system is generally defined as a model which consists of logical discrete event equations (with discrete states) coupled with differential equations (with continuous states) [14–16], while the hybrid dynamical system defined in Eqs. (1) – (3) is composed of difference equations (Eq. (2)) (with continuous states) coupled with differential equations (Eq. (1)) (with continuous states). Furthermore, the hybrid dynamical system in this paper is also constrained by the algebraic equations Eq. (3). In addition, from a continuous-time viewpoint, the hybrid dynamical system of Eqs. (1) – (3) can be viewed as a continuous time-varying system of Eq. (1) constrained on the manifold of Eq. (3), with time-varying parameters yn which change values at the instants t = nt due to Eq. (2).
Obviously, a stable equilibrium is more desirable for power systems because all variables are to be kept stably constant. Next, without confusion, t is occasionally dropped from the related variables, e.g. x(t) or z(t) is simply expressed by x or z. A significant characteristic of the hybrid dynamical system is that states vary continuously during the intersampling period for all variables but change discretely at the sampling instants for a part of variables. In other words, the system may have considerably different features at sampling and intersampling instants, in contrast to the continuous-time autonomous system whose analysis at equilibria is almost independent of instants.
3. Fundamental theorems This section gives three fundamental theorems for the analysis of the asymptotical stability, the singular perturbation and the Neimark–Sacker bifurcation, which are all proved in [8,9].
3.1. Asymptotical stability Assume pˆ = (xˆn, xˆn, yˆn, zˆn, zˆn ) satisfying Eq. (3) and hz + hzn pˆ " 0. Then, according to the implicit function theorem there exists a unique C mapping in open neighborhoods of pˆ such that at sampling instants zn = pn (xn, yn ).
(4)
Furthermore, let pˆ satisfy Eq. (3) and hz pˆ " 0. Then, again there exists a unique C mapping in open neighborhoods of pˆ such that at the intersampling instants z= p(x, xn, yn )
nt 5tB (n+1)t
(5)
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where pn (xn, yn )= p(xn, xn, yn ).Thus, the reduced system of Eqs. (1) and (3) at the neighborhood of pˆ becomes x; = fR(x, xn, yn ) f(x, xn, yn, p(x, xn, yn ), pn (xn, yn )). (6) We can prove that Eq. (6) can be expanded at the neighborhood of p¯ in the form of xn + 1 = x¯ +[e
tA
+ (e
tA
−E)A
−1
B, (e
tA
−E)A
C]
n
xn −x¯ +… yn −y¯
(7)
J= etA +(etA −E)A − 1B gxn −gzn (hz +hzn )
−1
(hx +hxn )
3.2. Singular perturbation theory Let us consider the numerical calculation of the trajectory for the hybrid dynamical system. Assume hz " 0 and hz + hzn " 0, then by the implicit function theorem, Eqs. (1)–(3) are locally equivalent to the following equations at nt5 tB(n+ 1)t x; (t)=f(x(t), xn, yn, x(t), zn )
(8)
yn + 1 = g(xn, yn, zn )
(9)
−1
Then by combining Eq. (7) with Eq. (2), we can have an equivalent discrete system of the original Eqs. (1)– (3) [8,9]. Assume A "0 and hz "0, hz +hzn "0 at p¯. Define a (nx +ny )× (nx +ny ) Jacobian matrix J of Eqs. (7) and (2) as,
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(etA −E)A − 1C gyn −gzn (hz +hzn ) − 1hyn
n
where −1 1 −1 A = fx −fzh − z hx, B = fxn −fzh z [hxn −hzn (hz +hzn ) −1 −1 hxn ] − fzn (hz + hzn ) (hx +hxn ), C =fyn −fzh z [hyn − hzn (hz +hzn ) − 1hyn ]− fzn (hz +hzn ) − 1hyn.
Actually, J is a Jacobian matrix of the equivalent system derived from the discretized Eqs. (7) and (2) [9], which reins the stability of the original system Eqs. (1) – (3). Theorem 3.1. Assume that p¯ =(x¯, x¯, y¯, z¯, z¯ ) is an equilibrium of hybrid dynamical system Eqs. (1) – (3), and hz p¯ "0, hz +hzn p¯ " 0. 1. If all the eigen6alues of J at p¯ ha6e moduli less than 1 and all the eigen6alues of A are negati6e then p¯ is an asymptotically stable equilibrium of hybrid dynamical system Eqs. (1) – (3). 2. If any one of the eigen6alues of J at p¯ has modulus more than 1 or any one of the eigen6alues of A is positi6e, the p¯ is an unstable equilibrium of hybrid dynamical system Eqs. (1) – (3).
In this paper, we only consider the asymptotical stability of the hyperbolic systems where no eigenvalue of their Jacobian matrix has modulus 1. For the nonhyperbolic systems, the Neimark – Sacker bifurcation is analyzed in Section 3.3. In addition, the nonlinear discretization of the hybrid dynamical system and generic bifurcations as well as the sufficient conditions of stability for quasi-equilibrium are given in [8,9].
1 z; (t)= − h − z hx f(x(t), xn, yn, z(t), zn ) with h(xn, xn, yn, zn, zn )= 0
(10)
where at t= (n+1)t, xn + 1 is equal to x((n +1)t) due to continuation of x(t), yn + 1 is equal to g(xn, yn, zn ) from Eq. (9), and zn + 1 is calculated by solving h(xn + 1, xn + 1, yn + 1, zn + 1, zn + 1)= 0 due to Eq. (4). The stability of an equilibrium of Eqs. (8)–(10) is the same as that of Eqs. (1)–(3), and the trajectory of the hybrid dynamical system can be numerically calculated from Eqs. (8)–(10) by the Runge–Kutta method, the Euler method etc. However, for the case where the trajectory intersects the surfaces {(x, xn, yn, z, zn ) hz = 0}or {(xn, xn, yn, zn, zn ) hz + hzn = 0}, the implicit function theorem does not apply and thereby Eqs. (8)–(10) cannot be used in the numerical calculations [6]. To overcome this problem, next we use singular perturbation theory to derive a new model. For the hybrid dynamical system Eqs. (1)–(3), we define an associated singularly perturbed system as x; (t)= f(x(t), xn, yn, x(t), zn )
(11)
yn + 1 = g(xn, yn, zn )
(12)
ez; (t)= h(x(t), xn, yn, z(t), zn ) (nt 5 tB(n+1)t; n= 0, 1, …)
(13)
where e is a sufficiently small positive number, i.e. the algebraic Eq. (3) corresponds to a limit of the fast dynamics Eq. (13). Obviously, Eq. (13) will approach the algebraic manifold when e 0. It is evident that Eqs. (11)–(13) have the same equilibrium as Eqs. (1)– (3) as far as h= 0. Theorem 3.2. Assume that p¯ = (x¯, x¯, y¯, z¯, z¯ ) is an equilibrium of the hybrid dynamical system Eqs. (1) – (3), and all real parts of eigen6alues for hz p¯ are negati6e (if all real parts of eigen6alues for hz are positi6e, change equation h= 0 into −h= 0 such that all real parts of eigen6alues for − hz are negati6e). Furthermore, assume that no eigen6alue of J at p¯ has the modulus 1 and all 1 eigen6alues of h − ¯ ha6e moduli less than 1, and z hznat p hz + hzn p¯ " 0, A p¯ " 0. Then there exists e¯ \0 such that for all positi6e eBe¯ stability of Eqs. (11) – (13) at p¯ is identical to that of Eqs. (1) – (3).
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Therefore, we can numerically simulate the hybrid dynamical system even if its trajectory may intersect the singular surfaces except the equilibrium p¯, by Eqs. (11) –(13) according to Theorem 3.2.
The Neimark–Sacker bifurcation generates oscillations which make the system unstable [7]. In Section 4, we use the sampling interval and the output of generator as the parameters to examine the stability and bifurcations.
3.3. Neimark –Sacker bifurcation When the eigenvalues of J have the moduli 1 or the real parts of the eigenvalues of A are zero, there may exist the bifurcations at p¯ of Eqs. (1) – (3) with the parameters changing. Furthermore, if hz has zero eigenvalues, the system may undergo bifurcations due to the singularity. Actually, we can show that there are five bifurcations under generic conditions for an equilibrium [9,10], i.e. fold, flip, Neimark – Sacker and singularity-2 induced bifurcations at the sampling instants, and one independent bifurcations, i.e. singularity-1 induced bifurcations at the intersampling instants, in contrast to two types at a equilibrium in continuous-time dynamical systems and three types at a fixed point in discrete-time dynamical systems. Since the Neimark – Sacker bifurcations are mostly observed in the simulations of a power system with digital controllers shown in Section 4, next we only summarize the conditions of the Neimark – Sacker bifurcation. The conditions for other four generic bifurcations are summarized in [9]. Theorem 3.3. Let a: R R be a bifurcation parameter of Eqs. (1) – (3). Assume that p¯ =(x¯, x¯, y¯, z¯, z¯ )is an equilibrium of hybrid dynamical system Eqs. (1) – (3) at a¯ , and hz , hz +hzn and A p¯ "0. Furthermore, at the equilibrium and for all sufficiently small a , assume that only a pair of eigen6alues of J(a) ha6e the form of l= r(a −a¯ )e 9 jf(a − a¯ ) where r(a¯ ) =1 for all sufficiently small a− a¯ and there is no other eigen6alue with modulus 1. If dr(a¯ )/da " 0 and e jkf(a¯ ) "1 for k = 1, 2, 3, 4 then there exists the Neimark– Sacker bifurcation which generates a unique closed in6ariant cur6e from the equilibrium point of Eqs. (1) – (3).
4. Digital PSS We show how a power system with digital PSS can be treated as a hybrid dynamical system in this section. For a power system, the dynamics of the generators and analog controllers (e.g. AVR) is defined as Eq. (1) of the hybrid dynamical system, while Eq. (3) is composed of the power flow balance equations. Then Eq. (2) corresponds to the discrete-time controllers (or digital PSS). In this paper, the parameters of the digital PSS are first designed by imitating the conventional analog PSS, and then adjusted and evaluated by using Theorem 3.1, Theorem 3.2 and Theorem 3.3. Next in order to obtain the digital PSS by imitating the designing of analog PSS, we derive Eq. (2) from an analog PSS with a hold circuit and a sampler. Generally, this process is not necessary if a digital PSS or Eq. (2) is given. Fig. 1 shows the basic diagram of the digital PSS, where H and S stand for the hold circuit and the sampler, respectively. Generally, the digital PSS consists of a sampler, a discrete-time controller and a hold circuit. In Fig. 1, ‘hold circuit control device sampler’ is the discrete-time controller. In this paper, we adopt three-order digital PSS with a zero-order hold circuit which is obtained by discretizing the analog PSS shown in Fig. 2, i.e. ‘control device’. The state y(t)eR3 and input un eR of the PSS follow y; (t)= D1y(t)+D2un un = u(xn, zn )= u(x(nt), z(nt)) (nt 5 tB (n+ 1)t; n =0, 1, …)
(14)
and the output un eR of the digital PSS can be expressed as 6(t)= Á6max if 6(t)]6max Â Ã Ã Í −(T3T5/T2T4)y1(t) +(T5/T4)y2(t) +y3(t) +(KT3T5/T1T2T4)un Ì Ã Ã Ä6min if 6(t) 5 6min Å Fig. 1. Diagram of a digital PSS.
6n = 6(nt) (nt 5tB (n+1)t; n= 0, 1, …),
Fig. 2. A three-order control device (corresponding to ‘control device’ of Fig. 1).
(15)
where, y(t)= (y1(t), y2(t), y3(t)/y(t))T, uR is the real power output of the generator which is also a given function of (x, z), 6 R (or 6n R) is the PSS output,
L. Chen et al. / Electric Power Systems Research 55 (2000) 79–86
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during nt5 tB (n+ 1)t for Eqs. (18) and (19) because 6n is constantly held during that period. Eqs. (18) and (19) are generally nonlinear. Therefore, combining Eqs. (16), (18) and (19), we can use Theorem 3.1, Theorem 3.2 and Theorem 3.3 to analyze the power system with the digital PSS numerically in a nonlinear way. Actually in Fig. 1, ‘ hold circuit control device sampler’ is a discrete-time controller for which both the input and output are impulse signals, and whose dynamics of state variables correspond to Eq. (16). Fig. 3. A power system model with a digital PSS (corresponding to Fig. 1).
5. Numerical examples and 6max, 6min are upper and lower bounds for the internal output of the PSS, un and 6n are the values of u(t) and 6(t) at instant t =nt, respectively, which are constantly hold during nt5 t B(n + 1)t. Define D1 and D2 as follows Æ −1/T1 Ã D1 = Ã −(T2 − T3)/T 22 Ã 2 È − (T3/T2)(T4 − T5)/T 4
0 − 1/T2 (T4 − T5)/T4
0 Ç Ã 0 Ã, Ã − 1/T4 É
Ç Æ 1/T 21 Ã Ã 2 D2 = Ã (K/T1)(T2 − T3)/T 2 Ã. Ã 2Ã È(KT3/T1T2)(T4 − T5)/T 4 É
Since Eq. (14) is a linear differential equation, it can be analytically solved. Therefore, we have following difference equation by integrating Eq. (15) from nt to (n+ 1)t, 1 D1t yn + 1 = eD1tyn + D − −E]D2u(xn, zn ) 1 [e
(16)
which has the same form with Eq. (2). That is, Eq. (16) is the discrete-time controller whose design imitates that of the continuous-time controller of Fig. 2, i.e. a threeorder analog PSS. Note that the real power output u(xn, zn ) of the generator is a given function of (xn, zn ). The PSS output 6n is obviously a function of yn and u(xn, zn ) according to Eq. (15), i.e. 6n = hn (xn, yn, zn ) h(yn, u(xn, zn ))
(17)
where 6n is not a linear function because of the saturation 6min, 6max of 6(t) shown in Eq. (15). On the other hand, the dynamics of generators and analog controllers corresponding to Eq. (1), the power flow balance equations corresponding to Eq. (3) can be written as follows x; (t) = f(x(t), hn (xn, yn, zn ), z(t))
(18)
h(x(t), hn (xn, yn, zn ), z(t)) = 0.
(19)
Note that xn, yn and zn can be viewed as constants
A power system with four buses and one generator shown in Fig. 3 (corresponding to Fig. 1) is used to illustrate the application, where bus-4 is an infinite-bus with constant voltage.The generator-1 is modeled by five ordinary differential equations, and automatic voltage regulator (AVR) of generator-1 is defined by three differential equations. The parameters of generator-1 and AVR are shown in Figs. 3 and 4, respectively. The designed digital power system stabilizer (PSS) is governed by three difference equations, i.e. Eq. (16). Besides, there are eight algebraic equations corresponding to four buses. Therefore, this model is a hybrid dynamical system including eight continuous-time and three discrete-time dynamical equations as well as eight algebraic equalities. Different from the continuous systems, the hybrid dynamical systems has a sampling interval t which may significantly affect the stability of the systems under certain circumstances, such as long transmission systems, heavy load conditions, etc. In addition, the stability may also be highly dependent on the parameters and sampling interval of digital PSS for the system with ultra high-response exciters because only AVR is generally not sufficient to stabilize the system. Next, we first use the sampling interval t and the output of generator1 (corresponding to power flow in transmission lines) as parameters to investigate their influences on the steadystate stability of the power system of Fig. 3, and then examine how the parameters of PSS affect the stability of the system. Notice that the power is mainly supplied from G1 to the load through a long transmission line (280 km) with voltage level 270 KV, and the open-loop poled of Fig. 3 are 0.00699j6.8707, − 2.90429 j3.4210, − 8.14708, − 10.4801, − 75.95199 j74.3825 where the first pair of poles are unstable poles. Using the sampling interval to study the stability is motivated by the design and field test of a digital PSS for Matsuura No. 2 generator in early 1997 which is a 1000 MW cross-compound turbine-generator. The PSS was originally designed by taking the discrete-time co
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ntroller approximately as the linear continuous-time one due to small sampling interval. During the test, however, it has been found that the PSS failed to stabilize the actual system even although the simulation by this model showed the system stable. One of the
main reasons is that the sampling interval (or time delay) and linearization significantly affects the dynamics of the concerned system even for tB 20 ms, which indicates the t as well as other PSS parameters should be carefully decided by using more exact model (such as the hybrid dynamical system) when designing a digital PSS [20]. The paper in reference [20] gives a detail report on the actual field testing of the digital PSS for Matsuura No. 2 generator. Certainly there are many reasons to cause power system unstable. In this section, we only focus on the sampling interval and output of generator-1 to study their influences on the dynamics of the power system although many other parameters, e.g. transformer tap, gains etc. may also affect the system stability.
5.1. Case-1 of digital PSS
Fig. 4. Diagram and parameters of an analog AVR.
Fig. 5. Stable and unstable regions according to numerical simulation for case-1.
For case-1, the digital PSS is designed as K=0.6, T1 = 1.0, T2 = 0.45, T3 = 7.5, T4 = 0.85, T5 =0.01, 6min = − 0.1, 6max = 0.1. For the numerical computation, x(t) and xn are calculated from numerical integration of Eq. (18) by the Euler or the Runge–Kutta methods, while z(t) and zn are obtain from Eq. (19) by the Newton method for intersampling and sampling instants, respectively. On the other hand, yn is computed by simply updating Eq. (16). Fig. 5 indicates the results of numerical simulation of the hybrid dynamical system by four-order Runge– Kutta method with increment Dt = 1 ms for Eq. (18), while Fig. 6 is the results obtained directly from Theorem 3.1. Obviously, the theoretical results coincide with those of the numerical simulation, which verifies the effectiveness of Theorem 3.1. From Fig. 6 or Fig. 5, it is evident that the system is apt to be unstable with the increasing t. For instance, when the real power of generator-1 is 1.0 p.u. the power system is stable if tB 9.2. However, at t: 9.2 ms, the system undergoes the Neimark–Sacker bifurcation shown in Fig. 7, so that it becomes unstable if t\ 9.2 ms. Actually, it has numerically been found that the boundary of stability in Fig. 6 are mainly composed of the points of the Neimark–Sacker bifurcation of Theorem 3.3 for this case. For the computation of the trajectory, Theorem 3.2 is used to undertake numerical calculations to overcome the singular surfaces.
5.2. Case-2 of digital PSS
Fig. 6. Stable and unstable regions according to Theorem 3.1 for case-1.
Next we modify the parameters of the digital PSS, i.e. let K= 0.28, T1 = 0.23, T2 = 0.11, T3 = 0.16, T4 = 1.18, T5 = 0.7, 6min = − 0.1, 6max = 0.1, to examine its performance. Fig. 8 shows the stable and unstable regions according to Theorem 3.1. Obviously, the stable
L. Chen et al. / Electric Power Systems Research 55 (2000) 79–86
Fig. 7. A Neimark–Sacker bifurcation for case-1.
85
to deal with nonlinear power systems in a more exact way. Then we applied these theorems to the analysis of a small power system with a digital PSS. Numerical simulations have verified that our theoretical results are useful and effectiveness for the analysis of the digital controls in power systems. It has been found that the sampling interval may significantly affect the stability of power system high-response excitation system, etc. As future works, more detail simulation for multimachine systems should be undertaken to investigate the basic characteristics of the digital controlled power systems.
7. List of symbols x(t),xn yn z(t), zn
Fig. 8. Stable and unstable regions according to Theorem 3.1 for case-2.
region has been considerably enlarged, compared to Fig. 6, as far as the real power of generator-1 is less than 1.18 p.u. and the sampling interval is taken as a changing parameter. It is interesting to notice that when the real power of generator-1 is more than 1.18 p.u., the power system is unstable even if t = 0 i.e. it is not stable even for the analog PSS. However, if we increase t, the system may become stable when 1.34 \ u\ 1.18 p.u., where u is the real power of generator-1. This case denotes that increasing sampling interval may enhance the asymptotical stability of the hybrid dynamical systems under certain circumstances. The results of case-1 and case-2 also show that the stability of the system is dependent on the parameters and sampling interval of the PSS for this long transmission line system. On the other hand, Fig. 8 also indicates that the stability generally deteriorates with the increase of the output of generator-1. Besides, we studied a system with digital PSS under heavy load conditions, which also show the similar results.
6. Conclusion We first described several theoretical results for the asymptotical stability, singular perturbation and the bifurcation for the hybrid dynamical systems, in order
f C 1 gC 1 hC 1 t fx A
· E eA
nx dimensional vectors Rnx at instant t and nt, respectively ny dimensional vector Rny at instant nt nz dimensional vectors Rnz at instants t and nt, respectively Rnx+nx+ny+nz+nz Rnx Rnx+ny+nz Rny Rnx+nx+ny+nz+nz Rnz a sampling interval (positive real number) (f(x, xn, yn, z, zn )/(x the determinant of a square matrix A the usual Euclidean norm of a vector or the induced matrix norm of a matrix an identity matrix with appropriate dimensions an exponential function of a matrix A
Acknowledgements This research was partially supported by both Special Science Fund of Osaka Sangyo University of Japan and National Key Basic Research Special Fund of China (No. G1998020304).
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[6] H.D. Chiang, C.C. Chu, G. Cauley, Direct stability analysis of electric power systems using energy functions, Proc. IEEE 83 (11) (1995) 1497 – 1529. [7] L. Chen, K. Aihara, Chaos and asymptotical stability in discretetime neural networks, Physica D 104 (1997) 286–325. [8] L. Chen, K. Aihara, Local stability in hybrid dynamical systems, International Symposium on Nonlinear Theory and its Applications (NOLTA’97), 1997, pp. 73–76. [9] L. Chen, K. Aihara, Stability and bifurcation analysis of hybrid dynamical systems, Technical Reports, METR 98-02, The University of Tokyo, 1998. [10] L. Chen, K. Aihara, Globally searching ability of chaotic neural networks, IEEE Trans. on Circuits and Systems, part I, 46 (8) (1999), 974 – 993. [11] Y. Yamamoto, A function space approach to sampled data control systems and tracking problems, IEEE Trans. AC 39 (4) (1994) 703 – 713. [12] A. Nerode, W. Kohn, Models for hybrid systems, Lect. Notes Comp. Sci. 736 (1993) 317–356. [13] S. Engell, I. Hoffmann, Modular hierarchiacal models of hybrid
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systems, The 35th Conference on Decision and Control, Kobe, Japan, 1996. A. Gollu, P.P. Varaiya, Hybrid dynamical systems, The 28th IEEE Conference on Decision and Control, Tampa, USA, 1989. R.L. Grossman, A. Nerode, A.P. Ravan, H. Rishel (Eds.), Hybrid Systems, Springer, New York, 1993. R.W. Brockett, Hybrid models for motion control systems, in: H.L. Trentelman, J.C. Willems (Eds.), Essays in Control, Birkhauser, Boston, 1993, pp. 29 – 53. P.M. Thompson, G. Stein, M.A. Athans, Conic sectors for sampled-data feedback systems, Syst. Control Lett. 3 (1983) 77 – 82. M. Morari, E. Zafiriou, Robust Process Control, Prentice-Hall, Englewood Cliffs, NJ, 1989. T. Chen, B.A. Francis, Input – output stability of sampled-data systems, IEEE Trans. Autom. Control 36 (1) (1991) 50–58. Y. Kida, T. Okabe, Y. Shimura, Design and the field test of a digital AVR/PSS for Matsuura No. 2 Generator, Tech. Meeting Papers on Power Engineering, IEE Japan, PE-97-68, (1997) pp. 31-36.
Stability analysis for digital controls of power systems Luonan Chen a,*, Hideya Tanaka b, Kazuo Katou c, Yoshiyuki Nakamura c a
Department of Electrical Engineering and Electronics, Osaka Sangyo Uni6ersity, Nakagaito 3 -1 -1, Daito, Osaka 574, Japan b KCC Ltd., Fukagawa 2 -2 -18, Koto-Ku, Tokyo 135, Japan c Electric Power De6elopment Co. Ltd., Ginza 6 -15 -1, Chuo-Ku, Tokyo 104, Japan Received 29 December 1998; received in revised form 28 June 1999; accepted 16 August 1999
Abstract The digital control systems of power grids are typical hybrid dynamical models or sampled data models which include not only differential–difference equations but also algebraic equalities. Therefore, they can be formulated as differential – difference–algebraic equation systems, which are generally nonlinear. This paper aims at analyzing the digital controls of power systems in a nonlinear manner. We first define a power system with digital controllers as a hybrid dynamical system, and then give a general analyzing methodology for the steady-state stability of digital controls in power systems with a special emphasis on the digital PSS. Numerical simulations for small power systems have verified our theoretical results. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Digital control; Hybrid dynamical system; Asymptotical stability; Bifurcation; Singular perturbation; PSS
1. Introduction It has been recognized that digital control provides various advantages over the usual analog feedback controls [11], and several kinds of digital controlled devices, such as digital AVR and digital PSS, have been put into practical use in power systems for the last decade. Power systems with digital devices yield mathematical descriptions with not only ordinary differential – algebraic equations, but also difference equations. For instance, the dynamics of the generators as well as their continuous-time controllers and the load dynamics together define the ordinary differential equations, while algebraic equalities are defined by the power balance equations of the transmission network [5]. On the other side, the difference equations govern the dynamics of the digital devices. Therefore, the digital controlled system of power grids is a hybrid dynamical model or a sampled-data model, which can mathematically be formulated as differential – difference – algebraic equations (DDA) where all of the states are continuous. * Corresponding author. Tel.: +81-728-753001; fax: + 81-728708189. E-mail address: [email protected] (L. Chen)
So far, both continuous-time and discrete-time nonlinear systems have attracted considerable attention and a variety of techniques have been developed [1–5,7]. In contrast, however, the researches for the hybrid systems are mainly for the systems defined by both differential equations with continuous states and logical-discreteevent equations with discrete states [12–16]. On the other hand, the sampled-data models or differential– difference equations where all of the states are continuous are investigated mostly for the linear systems [17–19]. Less attention has been focused on the nonlinear analysis of the DDA or the hybrid dynamical systems where the differential and the difference equations not only have continuous states but also are constrained by algebraic equations. This paper aims at analyzing the asymptotical stability of the digital controls in power systems with a special emphasis on the digital PSS, in a nonlinear manner. In other words, we intend to treat power systems with digital controllers as nonlinear hybrid dynamical systems so that the power systems can be analyzed in a more exact way, compared to the conventional linear analysis for digital controls (e.g. operation point linearized model, or small signal analysis). We first address the notation and give definitions of the
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equilibrium for general hybrid dynamical systems, and then describe three theorems for asymptotical stability, singular perturbation, and Neimark – Sacker bifurcation [8]. Finally, the theoretical results are applied to the analysis of the digital PSS in a one-machine – infinitebus power system with four buses, and numerical simulations have verified that our theoretical results are effective.
x (t) is continuous due to Eq. (1) while z(t) is generally discontinuous at t= nt, n= 0, 1, 2, …, namex(t) and lim ly lim x(t)=x(nt)= xn = t lim t nt − 0 nt + 0 t nt − 0 z(t)" z(nt)=zn = t lim z(t) as far as hz " 0 and nt + 0 hz + hzn " 0. In this paper, sampling instants mean t=nt while intersampling instants imply ntB tB (n+ 1)t where n= 0, 1, 2, …. Therefore, we have the following definition.
2. Hybrid dynamical system
Definition 2.1. Vectors p¯ = (x¯, x¯, y¯n, z¯, z¯ ) are called a equilibrium of hybrid dynamical system Eqs. (1)–(3) if the following equations hold
The following notation is used throughout the paper [8 – 10]. Let
f(x¯, x¯n, y¯n, z¯, z¯n )= 0
x; (t) = f(x(t), xn, yn, z(t), zn )
(1)
yn + 1 = g(xn, yn, zn )
(2)
g(x¯n, y¯n, z¯n )= y¯n
h(x(t), xn, yn, z(t), zn ) =0
(3)
h(x¯, x¯n, y¯n, z¯, z¯n )= 0
(nt 5tB(n+1)t;
n = 0, 1, 2, …)
be the hybrid dynamical system under study, where vectors x, xn, yn, z and zn belong to the Euclidean spaces Rnx, Rnx, Rny, Rnz and Rnz, respectively. f:Rnx + nx + ny + nz + nz Rnx, g:Rnx + ny + nz Rny and h:Rnx + nx + ny + nz + nz Rnz are all assumed to be functions of class C 1, which means that their partial derivatives of the first order with respect to x, xn, yn, z, zn exist and are continuous. t is a sampling interval (nonnegative real number). Define xn =x(nt), zn =z(nt) and yn is the value at the instant nt, where xn, yn and zn are constantly held during nt 5t B(n + 1)t for Eqs. (1) and (3). Then Eqs. (1) – (3) are an ordinary differential– difference–algebraic equation model (DDA). Let fx = (f(x, xn, yn, z, zn )/(x and fx is the determinant of fx. eA is the exponential function of a matrix A. Although Eqs. (1) – (3) are an autonomous DDA model with both continuous and discrete time, all variables take continuous values, which are different from the conventional hybrid systems [12 – 16] where the states of discrete-time equations are discrete. In other words, the conventional hybrid system is generally defined as a model which consists of logical discrete event equations (with discrete states) coupled with differential equations (with continuous states) [14–16], while the hybrid dynamical system defined in Eqs. (1) – (3) is composed of difference equations (Eq. (2)) (with continuous states) coupled with differential equations (Eq. (1)) (with continuous states). Furthermore, the hybrid dynamical system in this paper is also constrained by the algebraic equations Eq. (3). In addition, from a continuous-time viewpoint, the hybrid dynamical system of Eqs. (1) – (3) can be viewed as a continuous time-varying system of Eq. (1) constrained on the manifold of Eq. (3), with time-varying parameters yn which change values at the instants t = nt due to Eq. (2).
Obviously, a stable equilibrium is more desirable for power systems because all variables are to be kept stably constant. Next, without confusion, t is occasionally dropped from the related variables, e.g. x(t) or z(t) is simply expressed by x or z. A significant characteristic of the hybrid dynamical system is that states vary continuously during the intersampling period for all variables but change discretely at the sampling instants for a part of variables. In other words, the system may have considerably different features at sampling and intersampling instants, in contrast to the continuous-time autonomous system whose analysis at equilibria is almost independent of instants.
3. Fundamental theorems This section gives three fundamental theorems for the analysis of the asymptotical stability, the singular perturbation and the Neimark–Sacker bifurcation, which are all proved in [8,9].
3.1. Asymptotical stability Assume pˆ = (xˆn, xˆn, yˆn, zˆn, zˆn ) satisfying Eq. (3) and hz + hzn pˆ " 0. Then, according to the implicit function theorem there exists a unique C mapping in open neighborhoods of pˆ such that at sampling instants zn = pn (xn, yn ).
(4)
Furthermore, let pˆ satisfy Eq. (3) and hz pˆ " 0. Then, again there exists a unique C mapping in open neighborhoods of pˆ such that at the intersampling instants z= p(x, xn, yn )
nt 5tB (n+1)t
(5)
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where pn (xn, yn )= p(xn, xn, yn ).Thus, the reduced system of Eqs. (1) and (3) at the neighborhood of pˆ becomes x; = fR(x, xn, yn ) f(x, xn, yn, p(x, xn, yn ), pn (xn, yn )). (6) We can prove that Eq. (6) can be expanded at the neighborhood of p¯ in the form of xn + 1 = x¯ +[e
tA
+ (e
tA
−E)A
−1
B, (e
tA
−E)A
C]
n
xn −x¯ +… yn −y¯
(7)
J= etA +(etA −E)A − 1B gxn −gzn (hz +hzn )
−1
(hx +hxn )
3.2. Singular perturbation theory Let us consider the numerical calculation of the trajectory for the hybrid dynamical system. Assume hz " 0 and hz + hzn " 0, then by the implicit function theorem, Eqs. (1)–(3) are locally equivalent to the following equations at nt5 tB(n+ 1)t x; (t)=f(x(t), xn, yn, x(t), zn )
(8)
yn + 1 = g(xn, yn, zn )
(9)
−1
Then by combining Eq. (7) with Eq. (2), we can have an equivalent discrete system of the original Eqs. (1)– (3) [8,9]. Assume A "0 and hz "0, hz +hzn "0 at p¯. Define a (nx +ny )× (nx +ny ) Jacobian matrix J of Eqs. (7) and (2) as,
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(etA −E)A − 1C gyn −gzn (hz +hzn ) − 1hyn
n
where −1 1 −1 A = fx −fzh − z hx, B = fxn −fzh z [hxn −hzn (hz +hzn ) −1 −1 hxn ] − fzn (hz + hzn ) (hx +hxn ), C =fyn −fzh z [hyn − hzn (hz +hzn ) − 1hyn ]− fzn (hz +hzn ) − 1hyn.
Actually, J is a Jacobian matrix of the equivalent system derived from the discretized Eqs. (7) and (2) [9], which reins the stability of the original system Eqs. (1) – (3). Theorem 3.1. Assume that p¯ =(x¯, x¯, y¯, z¯, z¯ ) is an equilibrium of hybrid dynamical system Eqs. (1) – (3), and hz p¯ "0, hz +hzn p¯ " 0. 1. If all the eigen6alues of J at p¯ ha6e moduli less than 1 and all the eigen6alues of A are negati6e then p¯ is an asymptotically stable equilibrium of hybrid dynamical system Eqs. (1) – (3). 2. If any one of the eigen6alues of J at p¯ has modulus more than 1 or any one of the eigen6alues of A is positi6e, the p¯ is an unstable equilibrium of hybrid dynamical system Eqs. (1) – (3).
In this paper, we only consider the asymptotical stability of the hyperbolic systems where no eigenvalue of their Jacobian matrix has modulus 1. For the nonhyperbolic systems, the Neimark – Sacker bifurcation is analyzed in Section 3.3. In addition, the nonlinear discretization of the hybrid dynamical system and generic bifurcations as well as the sufficient conditions of stability for quasi-equilibrium are given in [8,9].
1 z; (t)= − h − z hx f(x(t), xn, yn, z(t), zn ) with h(xn, xn, yn, zn, zn )= 0
(10)
where at t= (n+1)t, xn + 1 is equal to x((n +1)t) due to continuation of x(t), yn + 1 is equal to g(xn, yn, zn ) from Eq. (9), and zn + 1 is calculated by solving h(xn + 1, xn + 1, yn + 1, zn + 1, zn + 1)= 0 due to Eq. (4). The stability of an equilibrium of Eqs. (8)–(10) is the same as that of Eqs. (1)–(3), and the trajectory of the hybrid dynamical system can be numerically calculated from Eqs. (8)–(10) by the Runge–Kutta method, the Euler method etc. However, for the case where the trajectory intersects the surfaces {(x, xn, yn, z, zn ) hz = 0}or {(xn, xn, yn, zn, zn ) hz + hzn = 0}, the implicit function theorem does not apply and thereby Eqs. (8)–(10) cannot be used in the numerical calculations [6]. To overcome this problem, next we use singular perturbation theory to derive a new model. For the hybrid dynamical system Eqs. (1)–(3), we define an associated singularly perturbed system as x; (t)= f(x(t), xn, yn, x(t), zn )
(11)
yn + 1 = g(xn, yn, zn )
(12)
ez; (t)= h(x(t), xn, yn, z(t), zn ) (nt 5 tB(n+1)t; n= 0, 1, …)
(13)
where e is a sufficiently small positive number, i.e. the algebraic Eq. (3) corresponds to a limit of the fast dynamics Eq. (13). Obviously, Eq. (13) will approach the algebraic manifold when e 0. It is evident that Eqs. (11)–(13) have the same equilibrium as Eqs. (1)– (3) as far as h= 0. Theorem 3.2. Assume that p¯ = (x¯, x¯, y¯, z¯, z¯ ) is an equilibrium of the hybrid dynamical system Eqs. (1) – (3), and all real parts of eigen6alues for hz p¯ are negati6e (if all real parts of eigen6alues for hz are positi6e, change equation h= 0 into −h= 0 such that all real parts of eigen6alues for − hz are negati6e). Furthermore, assume that no eigen6alue of J at p¯ has the modulus 1 and all 1 eigen6alues of h − ¯ ha6e moduli less than 1, and z hznat p hz + hzn p¯ " 0, A p¯ " 0. Then there exists e¯ \0 such that for all positi6e eBe¯ stability of Eqs. (11) – (13) at p¯ is identical to that of Eqs. (1) – (3).
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Therefore, we can numerically simulate the hybrid dynamical system even if its trajectory may intersect the singular surfaces except the equilibrium p¯, by Eqs. (11) –(13) according to Theorem 3.2.
The Neimark–Sacker bifurcation generates oscillations which make the system unstable [7]. In Section 4, we use the sampling interval and the output of generator as the parameters to examine the stability and bifurcations.
3.3. Neimark –Sacker bifurcation When the eigenvalues of J have the moduli 1 or the real parts of the eigenvalues of A are zero, there may exist the bifurcations at p¯ of Eqs. (1) – (3) with the parameters changing. Furthermore, if hz has zero eigenvalues, the system may undergo bifurcations due to the singularity. Actually, we can show that there are five bifurcations under generic conditions for an equilibrium [9,10], i.e. fold, flip, Neimark – Sacker and singularity-2 induced bifurcations at the sampling instants, and one independent bifurcations, i.e. singularity-1 induced bifurcations at the intersampling instants, in contrast to two types at a equilibrium in continuous-time dynamical systems and three types at a fixed point in discrete-time dynamical systems. Since the Neimark – Sacker bifurcations are mostly observed in the simulations of a power system with digital controllers shown in Section 4, next we only summarize the conditions of the Neimark – Sacker bifurcation. The conditions for other four generic bifurcations are summarized in [9]. Theorem 3.3. Let a: R R be a bifurcation parameter of Eqs. (1) – (3). Assume that p¯ =(x¯, x¯, y¯, z¯, z¯ )is an equilibrium of hybrid dynamical system Eqs. (1) – (3) at a¯ , and hz , hz +hzn and A p¯ "0. Furthermore, at the equilibrium and for all sufficiently small a , assume that only a pair of eigen6alues of J(a) ha6e the form of l= r(a −a¯ )e 9 jf(a − a¯ ) where r(a¯ ) =1 for all sufficiently small a− a¯ and there is no other eigen6alue with modulus 1. If dr(a¯ )/da " 0 and e jkf(a¯ ) "1 for k = 1, 2, 3, 4 then there exists the Neimark– Sacker bifurcation which generates a unique closed in6ariant cur6e from the equilibrium point of Eqs. (1) – (3).
4. Digital PSS We show how a power system with digital PSS can be treated as a hybrid dynamical system in this section. For a power system, the dynamics of the generators and analog controllers (e.g. AVR) is defined as Eq. (1) of the hybrid dynamical system, while Eq. (3) is composed of the power flow balance equations. Then Eq. (2) corresponds to the discrete-time controllers (or digital PSS). In this paper, the parameters of the digital PSS are first designed by imitating the conventional analog PSS, and then adjusted and evaluated by using Theorem 3.1, Theorem 3.2 and Theorem 3.3. Next in order to obtain the digital PSS by imitating the designing of analog PSS, we derive Eq. (2) from an analog PSS with a hold circuit and a sampler. Generally, this process is not necessary if a digital PSS or Eq. (2) is given. Fig. 1 shows the basic diagram of the digital PSS, where H and S stand for the hold circuit and the sampler, respectively. Generally, the digital PSS consists of a sampler, a discrete-time controller and a hold circuit. In Fig. 1, ‘hold circuit control device sampler’ is the discrete-time controller. In this paper, we adopt three-order digital PSS with a zero-order hold circuit which is obtained by discretizing the analog PSS shown in Fig. 2, i.e. ‘control device’. The state y(t)eR3 and input un eR of the PSS follow y; (t)= D1y(t)+D2un un = u(xn, zn )= u(x(nt), z(nt)) (nt 5 tB (n+ 1)t; n =0, 1, …)
(14)
and the output un eR of the digital PSS can be expressed as 6(t)= Á6max if 6(t)]6max Â Ã Ã Í −(T3T5/T2T4)y1(t) +(T5/T4)y2(t) +y3(t) +(KT3T5/T1T2T4)un Ì Ã Ã Ä6min if 6(t) 5 6min Å Fig. 1. Diagram of a digital PSS.
6n = 6(nt) (nt 5tB (n+1)t; n= 0, 1, …),
Fig. 2. A three-order control device (corresponding to ‘control device’ of Fig. 1).
(15)
where, y(t)= (y1(t), y2(t), y3(t)/y(t))T, uR is the real power output of the generator which is also a given function of (x, z), 6 R (or 6n R) is the PSS output,
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during nt5 tB (n+ 1)t for Eqs. (18) and (19) because 6n is constantly held during that period. Eqs. (18) and (19) are generally nonlinear. Therefore, combining Eqs. (16), (18) and (19), we can use Theorem 3.1, Theorem 3.2 and Theorem 3.3 to analyze the power system with the digital PSS numerically in a nonlinear way. Actually in Fig. 1, ‘ hold circuit control device sampler’ is a discrete-time controller for which both the input and output are impulse signals, and whose dynamics of state variables correspond to Eq. (16). Fig. 3. A power system model with a digital PSS (corresponding to Fig. 1).
5. Numerical examples and 6max, 6min are upper and lower bounds for the internal output of the PSS, un and 6n are the values of u(t) and 6(t) at instant t =nt, respectively, which are constantly hold during nt5 t B(n + 1)t. Define D1 and D2 as follows Æ −1/T1 Ã D1 = Ã −(T2 − T3)/T 22 Ã 2 È − (T3/T2)(T4 − T5)/T 4
0 − 1/T2 (T4 − T5)/T4
0 Ç Ã 0 Ã, Ã − 1/T4 É
Ç Æ 1/T 21 Ã Ã 2 D2 = Ã (K/T1)(T2 − T3)/T 2 Ã. Ã 2Ã È(KT3/T1T2)(T4 − T5)/T 4 É
Since Eq. (14) is a linear differential equation, it can be analytically solved. Therefore, we have following difference equation by integrating Eq. (15) from nt to (n+ 1)t, 1 D1t yn + 1 = eD1tyn + D − −E]D2u(xn, zn ) 1 [e
(16)
which has the same form with Eq. (2). That is, Eq. (16) is the discrete-time controller whose design imitates that of the continuous-time controller of Fig. 2, i.e. a threeorder analog PSS. Note that the real power output u(xn, zn ) of the generator is a given function of (xn, zn ). The PSS output 6n is obviously a function of yn and u(xn, zn ) according to Eq. (15), i.e. 6n = hn (xn, yn, zn ) h(yn, u(xn, zn ))
(17)
where 6n is not a linear function because of the saturation 6min, 6max of 6(t) shown in Eq. (15). On the other hand, the dynamics of generators and analog controllers corresponding to Eq. (1), the power flow balance equations corresponding to Eq. (3) can be written as follows x; (t) = f(x(t), hn (xn, yn, zn ), z(t))
(18)
h(x(t), hn (xn, yn, zn ), z(t)) = 0.
(19)
Note that xn, yn and zn can be viewed as constants
A power system with four buses and one generator shown in Fig. 3 (corresponding to Fig. 1) is used to illustrate the application, where bus-4 is an infinite-bus with constant voltage.The generator-1 is modeled by five ordinary differential equations, and automatic voltage regulator (AVR) of generator-1 is defined by three differential equations. The parameters of generator-1 and AVR are shown in Figs. 3 and 4, respectively. The designed digital power system stabilizer (PSS) is governed by three difference equations, i.e. Eq. (16). Besides, there are eight algebraic equations corresponding to four buses. Therefore, this model is a hybrid dynamical system including eight continuous-time and three discrete-time dynamical equations as well as eight algebraic equalities. Different from the continuous systems, the hybrid dynamical systems has a sampling interval t which may significantly affect the stability of the systems under certain circumstances, such as long transmission systems, heavy load conditions, etc. In addition, the stability may also be highly dependent on the parameters and sampling interval of digital PSS for the system with ultra high-response exciters because only AVR is generally not sufficient to stabilize the system. Next, we first use the sampling interval t and the output of generator1 (corresponding to power flow in transmission lines) as parameters to investigate their influences on the steadystate stability of the power system of Fig. 3, and then examine how the parameters of PSS affect the stability of the system. Notice that the power is mainly supplied from G1 to the load through a long transmission line (280 km) with voltage level 270 KV, and the open-loop poled of Fig. 3 are 0.00699j6.8707, − 2.90429 j3.4210, − 8.14708, − 10.4801, − 75.95199 j74.3825 where the first pair of poles are unstable poles. Using the sampling interval to study the stability is motivated by the design and field test of a digital PSS for Matsuura No. 2 generator in early 1997 which is a 1000 MW cross-compound turbine-generator. The PSS was originally designed by taking the discrete-time co
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L. Chen et al. / Electric Power Systems Research 55 (2000) 79–86
ntroller approximately as the linear continuous-time one due to small sampling interval. During the test, however, it has been found that the PSS failed to stabilize the actual system even although the simulation by this model showed the system stable. One of the
main reasons is that the sampling interval (or time delay) and linearization significantly affects the dynamics of the concerned system even for tB 20 ms, which indicates the t as well as other PSS parameters should be carefully decided by using more exact model (such as the hybrid dynamical system) when designing a digital PSS [20]. The paper in reference [20] gives a detail report on the actual field testing of the digital PSS for Matsuura No. 2 generator. Certainly there are many reasons to cause power system unstable. In this section, we only focus on the sampling interval and output of generator-1 to study their influences on the dynamics of the power system although many other parameters, e.g. transformer tap, gains etc. may also affect the system stability.
5.1. Case-1 of digital PSS
Fig. 4. Diagram and parameters of an analog AVR.
Fig. 5. Stable and unstable regions according to numerical simulation for case-1.
For case-1, the digital PSS is designed as K=0.6, T1 = 1.0, T2 = 0.45, T3 = 7.5, T4 = 0.85, T5 =0.01, 6min = − 0.1, 6max = 0.1. For the numerical computation, x(t) and xn are calculated from numerical integration of Eq. (18) by the Euler or the Runge–Kutta methods, while z(t) and zn are obtain from Eq. (19) by the Newton method for intersampling and sampling instants, respectively. On the other hand, yn is computed by simply updating Eq. (16). Fig. 5 indicates the results of numerical simulation of the hybrid dynamical system by four-order Runge– Kutta method with increment Dt = 1 ms for Eq. (18), while Fig. 6 is the results obtained directly from Theorem 3.1. Obviously, the theoretical results coincide with those of the numerical simulation, which verifies the effectiveness of Theorem 3.1. From Fig. 6 or Fig. 5, it is evident that the system is apt to be unstable with the increasing t. For instance, when the real power of generator-1 is 1.0 p.u. the power system is stable if tB 9.2. However, at t: 9.2 ms, the system undergoes the Neimark–Sacker bifurcation shown in Fig. 7, so that it becomes unstable if t\ 9.2 ms. Actually, it has numerically been found that the boundary of stability in Fig. 6 are mainly composed of the points of the Neimark–Sacker bifurcation of Theorem 3.3 for this case. For the computation of the trajectory, Theorem 3.2 is used to undertake numerical calculations to overcome the singular surfaces.
5.2. Case-2 of digital PSS
Fig. 6. Stable and unstable regions according to Theorem 3.1 for case-1.
Next we modify the parameters of the digital PSS, i.e. let K= 0.28, T1 = 0.23, T2 = 0.11, T3 = 0.16, T4 = 1.18, T5 = 0.7, 6min = − 0.1, 6max = 0.1, to examine its performance. Fig. 8 shows the stable and unstable regions according to Theorem 3.1. Obviously, the stable
L. Chen et al. / Electric Power Systems Research 55 (2000) 79–86
Fig. 7. A Neimark–Sacker bifurcation for case-1.
85
to deal with nonlinear power systems in a more exact way. Then we applied these theorems to the analysis of a small power system with a digital PSS. Numerical simulations have verified that our theoretical results are useful and effectiveness for the analysis of the digital controls in power systems. It has been found that the sampling interval may significantly affect the stability of power system high-response excitation system, etc. As future works, more detail simulation for multimachine systems should be undertaken to investigate the basic characteristics of the digital controlled power systems.
7. List of symbols x(t),xn yn z(t), zn
Fig. 8. Stable and unstable regions according to Theorem 3.1 for case-2.
region has been considerably enlarged, compared to Fig. 6, as far as the real power of generator-1 is less than 1.18 p.u. and the sampling interval is taken as a changing parameter. It is interesting to notice that when the real power of generator-1 is more than 1.18 p.u., the power system is unstable even if t = 0 i.e. it is not stable even for the analog PSS. However, if we increase t, the system may become stable when 1.34 \ u\ 1.18 p.u., where u is the real power of generator-1. This case denotes that increasing sampling interval may enhance the asymptotical stability of the hybrid dynamical systems under certain circumstances. The results of case-1 and case-2 also show that the stability of the system is dependent on the parameters and sampling interval of the PSS for this long transmission line system. On the other hand, Fig. 8 also indicates that the stability generally deteriorates with the increase of the output of generator-1. Besides, we studied a system with digital PSS under heavy load conditions, which also show the similar results.
6. Conclusion We first described several theoretical results for the asymptotical stability, singular perturbation and the bifurcation for the hybrid dynamical systems, in order
f C 1 gC 1 hC 1 t fx A
· E eA
nx dimensional vectors Rnx at instant t and nt, respectively ny dimensional vector Rny at instant nt nz dimensional vectors Rnz at instants t and nt, respectively Rnx+nx+ny+nz+nz Rnx Rnx+ny+nz Rny Rnx+nx+ny+nz+nz Rnz a sampling interval (positive real number) (f(x, xn, yn, z, zn )/(x the determinant of a square matrix A the usual Euclidean norm of a vector or the induced matrix norm of a matrix an identity matrix with appropriate dimensions an exponential function of a matrix A
Acknowledgements This research was partially supported by both Special Science Fund of Osaka Sangyo University of Japan and National Key Basic Research Special Fund of China (No. G1998020304).
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