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A two-loop excitation control system for synchronous generators
Electrical Power and Energy Systems 27 (2005) 556–566 www.elsevier.com/locate/ijepes
A two-loop excitation control system for synchronous generators Jose Alvarez-Ramirez a,b, Ilse Cervantes a,*, Rafael Escarela-Perez a,c, Gerardo Espinosa-Perez a,d a
Seccion de Estudios de Posgrado e Investigacion ESIME-C, Av. Santa Ana 1000 Col. San Francisco Culhuacan, Mexico D.F. 04430, Mexico1 b Division de Ciencias Basicas e Ingenieria, UAM-Iztapalapa, Mexico, Mexico c Departamento de Energia, Universidad Autonoma Metropolitana, Azcapotzalco, Mexico d DEPFI, UNAM, Mexico, Mexico Received 4 April 2003; revised 9 February 2005; accepted 11 August 2005
Abstract An excitation controller for a single generator based on modern multi-loop design methodology is presented in this paper. The proposed controller consists of two-loops: a stabilizing (damping injection) loop and a voltage regulating loop. The task of the stabilizing loop is to add damping in the face of voltage oscillations. The voltage regulating loop is basically a PI compensator whose objective is to obtain terminal voltage regulation about the prescribed reference. The main contribution of this paper is to give some insights into the systematic derivation of multi-loop controllers of power generators. Certain connections between the two-loop excitation controller and standard PSS–AVR schemes are discussed. In this way, some insight into the stability of the standard PSS scheme is obtained from the analysis of the proposed controller. The proposed controller is evaluated via numerical simulations on a full finite-element model. q 2005 Elsevier Ltd. All rights reserved. Keywords: Synchronous generator; Two-loop control; PSS; AVR
1. Introduction Excitation control of generators is a very important issue in the operation of power systems. A suitable designed excitation control has shown to be very efficient to support the voltage on the operation system, to enhance its transient performance and to damp its oscillations [1,2]. For damping oscillations, voltage regulators are not sufficient, and using additional signals feedback is necessary. Therefore, a good excitation controller should combine a voltage regulator with a feedback controller that adds damping into the control loop. In the last years, several new excitation control strategies have been proposed. In [3], a Lyapunov-based approach is used to propose a feedback function to reduce the swing oscillations.
* Corresponding author. Address: IPICyT, Division de Matematicas Aplicadas y Sistemas Computacionales, Camino a la Presa San Jose 2055, Col. Lomas 4a. seccio´n, San Luis Potosi 78216, Mexico. Tel.: C52 444 8342000; fax: C52 444 8342010. E-mail address: [email protected] (I. Cervantes). 1 Tel./fax: C52 55 56562058.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.08.001
The resulting controller has a simple structure and provides maximal damping in terms of the underlying Lyapunov function. In [4], a Lyapunov based design called LgV controller [5] is proposed. This static controller improve the system damping, while guaranteeing the enlargement of the estimate of the region of attraction of the equilibrium. However, this controller cannot be easily implemented and a dynamic extension is required. On the other hand, nonlinear controllers based on rigorous theoretical foundation have also been proposed in [6]. While these controllers have been successfully proven in numerical simulations, there are still open questions related to the effect of the controllers gains and structure on the closed-loop position of the equilibrium points. Passivation approaches have been also explored [4,7–9]. The main characteristic of this technique lies on the fact that a passive system can be stabilized only by static (proportional) output feedback. In the case of synchronous generators connected to an infinite bus, the passivity properties are currently well established [4]. Unfortunately, the passive output is a nonlinear function of the system states (some of them immeasurable) and the commonly unknown operating point makes unfeasible the direct implementation of passive controllers. Although attempts have been performed with
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
the aim of avoiding this difficulty by introducing state observers [8], passive controllers lack robustness guarantee in the face of model uncertainties. The main contribution of this paper is to explore some (partial) solution to the effective terminal voltage regulation, providing satisfactory transient performance. To this end, a twoloop approach seems to be the natural framework to address the excitation control objectives (see Fig. 1). The two-loop approach to design controllers aimed at inducing damped dynamics and regulating an output signal is commonly used in practice, ranging from power electronics [10] to chemical processes [11]. For instance, in current-mode control of boost converters [10], the current signal is used to damp the highly oscillatory behavior of the circuit, and the (non-minimum phase) capacitor voltage signal is used into a multi-loop framework to regulate the output voltage. In the case of synchronous generators, a measured signal (e.g. velocity) is used to provide damping in the face of system oscillations. Throughout this paper, we refer to this loop as the stabilizing loop, although authors acknowledge that the open-loop system is operated in stable conditions. The term is used this way to reflect the purpose of the loop: (a) improving performance and (b) enlarging stability domain. In general, the stabilizing loop suffices to obtain a sufficiently stable generator operation and acceptable dynamic performance. However, due to the presence of unavoidable uncertainties in system parameters and operating point components (e.g. power angle), the terminal voltage can display a significant departure (steady-state offset) from the prescribed operating value. To this end, the excitation controller is endowed with a voltage regulating loop that is aimed at keeping the terminal voltage equal to the reference value. In this work an excitation controller based on modern multi-loop control methodology is presented. The main contribution of this work is the systematic derivation of a simple two-loop controller that ensures both satisfactory transient performance and effective voltage regulation in spite of model uncertainties. It shown that the proposed controller is able to enlarge the region of attraction of the operating point by moving away the unstable equilibrium point. The proposed controller is evaluated via numerical simulation on a full finite-element model corresponding to Maxwell equations. These results shows the effectiveness of the proposed controller in spite of significant model uncertainty. 2. Basic model Notation. Throughout the paper, a ‘hat’ on top of symbols ^ The symbol s will denotes equilibrium values (e.g. d).
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indistinctly denote the Laplace variable or the time-derivative operator d/dt. For a given signal x(t), x_ denotes its timederivative. The following symbols will be used for the generator model. d is the power angle of the generator, u is the angular speed of the generator, us is the synchronous machine angular speed, vZuKus is the relative speed of the generator, Pm is the mechanical input power, Pe is the active electrical power delivered by the generator, PaZPe-Pm is the accelerating power, H is the per unit inertia constant, E 0 q is the transient EMF in the quadrature axis, Eq is the EMF in the quadrature axis, Ef is the equivalent EMF in the excitation coil, T 0 do is the direct axis transient open circuit time constant, VT is the generator terminal voltage, Vs is the infinite bus voltage, kc is the gain of the excitation amplifier, u is the input of the amplifier of the generator, xT is the reactance of the generator transformer, xd is the direct axis reactance, x 0 d is the direct axis transient reactance, xL is the reactance of one transmission line, and xad is the mutual reactance between the excitation coil and the stator coil. In addition, the following relationships are used: xds Z xT C 12 xL C xd ; x 0 ds Z xT C 12 xL C x 0 d and xs Z xT C 12 xL . The model considered for analysis and control derivation is the well-known third-order model [1] of a single generator connected to an infinite bus (SGIB) through a transmission line d_ Z v D u v_ ZK vK s Pa ðd; Eq0 Þ H H
(1)
1 0 E_ q Z 0 ðEf KEq ðd; Eq0 ÞÞ Tdo Pa ðd; Eq0 Þ Z Pe ðd; Eq0 ÞKPm Pe ðd; Eq0 Þ Z
1 V E ðd; Eq0 ÞsinðdÞ xds s q
(2)
x xd Kxd0 0 Eq ðd; Eq0 Þ Z ds E K Vs cosðdÞ q 0 0 xds xds Ef Z k c u where the effects of damping windings and mechanical damping are considered within the damping constant D. It should be noticed that the effects of electrical resistances, reluctance torques and shunt line reactances are neglected. The terminal voltage VT is given by Vt Z
1=2 1 2 2 x V C x2s Eq ðd; Eq0 Þ2 C 2xs xd xds Pe ðd; Eq0 ÞctgðdÞ xds d s (3)
Using (2) in (1) and (3), the SGIB model can be represented by d_ Z v v_ ZKb0 vKb1 Eq0 sinðdÞ C b2 sinðdÞcosðdÞ C P Fig. 1. Two-loop configuration controller.
0 E_ q
Z b3 cosðdÞKb4 Eq0
C hc u
(4)
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and 1=2 Vt Z b5 C b6 Eq ðd; Eq0 Þ2 C b7 Pe ðd; Eq0 ÞctgðdÞ
(5)
The relationship between the parameters in model (1) and those commonly used in synchronous machine modeling [1], [12] are given by 0 12 D x b5 Z @ d V s A b0 Z ; H xds 0 12 us V s x b1 Z 0 ; b6 Z @ s A Hxds xds b2 Z
us xd Kxd0 2 0 Vs ; H xds xds
b7 Z
2xs xd xds
x Kx 0 b 3 Z d 0 0 d Vs ; Tdo xds
u P Z s Pm H
x b4 Z 0 ds 0 ; xds Tdo
k hc Z 0c Tdo
(6)
^ where the fact that sinðdÞZ c has been considered. From (8), it is known that 0 ^ Z 1 ½hc u^ cosðdÞ ^ C b3 ð1Kc2 Þ E^ q cosðdÞ b4
It is easy to show that the following relationship is satisfied: b1 b3 Kb2 b4 Z 0
(7)
This equality will be important for the computation of the SGIB system equilibrium points. Although the third-order model (1)–(3) is quite simple, it retains the main characteristics of the power system dynamics; namely, power swings due to initial conditions, short circuits, line outages and other perturbations [1,12]. 2.1. Equilibrium points and stability Before proceeding with the derivation of the excitation control it is convenient to review some of the stability ^ where u^ is a properties of the open-loop system (i.e., uZ u, positive constant input). Due to physical considerations we are interested in the equilibria p2Dop, where Dop is the region of the state space defined as Dop Z fp 2R3 : d 2½0; p; Eq0 O 0g An equilibrium point pe 2R3 of the SGIB system is given ^ 0; Eq T , where d^ and E^ q are the solutions of the ^ ½d; by pZ nonlinear equations: 0 ^ ^ ^ b1 E^ q sinðdÞKb 2 sinðdÞ cosðdÞ Z P
^ C b4 E^ q0 Z hc u^ Kb3 cosðdÞ
(8)
0
By eliminating E^ q from the above equations and considering ^ the relationship (7), we see that the (d-component) equilibria satisfy the well-known relationship [1]: def
^ Zc sinðdÞ
b4 P b1 hc u^
0 equilibrium points contained in Dop; namely, p^ s Z ½d^s ; 0; E^ q;s T 0 and p^ u Z ½d^u ; 0; E^ q;u T with d^s 2½0; p=2Þ and d^u 2ðp=2; p. The value cZ 1 corresponds to a saddle-node bifurcation, where the two equilibrium points p^ s and p^ u collapse within a nonhyperbolic one [13]. For cO 1, the SGIB system does not have an operating point and trajectories diverge without bound. To establish the (local) stability of the equilibrium points p^ s and p^ u , the small-signal power system model around a given ^ 0; E^ q0 T is considered: cZ ^ ½d; _ Ao c, where equilibrium point pZ 0 ^ v; Eq0 KE^ q T and the matrix Ao is given by cZ ½dKd; 2 3 0 1 0 6 7 0 ^ C b2 ð1K2c2 Þ 0 Kb1 c 7 (10) Ao Z 6 4Kb1 E^ q cosðdÞ 5 Kb3 c 0 Kb4
(9)
Hence, c 2 ½0; 1 is a necessary and sufficient condition for the existence of equilibria. If c 2½0; 1Þ, there are only two
Then, the characteristic polynomial of Ao can be expressed as §o ðsÞZ s3 C a3 s2 C a2 sC a1 , where ^ a1 Z b1 hc u^ cosðdÞ 0
1 b ^ C b2 c2 a2 Z @ 1 Ahc u^ cosðdÞ b4
(11)
a 3 Z b4 The following stability results are obtained:
^ 0 and cosðd^u Þ! 0, one has that a1!0 at (i) Since bb14 hc uO ^dZ d^u . Consequently, the equilibrium point p^ u is unstable for all system parameters. ^ d^s are (ii) Since cosðd^u ÞO 0, one has that all the ai’s at dZ 2 positive. Moreover, a2 a3 Ka1 Z b2 b4 c O 0. Consequently, elementary Routh-Hurwitz analysis of the polynomial §o ðsÞ leads to the conclusion that the equilibrium point p^ s is stable. Example 1. To illustrate the results described above, consider the following parameter set taken from [14]: usZ 314.159, HZ8.0 s, Tdo Z 6:9 , k cZ1.0, x dZ1.863, xd0 Z 0:257, xTZ0.127, xLZ0.4853, VsZ1.0 p.u., PmZ0.9 p.u. and u^f Z 2:1162. These parameters give the 0 following computed parameters: xdsZ2.2327, xds Z 0:62665, xsZ0.36965, b1Z62.666, b2Z44.99, b3Z0.37143, b4Z 0.51636, b5Z0.69358, b6Z2.7411!10K2, b7Z0.61688, ^ 2:1. From these values, we have that PZ35.343 and uZ cZ 0:94938, so that the SGIB system has two equilibrium points corresponding to the stable d^s Z 728 and the unstable d^u Z 1088 angles. In practice, one is interested in the response of the system to a short circuit which consists of the connection of an impedance between the machine terminal and ground. The impedance is disconnected after the clearing time and the system is back to its predisturbance topology. In terms of model (1), this is equivalent to set
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
(c) the terminal voltage VT is kept equal to the reference value VT,ref.
Power Angle (rad)
1.6
As mentioned in the Introduction, we will follow a two-loop approach (see Fig. 1) to find an excitation control that meets the control requirements (a)–(c). In a first step, a stabilizing loop is proposed to add damping into the SGIB dynamics. It will be shown that this damping injection control has the capability of enlarging the region of attraction by moving away p^ u . In a second step, a PI voltage regulator is used to ensure that VT(t)/VT,ref asymptotically.
1.4
1.2
1.0
0
5
10
15
20
25
Time (sec)
3.1. Stabilizing loop
0.82 0.81 Transient EMF
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0.80 0.79 0.78 0.77 0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Power Angle (rad)
Fig. 2. Open-loop behavior of the SGIB system.
VsZ0 and lxL, 0%l%1, during the clearing time and then resetting it to its original value. Fig. 2 shows the dynamics of the load angle d(t) and the phase portrait (d, v) for a short-circuit with lZ0.5 at tZ10.0 s with clearing time equal to 125 ms. It is noted that, after the short-circuit event, the machine displays a highly oscillatory dynamics which eventually converges to the operating point ps. In practice, this oscillatory behavior is undesirable since can lead to irreversible degradation of electrical and mechanical components.
3. Excitation controller The system is designed to operate at the stable operating point p^s . The presence of the unstable equilibrium point p^ u restricts the size of the region of attraction of the operating point p^s , denoted in the sequel U. In fact, the boundary of the potential region of attraction U is the stable manifold of the unstable equilibrium p^ u . Eventhough the operating point p^s is stable, the dynamics of the SGIB around p^s commonly display a highly oscillatory behavior, which is undesirable in practice [1,12]. In view of these facts, the control objective is to find an excitation control input u such that (a) oscillations around the operating point p^ s are reduced by adding damping, (b) the (potential) region of attraction U is enlarged by moving away the unstable equilibrium point p^ u , and
The damping injection controller derivation can be best carried out using a small-signal power system model. Smallsignal analysis is used because the corrective actions aimed at reducing swing oscillations are limited to provide relatively small adjustments to the excitation level [1,12]. In fact, when the system trajectories depart far from the operating point p^ s , protection circuits are activated and the generation unit is isolated from the rest of the network. To simplify computations, it will be assumed that DZ0 (i.e., b1Z0). This assumption is taken without lost of generality since DO0 introduces additional damping to the SGIB system. ^ Du, where Du is taken for feedback control If uZ uC purposes, the small-signal power system model around the stable equilibrium point p^ s is given by: 2
Dd_
3
2
0
6 7 6 6 y_ 7 Z 6Kb E^ 0 cosðd^ Þ C b ð1K2c2 Þ s 2 4 5 4 1 q;s 0 DE_ q Kb3 c
32 3 2 3 0 Dd 76 7 6 7 7 y 5 C 4 0 5Du 0 Kb1 c 54 DEq0 hc 0 Kb4
1
0
(12)
Consider the following variable transformation: z1 Z Dd z2 Z v h i 0 z3 Z Kb1 E^ q;s cosðd^s Þ C b2 ð1K2c2 Þ DdKb1 cDEq0 System (12) can be described in z-coordinates as z_1 Z z2 z_2 Z z3
(13)
z_3 ZKa1 z1 Ka2 z2 Ka3 z3 C gDu where
x us P m g ZKb1 hc 8 ZK ds 0 0 Hu ^ xds Tdo
(14)
and the a’s are given by Eq. (11). Notice that the signal z1 satisfies the third-order differential equation ð2Þ (i) i i ð1Þ zð3Þ C a z C a z C a z Z gDu, where z Zd z/dt . 3 1 2 1 1 1 According to Fig. 1, the control input is given by DuZ DuSCDuR , where DuS and DuR correspond to the stabilizing (damping injection) and voltage regulating loop control inputs, respectively. For simplicity, in the derivation
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of the damping injection controller or stabilizing loop it will be considered that DuRZ0. The feedback function corresponding to the input DuR will be presented later in the paper. Consider the polynomial §3 ðsÞZ s3 C k3 s2 C k2 sC k1 such that all its roots {r1, r2, r3} are located in the open left-half complex plane. The following exact state-feedback function is proposed for the damping injection controller:
(a) Damping rate. Under the effect of (15), system dynamics are given by
DuS Z K1 ð3Þz1 C K2 ð3Þz2 C K3 ð3Þz3
To show that control law (15) provides arbitrarily damped dynamics, consider the scaled variables ziZ33-izi, iZ1–3. Then, system (17) can be rewritten as
(15)
where 3 is a positive parameter, and the gains K1(3) to K3(3) are given by 3
K1 ð3Þ ZKð3 k1 Ka1 Þ=g K2 ð3Þ ZKð32 k2 Ka2 Þ=g
(19)
z_3 ZK3 k1 z1 K3 k2 z2 K3k3 z3 3
2
2_ Z 3Ain 2
(20)
where
Controller (15) is basically a pole-placement feedback function, which cancels the open-loop dynamics (induced by Ka1z1Ka2z2Ka3z3) and impose the closed-loop poles {3r1, 3r2, 3r3}. In this way, by choosing suitable gains k1 to k3 , one can tune the parameter 3 to place the system poles {3r1, 3r2, 3r3} at positions very deep in the left-hand complex plane. One can recast the damping injection controller in physical variables in the following way. First, recall that z1ZDd and z2Zv. Since z3 is the mechanical acceleration _ one can write (15) as follows: (i.e., z3 Z v),
2
0
1
6 Ain Z 4 0
0
(17)
K1 ð3Þ zK33 k1 =g (18)
K3 ð3Þ zK3k3 =g which depends only on the parameter gZKb1 hc c! 0. On the other hand, one can notice that power angle deviation Dd of control law (17) is measured with respect to the operating angle d^s , which is not exactly known (this situation is equivalent to have unknown initial condition on the integral action of speed deviation). Consequently, the derivation of a practical excitation controller must rely on available estimates of the parameters and the operating point. Let us study the effects of the controller (15) on the damping rate, the location of the unstable operating point p^ s , and the robustness in the face of uncertain parameters and operating point.
0
Kk1 Kk2
3
1 7 5 Kk3
(21)
Since Ain is a stable matrix (in fact, its characteristic polynomial is §3 ðsÞ), there exists a positive-definite matrix Pin that satisfies the Lyapunov equation ATin Pin C Pin Ain ZKI3 , where I3 is the three-dimensional identity matrix. Consider the quadratic function V(z)ZzTPinz. The time-derivative of V(z) along the trajectories of system (20) is given by V_ ZK3j2j2 %K
Controller (17) constitutes a PID action on the speed deviation, that can be easily implemented with measurements of v. The advantage of the control configuration (16), (17) is that it depends only on a single parameter (3O0. Basically, 3 can be increased to enhance the damping capabilities of the SGIB system. It is interesting to note that in the high-gain limit (i.e., as 3 takes large values), the control gains are dominated by the imposed dynamics. In fact, for sufficiently large values of 3, one can obtain the approximation
K2 ð3Þ zK32 k2 =g
z_2 Z z3
(16)
K3 ð3Þ ZKð3k3 Ka3 Þ=g
Dus Z K1 ð3ÞDd C K2 ð3Þv C K3 ð3Þv_ ð Z K1 ð3Þ v ds C K2 ð3Þv C K3 ð3Þv_
z_1 Z z2
3 V lmin ðPin Þ
where lmin(Pin) denotes the minimum eigenvalue of Pin. From the above inequality one can get that V(t)%V(0)exp(K3t/ lmin(Pin)). Since lmin(Pin)jzj2%V(z)%lmax(Pin)jzj2, one has that jz(t)j%kjz(0)jexp(K3t/2lmin(Pin)), where kZlmax(Pin)/ lmin(Pin) is the condition number of matrix Pin. This implies that the trajectories z(t) converge arbitrarily fast with rate 3/ 2lmin(Pin), which is of the order of 3O0. That is, the damping rate induced by controller (15) can be made arbitrarily large by increasing the value of the single parameter 3O0. (b) Equilibrium points. Let us study effect of control law (15) on the location of the unstable equilibrium point p^s . Under (15), the equilibrium points of the SGIB system with DuouTZ0 are given by 0
^ ^ ^ b1 E^ q sinðdÞKb 2 sinðdÞcosðdÞ Z P 0
^ C b4 E^ q Khc K1in ð3ÞðdK ^ d^s Þ Z hc u^ Kb3 cosðdÞ
(22)
0 As before, one can eliminate E^ q to obtain the following relationship:
^ Z sinðdÞ
^ 2 b1 hc uc ^ d^s Þ ^ C ð33 k1 Ka1 ÞðdK b1 hc uc
(23)
^ Hence, the (d-component) equilibria of the controlled SGIB systems are given by the intersections of the sine function and a hyperbole. Notice that p^s is always an equilibrium point of the controlled SGIB system. To look for another equilibrium points, let us note that the hyperbole has a singularity at
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566 ^ c uc . In this way, depending on the sign of d^sg ð3ÞZ d^s K 3b31khK 1 a1 3 3 k1 Ka1 , three cases can be established: i) If 33 k1 Ka1 ! 0, then d^sg ð3ÞO d^s and the hyperbole has positive derivative with ^ In this case, there exists an additional intersection respect to d. ^da ð3Þ in the domain [0, p] corresponding to an unstable equilibrium point (see Fig. 3a); namely, 0 p^ a ð3ÞZ ½d^a ð3Þ; 0; E^ q;a ð3ÞT . ii) If 33 k^1 Ka1 Z 0, then ^ c, so that d^sg ð3ÞZ d^s . The equilibria are given by sin ðdÞZ the equilibria of the controlled SGIB system are the stable p^ u and the unstable p^ u ones of the uncontrolled system. In this case, controller (15) has no effects on the position of the equilibria. (iii) If 33 k1 Ka1 O 0, then d^sg ð3Þ! d^s and the ^ This hyperbole has negative derivative with respect to d. ^ ^ imply that there exists an intersection da ð3ÞO ds in the domain [p/2, p] corresponding to an unstable equilibrium point p^a ð3Þ. Moreover, d^sg ð3Þ/ p as 3/N, so that p^a ð3Þ converges to the boundary vDop of the operating region Dop (see Fig. 3b). From this analysis, one can conclude that control law (15) enlarges the potential region of attraction U of the operating point p^s in the sense that the unstable equilibrium point p^a ð3Þ is moved away from p^ s as the parameter 3 is increased. That is, although p^ s is not the only equilibrium point of the controlled SGIB system, the other (unstable) equilibrium point p^ a ð3Þ can be moved up to the boundary of the operating region. (c) Robustness. Unfortunately, the exact controller (15) cannot be implemented as it stands because in applications, the 0 parameters b1 to b4 and the operating point ps Z ½d^s ; 0; E^ q;s T are not exactly known [12,15]. Consequently, the derivation of a practical excitation controller must rely on available estimates of the parameters and the operating point. Let b1,e 0 to b4,e and ps;e Z ½d^s;e ; 0; E^ q;s;e T be estimates of b1 to b4 and 0 T ps Z ½d^s ; 0; E^ q;s , respectively. In particular, estimates the 0 0 voltage E^ q;s are rarely available in practice, so that E^ q;s;e Z 0 can be considered. These estimates yield the corresponding estimated parameters a1,e to a3,e and ge in the system (13). In the worst case, one can have that aieZ0, for iZ1–3, which
1.2
561
leads to the approximate control gains (18) with gZge. To study the robustness of the damping injection controller in the face of uncertain parameters and operating point, we will consider the system in z-coordinates (see Eq. (13)). The state z2Zv does not depend on the estimated parameters and _ operating point (i.e., z^2 Z 0). On the other hand, since z3 Z v, one has that z^3 Z 0. However, the state z1 is measured with respect to the operating angle d^s , which is not exactly known. The practical damping injection controller based on the estimates a1,e to a3,e, ge and d^s;e is given by DuS Z K1;e ð3Þz1 C K2;e ð3Þz2 C K3;e ð3Þz3 C K1;e ð3ÞDd^s where Dd^s Z d^s;e Kd^s and K1;e ð3Þ ZKð33 k1 Ka1;e Þ=ge K2;e ð3Þ ZKð32 k2 Ka2;e Þ=ge
(25)
K3;e ð3Þ ZKð3k3 Ka3;e Þ=ge Notice that controller (15) with gains given by (18) corresponds to the case where ai,eZ0, for iZ1–3. When the practical controller (24) is used in (5), one gets the controlled system z_1 Z z2 z_2 Z z3
(26)
z_3 ZKd1 ð3Þz1 Kd2 ð3Þz2 Kd3 ð3Þz3 C gK1;e ð3ÞDd^s where di(3)ZaiKgKi,e(3), iZ1–3. Notice that in the absence of uncertainties, system (26) reduces to the stable system (17). The 0 first effect of uncertainties is that ps Z ½d^s ; 0; E^ q;s T is not longer an equilibrium point of the controlled SGIB system (i.e., z^ s0). In fact, the equilibrium point of the linear system (26) is given by z^2 Z 0, z^3 Z 0 and z^1 ð3ÞZ gK1;e ð3ÞDd^s =d1 ð3Þ. Equivalently, the ^ is given by power angle equilibrium dð3Þ ^ Z d^s C gK1;e ð3ÞDd^s =d1 ð3Þ dð3Þ 1.2
(a) ε=5
(b)
δs
ε=4 0.9
δs
0.9 ε = 1.6
ε=3
δc
δsg < δs
0.6
0.6
δc
δsg > δs 0.3
0.3
0.0
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Equilibrium Power Angle (rad)
3.0
3.5
(24)
0.0
0.5
1.0
1.5
2.0
2.5
Equilibrium Power Angle (rad)
Fig. 3. Equilibrium points under damping injection controller.
3.0
3.5
8 min Du > > < S Dusat DuS S Z > > : max DuS
if
Dumin S % DuS
if
max Dumin S ! DuS ! DuS
if
Dumax S % DuS
(27)
The upper and lower limits of the stabilizing loop action was min chosen as Dumax S Z 6:0 and DuS ZK6:0, respectively. Fig. 4 shows the behavior of the SGIB system under controller (15) for three different values of the parameter 3 (namely 3Z3–5).
40 30 20 10 0 -10 -20 -30 -40 0
Vt (Volts)
That is, there is a steady-state offset gK1;e ð3ÞDd^s =d1 ð3Þ induced by uncertain parameters and operating point. Notice that ^ lim3/Njdð3ÞK d^s j s0, which means that the steady-state offset cannot be removed even with a high-gain feedback. On the other hand, the stability of the equilibrium point corresponding to the ^ angle dð3Þ is governed by the stability of the polynomial §e ðsÞZ s3 C d3 ð3Þs2 C d2 ð3ÞsC d1 ð3Þ. Some sufficient stability conditions can be derived for sufficiently values of 3. From elementary Routh-Hurwitz analysis, the following two conditions are sufficient to ensure stability of §e ðsÞ: (i) di(3)O0 for iZ1–3, and (ii) d3(3)d2(3)Kd1(3)O0. Since sign(ge)Zsign(g)ZK1 (in d^s and d^s;e are in (0, p/2)), di ð3ÞZ ai C ðg=ge Þ fact, 3KiC1 3 ki Kai;e and ki O 0, there exists a positive constant 3min such that di(3)O0, iZ1–3, for all 3O3min. This result implies that the damping injection controller is (locally) robust against uncertainties in the SGIB parameters and operating point. It should be stressed that an estimate ge can be obtained from available SGIB system information (see Eq. (14)). Remark 1. From the analysis above it is clear that the stabilizing loop or damping injection controller is of high-gain nature. Since in practice the control actuator is limited it becomes necessary to discuss the effect of bounded control actions on the closed-loop stability and performance. Please note that the first effect of the bounded input is to restrict the enlargement of the region of attraction. That is, the maximum size of the attraction domain will be determined by the control bounds. On the other hand, the transient performance of the system may also be deteriorated since the system cannot be arbitrary damped and the place of the poles cannot be arbitrary situated in the left-hand complex plane. Hence the damping added to the system will depend (again) on the control bounds. This result resembles the necessity of appropriate control actuator design in order to guarantee both the closed-loop stability and performance of the system. Example 1 (continued). Let us illustrate the damping capabilities of the stabilizing loop (15). To this end, a full finite-element model of a generator corresponding to Maxwell equations will be used. A detailed description of the finiteelement model used here can be found in [16]. The estimates ai,eZ0, for iZ1 to 3 (see Eq. (18)) are used. In this way, geZK0.67 is the only estimated parameter required to implement the stabilizing loop. The gains k1 Z 3:0, k2 Z 3:0 and k3 Z 1:0 has been chosen, so that all the roots of the polynomial §3 ðsÞ are located at K1 s-1. In practice, due to physical limitations, control action is subjected to take values into a certain domain Dumin ; Dumax . Hence, the actual control S S sat input, denoted by DuS , is computed as
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J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
1
2
3
16000 14000 12000 10000 8000 6000 4000
Delta (DEG)
562
4
Open-loop ε=5 ε=4 ε=3 0
1
0
1
2
3
4
2
3
4
90 80 70 60 50 40 30 20
Time (s)
Fig. 4. Effect of the tuning parameter 3 of the stabilizing loop on the SGIB system.
As expected, the larger the value of 3, the faster the convergence of the system dynamics. Furthermore, the equilibrium power angle of the system in these three cases are given by d^a ð3ÞZ 0:8785p, d^a ð4ÞZ 0:9485p and d^a ð5ÞZ 0:9758p: This shows how the proposed damping injection controller moves the unstable power angle away from d^s as 3 is increased. In this way, the proposed controller enlarges the potential region of stability as 3/N However, the achievable damping rate and the size of the stability region is limited by and lower Dumin limits imposed to the the upper Dumax S S stabilizing loop control action. It is interesting to notice that controller gains k1 , k2 and k3 ; can be defined to satisfy a prescribed (required) dynamic performance. Such gains can be computed for example, by minimizing a index performance criterium as in LQR or LQG (see also [15] for a robust design) By restricting controller parameter 3 to take values equal or greater than one and considering the case when exact parameters are available for feedback, gains k1 , k2 and k3 can be interpreted as the worst possible performance obtained by controller (15) (i.e. for 3Z1). Similarly as in the example above, one can obtain more damped responses by increasing parameter 3. On the other hand, in the uncertain case, there exists a minimum value 3pmin that can be used to satisfy the damping requirements. This is illustrated in Fig. 5, where the controller proposed by [15] is compared with control law (24). It can be observed that as 3 increases the proposed controller is able to damp more adequately the system. 3.2. Voltage regulating loop When the stabilizing loop controller is applied to the SGIB system, swing oscillations are well damped so that the SGIB
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J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
Consider the following PI controller
40 30 20 10 0 -10 -20 -30 -40
ðt
DuR Z KP eV C KI eV ðtÞdt
Delta (DEG)
Vt (Volt)
0.0
0.5
1.0
1.5
ε=2 ε=5 bourles 0.0
0.5
0.0
0.5
1.0
1.5
1.0
1.5
90 80 70 60 50 40 30 20 Time (s)
Fig. 5. Comparison of controller in [15] and the proposed damping injection controller.
system can operate under safe conditions. However, due to uncertain system parameters and unknown operating point, the terminal voltage can have a steady-state offset ðV^ t sVt;ref Þ. Under this situation, the voltage loop objective is to regulate the voltage VT at a given reference value Vt,ref. In this paper, a linear PI compensator will be used as the regulating loop controller. To this end, results on PI control stability in [17] u^R Þ will be used. The first step is to compute the gain KuV Z d4ðD dDu^R of the steady-state map V^ t Z 4ðDu^R Þ at the operating point p^s . ^ DuS C DuR and Du^in Z 0 at p^s (see Eq. (15)), one Since uZ uC has that ^ d4ðDu^ R Þ d4ðuÞ Z dDu^ R du^
(28)
That is, the steady-state gain KuV can be computed from the ^ with the stabilizing loop in off steady-state map V^ t Z 4ðuÞ position (i.e. Du^ S Z 0). From (1) and (2), we know that E^ q Z V s kc ^ E^ q ÞctgðdÞZ ^ ^ Hence, ^ sinðdÞ. kc u^ and Pe ðd; xds u 1=2 bVk ^ ^ V^ t Z 4ðuÞdef def b5 C b6 kc2 u^ 2 C 7 s c u^ cosðdÞ xds
(29)
The computation of the derivative (28) gives "
KuV Z
(31)
0
16000 14000 12000 10000 8000 6000 4000
KuV Z
563
1 bVk 2b6 kc2 u^ C 7 s c xds 2V^ t
^ u^ sinðdÞ ^ cosðdÞK
dd^ du^
!# (30)
b4 P ^ From (9), we know that sinðdÞZ b1 hc u^ . Moreover, ^ d d ^ ! 0 at dZ ^ d^s . Consed^s 2 ½0; p=2. This implies that sinðdÞ du^ quently, for all p^ s , (i) KuVO0 and (ii) the steady-state map ^ is invertible (for each u^ there exists one and only one V^ t Z 4ðuÞ V^ t , and vice versa).
where eV def defVt;ref KVt is the voltage regulation error, and KPO0 and KIO0 are the proportional and integral control gains. It is noted that the sign of the steady-state gain determines the sign of the PI control gains (i.e. sign(KP)Z sign(KuV)). Theorem 2.1 in [17] implies that there exist small enough gains KP and KI such that, for all initial conditions ½d0 ; v0 ; E 0 q;0 in a neighborhood of the operating point p^s , the corresponding states ½dðtÞ; vðtÞ; E 0 q ðtÞ are bounded and Vt ðtÞ/ Vt;ref as t/N. That is, there exist PI control gains KPO0 and KIO0 such that the controlled SGIB is (locally) stable and the generator terminal voltage VT converge to the prescribed reference value VT,ref. It should be stressed that the above is only a robustness result in the sense that it only states the existence of controller gains KP and KI that provides asymptotic regulation of the terminal voltage. In practice, one should use a suitable procedure to tune the PI control (31), which can be borrowed from Internal Model Control (IMC) ideas [18]. This can be done by recalling that the regulating loop action is supported on the stabilizing loop action, which introduces a strong damping effect into the controlled SGIB system. In this way, input– output dynamics DuR(t)/DVT(t) can be approximated as a stable first-order system DuR ðsÞ KuV Z DVt ðsÞ Ts C1
(32)
where s is the Laplace variable (i.e. sZd/dt) and TuVO0 is a given time-constant. Two procedures can be proposed to obtain estimates of KuV and TuV: (i) the steady-state gain KuV can be estimated from (30) and the time-constant TuV can be taken of the order of Tdo , which determines the worst damping conditions under the stabilizing loop control action; and ii) KuV and TuV can be estimated from standard step or frequency responses. Given the estimated values KuV and TuV, IMC tuning rules give: 0 1 1 @ TuV A KP Z KK uV TR (33) K1 1 KI Z KuV TR where TRO0 is a tuning parameter corresponding to the timeconstant of the regulating loop. In this way, the tuning of the PI control (31) depends only on the single parameter TR. Moreover, the smaller the value of TR, the faster the convergence VT(t)/VT,ref [18]. Example 1 (continued). Let us illustrate the damping capabilities of the proposed two-loop excitation control scheme with stabilizing loop gains given by Eq. (18). As in the stabilizing loop case, the regulating loop control action is max subjected to take values into a certain domain ½Dumin R ; DuR .
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Hence, the actual regulating control input, denoted by Dusat R , is computed as 8 min Du > > < R sat DuR Z DuR > > : max DuR
if
Dumin R R DuR
if
max Dumin R ! DuR ! DuR
if
Dumax R % DuR
(34)
w - ws (RPM)
The upper and lower limits of the regulating loop control min action was chosen as Dumax R Z 2:0 and DuR ZK2:0, respectively. The transfer function (32) was computed from a firstorder approximation of a step perturbation on the SGIB system under the stabilizing loop control action. In this way, the computed parameters are the following: KuVZ0.162, TuVZ 2.5 s. The IMC parameter TR was set equal to 1 s. Fig. 6 shows the behavior of the SGIB system under a short circuit event under the controller (24), (25) with 3Z4 sK1 and the voltage controller (31) tuned with the IMC rules (33). It is noted that the terminal voltage presents a steady-state offset (i.e. V^ t sVt;ref ) when the regulator is not acting on the system. This offset is removed when the regulating loop controller is connected, such that VT(t)/VT,ref. To show that the proposed two-loop scheme is able to provide stable operation with asymptotic terminal voltage regulation, Fig. 6 presents the SGIB dynamics under setpoint changes in the terminal voltage. Notice that the two-loop scheme provides smooth setpoint ^ 728 does not tracking in spite that the nominal angle dZ correspond to the prescribed terminal voltage conditions. 40 30 20 10 0 -10 -20 -30 -40
Vt (Volt)
It has been shown that the two-loop excitation control scheme composed by the feedback function (15) (stabilizing loop) and the PI compensator (31), (33) (regulating loop) is able to provide (local) damping of swing oscillations and asymptotic terminal voltage regulation about a given setpoint. Besides, the potential region of attraction of the operating point is enlarged in the sense that the unstable equilibrium point is displaced far from the (stable) operating point. The objective of this part of the paper is to reveal the control structure of the two-loop excitation control. The structure of the stabilizing loop depends on the choice of the input quantities, with the most commonly used quantities being: shaft speed deviation v, the accelerating power Pa [1], the transient EMF Eq and the frequency deviation Df. As each of these signals has its advantages and disadvantages, the damping injection loop is often designed to operate on a number (usually two) of these input signals [1,12]. In particular, the angle d is rarely available from measurements. Let CR(s) and CS(s) be the transfer functions of the regulating and stabilizing loops, respectively. Then, DuðsÞ Z CR ðsÞeV ðsÞ C CS ðsÞvðsÞ Z CR ðsÞ½eV ðsÞ C CR ðsÞK1 CS ðsÞvðsÞ
1
2
3
4
washout filter ffl{ stabilizing gain zfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflffl
z}|{ K
stabilizing and regulating loops stabilizing loop
0
1
2
3
4
90 80 70 60 50 40 30
when the shaft velocity deviation v is the signal used for control input computation. From Eq. (24), one has that
1
2
3
(37)
It is noted that (37) contains an ideal integrator 1/s and an ideal differentiator s. Controllers in industrial power systems has been traditionally implemented as analog controllers. The implementation of the ideal integrator and the differentiator on an analog framework led commonly to expensive devices. Frequently, a cheaper static electric network consisting of resistors and capacitors sufficiently approximates the differentiator s and the integrator 1/s. This is made as follows [19] 1 Ta s C 1 z ; Ta / Tb s Tb s C 1
0
lead lag blocks
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ Tw s T1 s C 1 T3 s C 1 Tw s C 1 T2 s C 1 T4 s C 1 (36)
DuS ðsÞ 1 Z K1;e ð3Þ C K2;e ð3Þ C K3;e ð3Þs vðsÞ s
16000 14000 12000 10000 8000 6000 4000
(35)
Secondly, let us show that KCR(s)-1CS(s), displays the following structure
KCR ðsÞK1 CS ðsÞZ
0
delta (DEG)
4. Controller structure
(38)
4
Time (s)
Fig. 6. SGIB system during a short circuit event under the proposed two-loop controller.
and sz
Tc s C 1 ; Tc [ T d Td s C 1
(39)
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
where Ta to Td are suitable time-constants. In this way, the practical damping injection controller is given by DuS ðsÞ T s C1 T s C1 Z K1;e ð3Þ a C K2;e ð3Þ C K3;e ð3Þ c vðsÞ Tb s C 1 Td s C 1
1.1
1.0
After some algebraic manipulations, the following practical transfer function is obtained:
0.9
(41)
where
Voltage, V (p.u.)
(40)
uS ðsÞ b1 s2 C b2 s C 1 Z CS ðsÞ Z KS vðsÞ ðTb s C 1ÞðTd s C 1Þ
565
Lead-Lag Exact
0.8
0.7
KS Z K1;e ð3Þ C K2;e ð3Þ C K2;e ð3Þ 1 b1 Z KK S ½Ta Td K1;e ð3Þ C Tb Td K2;e ð3Þ C Tb Tc K3;e ð3Þ 1 b2 Z KK S ½ðTa
(42)
C Td ÞK1;e ð3Þ C ðTb C Td ÞK2;e ð3Þ
C ðTb C Tc ÞK3;e ð3Þ
0.5 0
Notice that b1O0 and b2O0. If b22 K4b1 O 0, the numerator in (41) can be written as (TesC1)(TfsC1), 1 K1 where KTK e ! 0 and KTf ! 0 are the roots of the 2 polynomial b1 s C b2 sC 1Z 0. That is, the transfer function CS(s) can be written as the composition of a stabilizing gain Kin and two lead-lag blocks as follows: CS ðsÞ Z KS
0.6
Te s C 1 Tf s C 1 Tb s C 1 Td s C 1
On the other hand, the transfer function of the PI compensator (31)–(33) can be written as TuV s C 1 CR ðsÞ Z KP (43) TuV s In this way, one obtains KCR ðsÞK1 CS ðsÞ TuV s Te s C 1 Tf s C 1 ZKKP KS TuV s C 1 Tb s C 1 Td s C 1
(44)
Summarizing, it has been shown that the proposed twoloop excitation control configuration has structures of traditional control of power systems (i.e. wash-out filters, lead-lag blocks). 5. Implementation issues The practical two-loop controller is given by the stabilizing loop controller (24) and the regulator (31). The implementation of this practical excitation controller depends on: (a) the relative speed of the generator vZuKus and the terminal voltage VT, which are always available from measurements, (b) an estimate d^s;e of the operating power angle d^s , which is also available in practice, (c) an estimate ge of the high-frequency gain g, which can be estimated from standard synchronous machine modeling, and d) an estimate of the steady-state gain KuV and the timeconstant TuV, which can be estimated from plant data or system step response. In this way, the implementation of the proposed two-loop excitation controller does not require the exact
3
6
9
12
15
Time, t (s) Fig. 7. Performance of the two-loop configuration under lead-lag approximation when a short circuit occurs.
knowledge of the operating point or measurement of the power angle d(t). In this way, the implementation of the twoloop excitation controller can be made on the basis of available practical information of the SGIB system. The tuning of the two-loop excitation controller is particularly easy to carry out. The damping injection tuning depends only on the single parameter 3O0, which can be increased to enhance the damping of the controlled SGIB system. Numerical simulations suggest that values of 3 about 3–5 sK1 give good performance with acceptable damping rate. The regulating loop tuning depends only on the single timeconstant TuV, which can be reduced to accelerate the terminal voltage convergence. Numerical simulations in this paper suggest that values of TuV of the order of 0.5–1.0 s provide fast voltage convergence. These features facilitate the possible practical implementation of the two-loop controller. Example 1 (continued). Let us illustrate the performance of the two-loop excitation control configuration under the lead-lag approximations (40) and (41). The controller was tuned as follows, TbZ20 s and TdZ1.4 s were chosen [1,15]. Moreover, TaZ0.5 and TcZ20 s were chosen to meet the constraints Ta/Tb and Tc[Td. Fig. 7 presents the dynamics of the terminal voltage with exact and approximate integrator and differentiator. It is noted that the lead-lag filters (40) and (41) induce a small delay in the convergence of the SGIB system to the operating conditions, although the two-loop control performance is still acceptable. This simulations show that the practical approximations (40) and (41) do not degrade significantly the performance of damping injection controller (15). 6. Conclusions A two-loop excitation controller for synchronous machines has been proposed in this work. The stabilizing loop is a simple
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linear state-feedback excitation controller that increases the damping and enlarges the potential region of attraction of the operating point by moving away the unstable equilibrium point. The voltage regulator is composed basically by a PI compensator, which provides asymptotic regulation of the terminal voltage while maintaining ultimate boundedness of all trajectories in the domain of operation. Since the damping injection controller only requires measurements of the speed deviation and an estimate d^e of the operating power angle, it can be implemented in practice. The results in this paper show that it is possible to propose simple excitation control schemes by departing from modern state-space methods. Maybe one of the main advantages of this approach is that the tuning of the controller parameters is very systematic and straightforward. Specifically, the tuning of the damping action depends on the parameter 3 and an estimate of the high-frequency gain g, which is always available in practice. On the other hand, the tuning of the voltage controller requires estimates of a timeconstant and a steady-state gain, which can be obtained from plant data (e.g. step response). References [1] Kundur P. Power system stability and control. McGraw Hill; 1994. [2] Law KT, Hill DJ, Godfrey NR. Robust co-ordinated PSS-AVR design. IEEE Trans Power Syst 1994;9:1218–25. [3] Machowski J, Bialek JW, Robak S, Bumby JR. Excitation control system for use with synchronous generators. IEE Proc-Gener Transm Distrib 1998;145:537–46. [4] Bazanella AS, Silva AS, Kokotovic PV. Lyapunov design of excitation control for synchronous machines. Proceedings of 1997 conference on decision and control, San Diego California, USA 1997. [5] Sepulcre R, Jankovic M, Kokotovic PV. Constructive nonlinear control. Springer; 1996.
[6] Wang Y, Xie L, Hill DJ, Middleton RH. Robust nonlinear controler design for transient stability enhacement of power systems Proceedings of 1992 conference on decision and control, Tucson Arisona, USA 1992. [7] Galaz M, Ortega R, Bazanella A, Stankovic A. An energy-shaping approach to excitation control of synchronous generators Proceedings of 2001 American control conference, Arlington, Virginia, USA 2001. [8] de Leon Jesus, Gerardo Espinosa-Perez, Ivan Macias. Observer-based Control of a synchronous generator: a Hamiltonian approach. Int J Electric Power Energy Syst 2002;24(8):655–63. [9] Pogromsky AY, Fradkov AL, Hill DJ. Passivity based damping of power system oscillations Proceedings of 1996 conference on decision and control, Kobe Japan 1996. [10] Middlebrook RD. Topics in multi-loop regulators and current-mode programming. IEEE Trans Power Electron 1987;2:109–24. [11] Luyben WL. Parallel cascade control. Ind Eng Chem Fundam 1973;463: 1973. [12] Machowski J, Bialek JW, Bumby JR. Power system dynamics and stability. Wiley; 1997. [13] Wiggins S, Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer; 1990. [14] Bergan AR, Bergan AR. Power systems analysis. New Jersey: PrenticeHall; 1986. [15] Bourles H, Peres S, Margotin T, Houry MP. Analysis and design of a robust coordinated AVR/PSS. IEEE Trans Power Syst 1998;13(2): 568–75. [16] Escarela-Perez R, Arjona-Lopez MA, Melgoza-Vazquez E, CamperoLittlewood E, Aviles-Cruz C. A comprehensive finite-element model of a turbine-generator infinite-busbar system. Int J Finite Elem Anal Des 2004; 40(5):485–509. [17] Desoer ChA, Lin ChA. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans Autom Control 1985; 30(9):861–7. [18] Morari M, Zafiriou E. Robust process control. New Jersey: Prectice-Hall; 1989. [19] D’Azzo JJ, Houpis CH. Feedback control systems analysis and synthesis. 2nd ed. Tokyo: McGraw Hill; 1966.
A two-loop excitation control system for synchronous generators Jose Alvarez-Ramirez a,b, Ilse Cervantes a,*, Rafael Escarela-Perez a,c, Gerardo Espinosa-Perez a,d a
Seccion de Estudios de Posgrado e Investigacion ESIME-C, Av. Santa Ana 1000 Col. San Francisco Culhuacan, Mexico D.F. 04430, Mexico1 b Division de Ciencias Basicas e Ingenieria, UAM-Iztapalapa, Mexico, Mexico c Departamento de Energia, Universidad Autonoma Metropolitana, Azcapotzalco, Mexico d DEPFI, UNAM, Mexico, Mexico Received 4 April 2003; revised 9 February 2005; accepted 11 August 2005
Abstract An excitation controller for a single generator based on modern multi-loop design methodology is presented in this paper. The proposed controller consists of two-loops: a stabilizing (damping injection) loop and a voltage regulating loop. The task of the stabilizing loop is to add damping in the face of voltage oscillations. The voltage regulating loop is basically a PI compensator whose objective is to obtain terminal voltage regulation about the prescribed reference. The main contribution of this paper is to give some insights into the systematic derivation of multi-loop controllers of power generators. Certain connections between the two-loop excitation controller and standard PSS–AVR schemes are discussed. In this way, some insight into the stability of the standard PSS scheme is obtained from the analysis of the proposed controller. The proposed controller is evaluated via numerical simulations on a full finite-element model. q 2005 Elsevier Ltd. All rights reserved. Keywords: Synchronous generator; Two-loop control; PSS; AVR
1. Introduction Excitation control of generators is a very important issue in the operation of power systems. A suitable designed excitation control has shown to be very efficient to support the voltage on the operation system, to enhance its transient performance and to damp its oscillations [1,2]. For damping oscillations, voltage regulators are not sufficient, and using additional signals feedback is necessary. Therefore, a good excitation controller should combine a voltage regulator with a feedback controller that adds damping into the control loop. In the last years, several new excitation control strategies have been proposed. In [3], a Lyapunov-based approach is used to propose a feedback function to reduce the swing oscillations.
* Corresponding author. Address: IPICyT, Division de Matematicas Aplicadas y Sistemas Computacionales, Camino a la Presa San Jose 2055, Col. Lomas 4a. seccio´n, San Luis Potosi 78216, Mexico. Tel.: C52 444 8342000; fax: C52 444 8342010. E-mail address: [email protected] (I. Cervantes). 1 Tel./fax: C52 55 56562058.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.08.001
The resulting controller has a simple structure and provides maximal damping in terms of the underlying Lyapunov function. In [4], a Lyapunov based design called LgV controller [5] is proposed. This static controller improve the system damping, while guaranteeing the enlargement of the estimate of the region of attraction of the equilibrium. However, this controller cannot be easily implemented and a dynamic extension is required. On the other hand, nonlinear controllers based on rigorous theoretical foundation have also been proposed in [6]. While these controllers have been successfully proven in numerical simulations, there are still open questions related to the effect of the controllers gains and structure on the closed-loop position of the equilibrium points. Passivation approaches have been also explored [4,7–9]. The main characteristic of this technique lies on the fact that a passive system can be stabilized only by static (proportional) output feedback. In the case of synchronous generators connected to an infinite bus, the passivity properties are currently well established [4]. Unfortunately, the passive output is a nonlinear function of the system states (some of them immeasurable) and the commonly unknown operating point makes unfeasible the direct implementation of passive controllers. Although attempts have been performed with
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
the aim of avoiding this difficulty by introducing state observers [8], passive controllers lack robustness guarantee in the face of model uncertainties. The main contribution of this paper is to explore some (partial) solution to the effective terminal voltage regulation, providing satisfactory transient performance. To this end, a twoloop approach seems to be the natural framework to address the excitation control objectives (see Fig. 1). The two-loop approach to design controllers aimed at inducing damped dynamics and regulating an output signal is commonly used in practice, ranging from power electronics [10] to chemical processes [11]. For instance, in current-mode control of boost converters [10], the current signal is used to damp the highly oscillatory behavior of the circuit, and the (non-minimum phase) capacitor voltage signal is used into a multi-loop framework to regulate the output voltage. In the case of synchronous generators, a measured signal (e.g. velocity) is used to provide damping in the face of system oscillations. Throughout this paper, we refer to this loop as the stabilizing loop, although authors acknowledge that the open-loop system is operated in stable conditions. The term is used this way to reflect the purpose of the loop: (a) improving performance and (b) enlarging stability domain. In general, the stabilizing loop suffices to obtain a sufficiently stable generator operation and acceptable dynamic performance. However, due to the presence of unavoidable uncertainties in system parameters and operating point components (e.g. power angle), the terminal voltage can display a significant departure (steady-state offset) from the prescribed operating value. To this end, the excitation controller is endowed with a voltage regulating loop that is aimed at keeping the terminal voltage equal to the reference value. In this work an excitation controller based on modern multi-loop control methodology is presented. The main contribution of this work is the systematic derivation of a simple two-loop controller that ensures both satisfactory transient performance and effective voltage regulation in spite of model uncertainties. It shown that the proposed controller is able to enlarge the region of attraction of the operating point by moving away the unstable equilibrium point. The proposed controller is evaluated via numerical simulation on a full finite-element model corresponding to Maxwell equations. These results shows the effectiveness of the proposed controller in spite of significant model uncertainty. 2. Basic model Notation. Throughout the paper, a ‘hat’ on top of symbols ^ The symbol s will denotes equilibrium values (e.g. d).
557
indistinctly denote the Laplace variable or the time-derivative operator d/dt. For a given signal x(t), x_ denotes its timederivative. The following symbols will be used for the generator model. d is the power angle of the generator, u is the angular speed of the generator, us is the synchronous machine angular speed, vZuKus is the relative speed of the generator, Pm is the mechanical input power, Pe is the active electrical power delivered by the generator, PaZPe-Pm is the accelerating power, H is the per unit inertia constant, E 0 q is the transient EMF in the quadrature axis, Eq is the EMF in the quadrature axis, Ef is the equivalent EMF in the excitation coil, T 0 do is the direct axis transient open circuit time constant, VT is the generator terminal voltage, Vs is the infinite bus voltage, kc is the gain of the excitation amplifier, u is the input of the amplifier of the generator, xT is the reactance of the generator transformer, xd is the direct axis reactance, x 0 d is the direct axis transient reactance, xL is the reactance of one transmission line, and xad is the mutual reactance between the excitation coil and the stator coil. In addition, the following relationships are used: xds Z xT C 12 xL C xd ; x 0 ds Z xT C 12 xL C x 0 d and xs Z xT C 12 xL . The model considered for analysis and control derivation is the well-known third-order model [1] of a single generator connected to an infinite bus (SGIB) through a transmission line d_ Z v D u v_ ZK vK s Pa ðd; Eq0 Þ H H
(1)
1 0 E_ q Z 0 ðEf KEq ðd; Eq0 ÞÞ Tdo Pa ðd; Eq0 Þ Z Pe ðd; Eq0 ÞKPm Pe ðd; Eq0 Þ Z
1 V E ðd; Eq0 ÞsinðdÞ xds s q
(2)
x xd Kxd0 0 Eq ðd; Eq0 Þ Z ds E K Vs cosðdÞ q 0 0 xds xds Ef Z k c u where the effects of damping windings and mechanical damping are considered within the damping constant D. It should be noticed that the effects of electrical resistances, reluctance torques and shunt line reactances are neglected. The terminal voltage VT is given by Vt Z
1=2 1 2 2 x V C x2s Eq ðd; Eq0 Þ2 C 2xs xd xds Pe ðd; Eq0 ÞctgðdÞ xds d s (3)
Using (2) in (1) and (3), the SGIB model can be represented by d_ Z v v_ ZKb0 vKb1 Eq0 sinðdÞ C b2 sinðdÞcosðdÞ C P Fig. 1. Two-loop configuration controller.
0 E_ q
Z b3 cosðdÞKb4 Eq0
C hc u
(4)
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J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
and 1=2 Vt Z b5 C b6 Eq ðd; Eq0 Þ2 C b7 Pe ðd; Eq0 ÞctgðdÞ
(5)
The relationship between the parameters in model (1) and those commonly used in synchronous machine modeling [1], [12] are given by 0 12 D x b5 Z @ d V s A b0 Z ; H xds 0 12 us V s x b1 Z 0 ; b6 Z @ s A Hxds xds b2 Z
us xd Kxd0 2 0 Vs ; H xds xds
b7 Z
2xs xd xds
x Kx 0 b 3 Z d 0 0 d Vs ; Tdo xds
u P Z s Pm H
x b4 Z 0 ds 0 ; xds Tdo
k hc Z 0c Tdo
(6)
^ where the fact that sinðdÞZ c has been considered. From (8), it is known that 0 ^ Z 1 ½hc u^ cosðdÞ ^ C b3 ð1Kc2 Þ E^ q cosðdÞ b4
It is easy to show that the following relationship is satisfied: b1 b3 Kb2 b4 Z 0
(7)
This equality will be important for the computation of the SGIB system equilibrium points. Although the third-order model (1)–(3) is quite simple, it retains the main characteristics of the power system dynamics; namely, power swings due to initial conditions, short circuits, line outages and other perturbations [1,12]. 2.1. Equilibrium points and stability Before proceeding with the derivation of the excitation control it is convenient to review some of the stability ^ where u^ is a properties of the open-loop system (i.e., uZ u, positive constant input). Due to physical considerations we are interested in the equilibria p2Dop, where Dop is the region of the state space defined as Dop Z fp 2R3 : d 2½0; p; Eq0 O 0g An equilibrium point pe 2R3 of the SGIB system is given ^ 0; Eq T , where d^ and E^ q are the solutions of the ^ ½d; by pZ nonlinear equations: 0 ^ ^ ^ b1 E^ q sinðdÞKb 2 sinðdÞ cosðdÞ Z P
^ C b4 E^ q0 Z hc u^ Kb3 cosðdÞ
(8)
0
By eliminating E^ q from the above equations and considering ^ the relationship (7), we see that the (d-component) equilibria satisfy the well-known relationship [1]: def
^ Zc sinðdÞ
b4 P b1 hc u^
0 equilibrium points contained in Dop; namely, p^ s Z ½d^s ; 0; E^ q;s T 0 and p^ u Z ½d^u ; 0; E^ q;u T with d^s 2½0; p=2Þ and d^u 2ðp=2; p. The value cZ 1 corresponds to a saddle-node bifurcation, where the two equilibrium points p^ s and p^ u collapse within a nonhyperbolic one [13]. For cO 1, the SGIB system does not have an operating point and trajectories diverge without bound. To establish the (local) stability of the equilibrium points p^ s and p^ u , the small-signal power system model around a given ^ 0; E^ q0 T is considered: cZ ^ ½d; _ Ao c, where equilibrium point pZ 0 ^ v; Eq0 KE^ q T and the matrix Ao is given by cZ ½dKd; 2 3 0 1 0 6 7 0 ^ C b2 ð1K2c2 Þ 0 Kb1 c 7 (10) Ao Z 6 4Kb1 E^ q cosðdÞ 5 Kb3 c 0 Kb4
(9)
Hence, c 2 ½0; 1 is a necessary and sufficient condition for the existence of equilibria. If c 2½0; 1Þ, there are only two
Then, the characteristic polynomial of Ao can be expressed as §o ðsÞZ s3 C a3 s2 C a2 sC a1 , where ^ a1 Z b1 hc u^ cosðdÞ 0
1 b ^ C b2 c2 a2 Z @ 1 Ahc u^ cosðdÞ b4
(11)
a 3 Z b4 The following stability results are obtained:
^ 0 and cosðd^u Þ! 0, one has that a1!0 at (i) Since bb14 hc uO ^dZ d^u . Consequently, the equilibrium point p^ u is unstable for all system parameters. ^ d^s are (ii) Since cosðd^u ÞO 0, one has that all the ai’s at dZ 2 positive. Moreover, a2 a3 Ka1 Z b2 b4 c O 0. Consequently, elementary Routh-Hurwitz analysis of the polynomial §o ðsÞ leads to the conclusion that the equilibrium point p^ s is stable. Example 1. To illustrate the results described above, consider the following parameter set taken from [14]: usZ 314.159, HZ8.0 s, Tdo Z 6:9 , k cZ1.0, x dZ1.863, xd0 Z 0:257, xTZ0.127, xLZ0.4853, VsZ1.0 p.u., PmZ0.9 p.u. and u^f Z 2:1162. These parameters give the 0 following computed parameters: xdsZ2.2327, xds Z 0:62665, xsZ0.36965, b1Z62.666, b2Z44.99, b3Z0.37143, b4Z 0.51636, b5Z0.69358, b6Z2.7411!10K2, b7Z0.61688, ^ 2:1. From these values, we have that PZ35.343 and uZ cZ 0:94938, so that the SGIB system has two equilibrium points corresponding to the stable d^s Z 728 and the unstable d^u Z 1088 angles. In practice, one is interested in the response of the system to a short circuit which consists of the connection of an impedance between the machine terminal and ground. The impedance is disconnected after the clearing time and the system is back to its predisturbance topology. In terms of model (1), this is equivalent to set
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
(c) the terminal voltage VT is kept equal to the reference value VT,ref.
Power Angle (rad)
1.6
As mentioned in the Introduction, we will follow a two-loop approach (see Fig. 1) to find an excitation control that meets the control requirements (a)–(c). In a first step, a stabilizing loop is proposed to add damping into the SGIB dynamics. It will be shown that this damping injection control has the capability of enlarging the region of attraction by moving away p^ u . In a second step, a PI voltage regulator is used to ensure that VT(t)/VT,ref asymptotically.
1.4
1.2
1.0
0
5
10
15
20
25
Time (sec)
3.1. Stabilizing loop
0.82 0.81 Transient EMF
559
0.80 0.79 0.78 0.77 0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Power Angle (rad)
Fig. 2. Open-loop behavior of the SGIB system.
VsZ0 and lxL, 0%l%1, during the clearing time and then resetting it to its original value. Fig. 2 shows the dynamics of the load angle d(t) and the phase portrait (d, v) for a short-circuit with lZ0.5 at tZ10.0 s with clearing time equal to 125 ms. It is noted that, after the short-circuit event, the machine displays a highly oscillatory dynamics which eventually converges to the operating point ps. In practice, this oscillatory behavior is undesirable since can lead to irreversible degradation of electrical and mechanical components.
3. Excitation controller The system is designed to operate at the stable operating point p^s . The presence of the unstable equilibrium point p^ u restricts the size of the region of attraction of the operating point p^s , denoted in the sequel U. In fact, the boundary of the potential region of attraction U is the stable manifold of the unstable equilibrium p^ u . Eventhough the operating point p^s is stable, the dynamics of the SGIB around p^s commonly display a highly oscillatory behavior, which is undesirable in practice [1,12]. In view of these facts, the control objective is to find an excitation control input u such that (a) oscillations around the operating point p^ s are reduced by adding damping, (b) the (potential) region of attraction U is enlarged by moving away the unstable equilibrium point p^ u , and
The damping injection controller derivation can be best carried out using a small-signal power system model. Smallsignal analysis is used because the corrective actions aimed at reducing swing oscillations are limited to provide relatively small adjustments to the excitation level [1,12]. In fact, when the system trajectories depart far from the operating point p^ s , protection circuits are activated and the generation unit is isolated from the rest of the network. To simplify computations, it will be assumed that DZ0 (i.e., b1Z0). This assumption is taken without lost of generality since DO0 introduces additional damping to the SGIB system. ^ Du, where Du is taken for feedback control If uZ uC purposes, the small-signal power system model around the stable equilibrium point p^ s is given by: 2
Dd_
3
2
0
6 7 6 6 y_ 7 Z 6Kb E^ 0 cosðd^ Þ C b ð1K2c2 Þ s 2 4 5 4 1 q;s 0 DE_ q Kb3 c
32 3 2 3 0 Dd 76 7 6 7 7 y 5 C 4 0 5Du 0 Kb1 c 54 DEq0 hc 0 Kb4
1
0
(12)
Consider the following variable transformation: z1 Z Dd z2 Z v h i 0 z3 Z Kb1 E^ q;s cosðd^s Þ C b2 ð1K2c2 Þ DdKb1 cDEq0 System (12) can be described in z-coordinates as z_1 Z z2 z_2 Z z3
(13)
z_3 ZKa1 z1 Ka2 z2 Ka3 z3 C gDu where
x us P m g ZKb1 hc 8 ZK ds 0 0 Hu ^ xds Tdo
(14)
and the a’s are given by Eq. (11). Notice that the signal z1 satisfies the third-order differential equation ð2Þ (i) i i ð1Þ zð3Þ C a z C a z C a z Z gDu, where z Zd z/dt . 3 1 2 1 1 1 According to Fig. 1, the control input is given by DuZ DuSCDuR , where DuS and DuR correspond to the stabilizing (damping injection) and voltage regulating loop control inputs, respectively. For simplicity, in the derivation
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J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
of the damping injection controller or stabilizing loop it will be considered that DuRZ0. The feedback function corresponding to the input DuR will be presented later in the paper. Consider the polynomial §3 ðsÞZ s3 C k3 s2 C k2 sC k1 such that all its roots {r1, r2, r3} are located in the open left-half complex plane. The following exact state-feedback function is proposed for the damping injection controller:
(a) Damping rate. Under the effect of (15), system dynamics are given by
DuS Z K1 ð3Þz1 C K2 ð3Þz2 C K3 ð3Þz3
To show that control law (15) provides arbitrarily damped dynamics, consider the scaled variables ziZ33-izi, iZ1–3. Then, system (17) can be rewritten as
(15)
where 3 is a positive parameter, and the gains K1(3) to K3(3) are given by 3
K1 ð3Þ ZKð3 k1 Ka1 Þ=g K2 ð3Þ ZKð32 k2 Ka2 Þ=g
(19)
z_3 ZK3 k1 z1 K3 k2 z2 K3k3 z3 3
2
2_ Z 3Ain 2
(20)
where
Controller (15) is basically a pole-placement feedback function, which cancels the open-loop dynamics (induced by Ka1z1Ka2z2Ka3z3) and impose the closed-loop poles {3r1, 3r2, 3r3}. In this way, by choosing suitable gains k1 to k3 , one can tune the parameter 3 to place the system poles {3r1, 3r2, 3r3} at positions very deep in the left-hand complex plane. One can recast the damping injection controller in physical variables in the following way. First, recall that z1ZDd and z2Zv. Since z3 is the mechanical acceleration _ one can write (15) as follows: (i.e., z3 Z v),
2
0
1
6 Ain Z 4 0
0
(17)
K1 ð3Þ zK33 k1 =g (18)
K3 ð3Þ zK3k3 =g which depends only on the parameter gZKb1 hc c! 0. On the other hand, one can notice that power angle deviation Dd of control law (17) is measured with respect to the operating angle d^s , which is not exactly known (this situation is equivalent to have unknown initial condition on the integral action of speed deviation). Consequently, the derivation of a practical excitation controller must rely on available estimates of the parameters and the operating point. Let us study the effects of the controller (15) on the damping rate, the location of the unstable operating point p^ s , and the robustness in the face of uncertain parameters and operating point.
0
Kk1 Kk2
3
1 7 5 Kk3
(21)
Since Ain is a stable matrix (in fact, its characteristic polynomial is §3 ðsÞ), there exists a positive-definite matrix Pin that satisfies the Lyapunov equation ATin Pin C Pin Ain ZKI3 , where I3 is the three-dimensional identity matrix. Consider the quadratic function V(z)ZzTPinz. The time-derivative of V(z) along the trajectories of system (20) is given by V_ ZK3j2j2 %K
Controller (17) constitutes a PID action on the speed deviation, that can be easily implemented with measurements of v. The advantage of the control configuration (16), (17) is that it depends only on a single parameter (3O0. Basically, 3 can be increased to enhance the damping capabilities of the SGIB system. It is interesting to note that in the high-gain limit (i.e., as 3 takes large values), the control gains are dominated by the imposed dynamics. In fact, for sufficiently large values of 3, one can obtain the approximation
K2 ð3Þ zK32 k2 =g
z_2 Z z3
(16)
K3 ð3Þ ZKð3k3 Ka3 Þ=g
Dus Z K1 ð3ÞDd C K2 ð3Þv C K3 ð3Þv_ ð Z K1 ð3Þ v ds C K2 ð3Þv C K3 ð3Þv_
z_1 Z z2
3 V lmin ðPin Þ
where lmin(Pin) denotes the minimum eigenvalue of Pin. From the above inequality one can get that V(t)%V(0)exp(K3t/ lmin(Pin)). Since lmin(Pin)jzj2%V(z)%lmax(Pin)jzj2, one has that jz(t)j%kjz(0)jexp(K3t/2lmin(Pin)), where kZlmax(Pin)/ lmin(Pin) is the condition number of matrix Pin. This implies that the trajectories z(t) converge arbitrarily fast with rate 3/ 2lmin(Pin), which is of the order of 3O0. That is, the damping rate induced by controller (15) can be made arbitrarily large by increasing the value of the single parameter 3O0. (b) Equilibrium points. Let us study effect of control law (15) on the location of the unstable equilibrium point p^s . Under (15), the equilibrium points of the SGIB system with DuouTZ0 are given by 0
^ ^ ^ b1 E^ q sinðdÞKb 2 sinðdÞcosðdÞ Z P 0
^ C b4 E^ q Khc K1in ð3ÞðdK ^ d^s Þ Z hc u^ Kb3 cosðdÞ
(22)
0 As before, one can eliminate E^ q to obtain the following relationship:
^ Z sinðdÞ
^ 2 b1 hc uc ^ d^s Þ ^ C ð33 k1 Ka1 ÞðdK b1 hc uc
(23)
^ Hence, the (d-component) equilibria of the controlled SGIB systems are given by the intersections of the sine function and a hyperbole. Notice that p^s is always an equilibrium point of the controlled SGIB system. To look for another equilibrium points, let us note that the hyperbole has a singularity at
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566 ^ c uc . In this way, depending on the sign of d^sg ð3ÞZ d^s K 3b31khK 1 a1 3 3 k1 Ka1 , three cases can be established: i) If 33 k1 Ka1 ! 0, then d^sg ð3ÞO d^s and the hyperbole has positive derivative with ^ In this case, there exists an additional intersection respect to d. ^da ð3Þ in the domain [0, p] corresponding to an unstable equilibrium point (see Fig. 3a); namely, 0 p^ a ð3ÞZ ½d^a ð3Þ; 0; E^ q;a ð3ÞT . ii) If 33 k^1 Ka1 Z 0, then ^ c, so that d^sg ð3ÞZ d^s . The equilibria are given by sin ðdÞZ the equilibria of the controlled SGIB system are the stable p^ u and the unstable p^ u ones of the uncontrolled system. In this case, controller (15) has no effects on the position of the equilibria. (iii) If 33 k1 Ka1 O 0, then d^sg ð3Þ! d^s and the ^ This hyperbole has negative derivative with respect to d. ^ ^ imply that there exists an intersection da ð3ÞO ds in the domain [p/2, p] corresponding to an unstable equilibrium point p^a ð3Þ. Moreover, d^sg ð3Þ/ p as 3/N, so that p^a ð3Þ converges to the boundary vDop of the operating region Dop (see Fig. 3b). From this analysis, one can conclude that control law (15) enlarges the potential region of attraction U of the operating point p^s in the sense that the unstable equilibrium point p^a ð3Þ is moved away from p^ s as the parameter 3 is increased. That is, although p^ s is not the only equilibrium point of the controlled SGIB system, the other (unstable) equilibrium point p^ a ð3Þ can be moved up to the boundary of the operating region. (c) Robustness. Unfortunately, the exact controller (15) cannot be implemented as it stands because in applications, the 0 parameters b1 to b4 and the operating point ps Z ½d^s ; 0; E^ q;s T are not exactly known [12,15]. Consequently, the derivation of a practical excitation controller must rely on available estimates of the parameters and the operating point. Let b1,e 0 to b4,e and ps;e Z ½d^s;e ; 0; E^ q;s;e T be estimates of b1 to b4 and 0 T ps Z ½d^s ; 0; E^ q;s , respectively. In particular, estimates the 0 0 voltage E^ q;s are rarely available in practice, so that E^ q;s;e Z 0 can be considered. These estimates yield the corresponding estimated parameters a1,e to a3,e and ge in the system (13). In the worst case, one can have that aieZ0, for iZ1–3, which
1.2
561
leads to the approximate control gains (18) with gZge. To study the robustness of the damping injection controller in the face of uncertain parameters and operating point, we will consider the system in z-coordinates (see Eq. (13)). The state z2Zv does not depend on the estimated parameters and _ operating point (i.e., z^2 Z 0). On the other hand, since z3 Z v, one has that z^3 Z 0. However, the state z1 is measured with respect to the operating angle d^s , which is not exactly known. The practical damping injection controller based on the estimates a1,e to a3,e, ge and d^s;e is given by DuS Z K1;e ð3Þz1 C K2;e ð3Þz2 C K3;e ð3Þz3 C K1;e ð3ÞDd^s where Dd^s Z d^s;e Kd^s and K1;e ð3Þ ZKð33 k1 Ka1;e Þ=ge K2;e ð3Þ ZKð32 k2 Ka2;e Þ=ge
(25)
K3;e ð3Þ ZKð3k3 Ka3;e Þ=ge Notice that controller (15) with gains given by (18) corresponds to the case where ai,eZ0, for iZ1–3. When the practical controller (24) is used in (5), one gets the controlled system z_1 Z z2 z_2 Z z3
(26)
z_3 ZKd1 ð3Þz1 Kd2 ð3Þz2 Kd3 ð3Þz3 C gK1;e ð3ÞDd^s where di(3)ZaiKgKi,e(3), iZ1–3. Notice that in the absence of uncertainties, system (26) reduces to the stable system (17). The 0 first effect of uncertainties is that ps Z ½d^s ; 0; E^ q;s T is not longer an equilibrium point of the controlled SGIB system (i.e., z^ s0). In fact, the equilibrium point of the linear system (26) is given by z^2 Z 0, z^3 Z 0 and z^1 ð3ÞZ gK1;e ð3ÞDd^s =d1 ð3Þ. Equivalently, the ^ is given by power angle equilibrium dð3Þ ^ Z d^s C gK1;e ð3ÞDd^s =d1 ð3Þ dð3Þ 1.2
(a) ε=5
(b)
δs
ε=4 0.9
δs
0.9 ε = 1.6
ε=3
δc
δsg < δs
0.6
0.6
δc
δsg > δs 0.3
0.3
0.0
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Equilibrium Power Angle (rad)
3.0
3.5
(24)
0.0
0.5
1.0
1.5
2.0
2.5
Equilibrium Power Angle (rad)
Fig. 3. Equilibrium points under damping injection controller.
3.0
3.5
8 min Du > > < S Dusat DuS S Z > > : max DuS
if
Dumin S % DuS
if
max Dumin S ! DuS ! DuS
if
Dumax S % DuS
(27)
The upper and lower limits of the stabilizing loop action was min chosen as Dumax S Z 6:0 and DuS ZK6:0, respectively. Fig. 4 shows the behavior of the SGIB system under controller (15) for three different values of the parameter 3 (namely 3Z3–5).
40 30 20 10 0 -10 -20 -30 -40 0
Vt (Volts)
That is, there is a steady-state offset gK1;e ð3ÞDd^s =d1 ð3Þ induced by uncertain parameters and operating point. Notice that ^ lim3/Njdð3ÞK d^s j s0, which means that the steady-state offset cannot be removed even with a high-gain feedback. On the other hand, the stability of the equilibrium point corresponding to the ^ angle dð3Þ is governed by the stability of the polynomial §e ðsÞZ s3 C d3 ð3Þs2 C d2 ð3ÞsC d1 ð3Þ. Some sufficient stability conditions can be derived for sufficiently values of 3. From elementary Routh-Hurwitz analysis, the following two conditions are sufficient to ensure stability of §e ðsÞ: (i) di(3)O0 for iZ1–3, and (ii) d3(3)d2(3)Kd1(3)O0. Since sign(ge)Zsign(g)ZK1 (in d^s and d^s;e are in (0, p/2)), di ð3ÞZ ai C ðg=ge Þ fact, 3KiC1 3 ki Kai;e and ki O 0, there exists a positive constant 3min such that di(3)O0, iZ1–3, for all 3O3min. This result implies that the damping injection controller is (locally) robust against uncertainties in the SGIB parameters and operating point. It should be stressed that an estimate ge can be obtained from available SGIB system information (see Eq. (14)). Remark 1. From the analysis above it is clear that the stabilizing loop or damping injection controller is of high-gain nature. Since in practice the control actuator is limited it becomes necessary to discuss the effect of bounded control actions on the closed-loop stability and performance. Please note that the first effect of the bounded input is to restrict the enlargement of the region of attraction. That is, the maximum size of the attraction domain will be determined by the control bounds. On the other hand, the transient performance of the system may also be deteriorated since the system cannot be arbitrary damped and the place of the poles cannot be arbitrary situated in the left-hand complex plane. Hence the damping added to the system will depend (again) on the control bounds. This result resembles the necessity of appropriate control actuator design in order to guarantee both the closed-loop stability and performance of the system. Example 1 (continued). Let us illustrate the damping capabilities of the stabilizing loop (15). To this end, a full finite-element model of a generator corresponding to Maxwell equations will be used. A detailed description of the finiteelement model used here can be found in [16]. The estimates ai,eZ0, for iZ1 to 3 (see Eq. (18)) are used. In this way, geZK0.67 is the only estimated parameter required to implement the stabilizing loop. The gains k1 Z 3:0, k2 Z 3:0 and k3 Z 1:0 has been chosen, so that all the roots of the polynomial §3 ðsÞ are located at K1 s-1. In practice, due to physical limitations, control action is subjected to take values into a certain domain Dumin ; Dumax . Hence, the actual control S S sat input, denoted by DuS , is computed as
w - ws (RPM)
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
1
2
3
16000 14000 12000 10000 8000 6000 4000
Delta (DEG)
562
4
Open-loop ε=5 ε=4 ε=3 0
1
0
1
2
3
4
2
3
4
90 80 70 60 50 40 30 20
Time (s)
Fig. 4. Effect of the tuning parameter 3 of the stabilizing loop on the SGIB system.
As expected, the larger the value of 3, the faster the convergence of the system dynamics. Furthermore, the equilibrium power angle of the system in these three cases are given by d^a ð3ÞZ 0:8785p, d^a ð4ÞZ 0:9485p and d^a ð5ÞZ 0:9758p: This shows how the proposed damping injection controller moves the unstable power angle away from d^s as 3 is increased. In this way, the proposed controller enlarges the potential region of stability as 3/N However, the achievable damping rate and the size of the stability region is limited by and lower Dumin limits imposed to the the upper Dumax S S stabilizing loop control action. It is interesting to notice that controller gains k1 , k2 and k3 ; can be defined to satisfy a prescribed (required) dynamic performance. Such gains can be computed for example, by minimizing a index performance criterium as in LQR or LQG (see also [15] for a robust design) By restricting controller parameter 3 to take values equal or greater than one and considering the case when exact parameters are available for feedback, gains k1 , k2 and k3 can be interpreted as the worst possible performance obtained by controller (15) (i.e. for 3Z1). Similarly as in the example above, one can obtain more damped responses by increasing parameter 3. On the other hand, in the uncertain case, there exists a minimum value 3pmin that can be used to satisfy the damping requirements. This is illustrated in Fig. 5, where the controller proposed by [15] is compared with control law (24). It can be observed that as 3 increases the proposed controller is able to damp more adequately the system. 3.2. Voltage regulating loop When the stabilizing loop controller is applied to the SGIB system, swing oscillations are well damped so that the SGIB
w - ws (RPM)
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
Consider the following PI controller
40 30 20 10 0 -10 -20 -30 -40
ðt
DuR Z KP eV C KI eV ðtÞdt
Delta (DEG)
Vt (Volt)
0.0
0.5
1.0
1.5
ε=2 ε=5 bourles 0.0
0.5
0.0
0.5
1.0
1.5
1.0
1.5
90 80 70 60 50 40 30 20 Time (s)
Fig. 5. Comparison of controller in [15] and the proposed damping injection controller.
system can operate under safe conditions. However, due to uncertain system parameters and unknown operating point, the terminal voltage can have a steady-state offset ðV^ t sVt;ref Þ. Under this situation, the voltage loop objective is to regulate the voltage VT at a given reference value Vt,ref. In this paper, a linear PI compensator will be used as the regulating loop controller. To this end, results on PI control stability in [17] u^R Þ will be used. The first step is to compute the gain KuV Z d4ðD dDu^R of the steady-state map V^ t Z 4ðDu^R Þ at the operating point p^s . ^ DuS C DuR and Du^in Z 0 at p^s (see Eq. (15)), one Since uZ uC has that ^ d4ðDu^ R Þ d4ðuÞ Z dDu^ R du^
(28)
That is, the steady-state gain KuV can be computed from the ^ with the stabilizing loop in off steady-state map V^ t Z 4ðuÞ position (i.e. Du^ S Z 0). From (1) and (2), we know that E^ q Z V s kc ^ E^ q ÞctgðdÞZ ^ ^ Hence, ^ sinðdÞ. kc u^ and Pe ðd; xds u 1=2 bVk ^ ^ V^ t Z 4ðuÞdef def b5 C b6 kc2 u^ 2 C 7 s c u^ cosðdÞ xds
(29)
The computation of the derivative (28) gives "
KuV Z
(31)
0
16000 14000 12000 10000 8000 6000 4000
KuV Z
563
1 bVk 2b6 kc2 u^ C 7 s c xds 2V^ t
^ u^ sinðdÞ ^ cosðdÞK
dd^ du^
!# (30)
b4 P ^ From (9), we know that sinðdÞZ b1 hc u^ . Moreover, ^ d d ^ ! 0 at dZ ^ d^s . Consed^s 2 ½0; p=2. This implies that sinðdÞ du^ quently, for all p^ s , (i) KuVO0 and (ii) the steady-state map ^ is invertible (for each u^ there exists one and only one V^ t Z 4ðuÞ V^ t , and vice versa).
where eV def defVt;ref KVt is the voltage regulation error, and KPO0 and KIO0 are the proportional and integral control gains. It is noted that the sign of the steady-state gain determines the sign of the PI control gains (i.e. sign(KP)Z sign(KuV)). Theorem 2.1 in [17] implies that there exist small enough gains KP and KI such that, for all initial conditions ½d0 ; v0 ; E 0 q;0 in a neighborhood of the operating point p^s , the corresponding states ½dðtÞ; vðtÞ; E 0 q ðtÞ are bounded and Vt ðtÞ/ Vt;ref as t/N. That is, there exist PI control gains KPO0 and KIO0 such that the controlled SGIB is (locally) stable and the generator terminal voltage VT converge to the prescribed reference value VT,ref. It should be stressed that the above is only a robustness result in the sense that it only states the existence of controller gains KP and KI that provides asymptotic regulation of the terminal voltage. In practice, one should use a suitable procedure to tune the PI control (31), which can be borrowed from Internal Model Control (IMC) ideas [18]. This can be done by recalling that the regulating loop action is supported on the stabilizing loop action, which introduces a strong damping effect into the controlled SGIB system. In this way, input– output dynamics DuR(t)/DVT(t) can be approximated as a stable first-order system DuR ðsÞ KuV Z DVt ðsÞ Ts C1
(32)
where s is the Laplace variable (i.e. sZd/dt) and TuVO0 is a given time-constant. Two procedures can be proposed to obtain estimates of KuV and TuV: (i) the steady-state gain KuV can be estimated from (30) and the time-constant TuV can be taken of the order of Tdo , which determines the worst damping conditions under the stabilizing loop control action; and ii) KuV and TuV can be estimated from standard step or frequency responses. Given the estimated values KuV and TuV, IMC tuning rules give: 0 1 1 @ TuV A KP Z KK uV TR (33) K1 1 KI Z KuV TR where TRO0 is a tuning parameter corresponding to the timeconstant of the regulating loop. In this way, the tuning of the PI control (31) depends only on the single parameter TR. Moreover, the smaller the value of TR, the faster the convergence VT(t)/VT,ref [18]. Example 1 (continued). Let us illustrate the damping capabilities of the proposed two-loop excitation control scheme with stabilizing loop gains given by Eq. (18). As in the stabilizing loop case, the regulating loop control action is max subjected to take values into a certain domain ½Dumin R ; DuR .
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Hence, the actual regulating control input, denoted by Dusat R , is computed as 8 min Du > > < R sat DuR Z DuR > > : max DuR
if
Dumin R R DuR
if
max Dumin R ! DuR ! DuR
if
Dumax R % DuR
(34)
w - ws (RPM)
The upper and lower limits of the regulating loop control min action was chosen as Dumax R Z 2:0 and DuR ZK2:0, respectively. The transfer function (32) was computed from a firstorder approximation of a step perturbation on the SGIB system under the stabilizing loop control action. In this way, the computed parameters are the following: KuVZ0.162, TuVZ 2.5 s. The IMC parameter TR was set equal to 1 s. Fig. 6 shows the behavior of the SGIB system under a short circuit event under the controller (24), (25) with 3Z4 sK1 and the voltage controller (31) tuned with the IMC rules (33). It is noted that the terminal voltage presents a steady-state offset (i.e. V^ t sVt;ref ) when the regulator is not acting on the system. This offset is removed when the regulating loop controller is connected, such that VT(t)/VT,ref. To show that the proposed two-loop scheme is able to provide stable operation with asymptotic terminal voltage regulation, Fig. 6 presents the SGIB dynamics under setpoint changes in the terminal voltage. Notice that the two-loop scheme provides smooth setpoint ^ 728 does not tracking in spite that the nominal angle dZ correspond to the prescribed terminal voltage conditions. 40 30 20 10 0 -10 -20 -30 -40
Vt (Volt)
It has been shown that the two-loop excitation control scheme composed by the feedback function (15) (stabilizing loop) and the PI compensator (31), (33) (regulating loop) is able to provide (local) damping of swing oscillations and asymptotic terminal voltage regulation about a given setpoint. Besides, the potential region of attraction of the operating point is enlarged in the sense that the unstable equilibrium point is displaced far from the (stable) operating point. The objective of this part of the paper is to reveal the control structure of the two-loop excitation control. The structure of the stabilizing loop depends on the choice of the input quantities, with the most commonly used quantities being: shaft speed deviation v, the accelerating power Pa [1], the transient EMF Eq and the frequency deviation Df. As each of these signals has its advantages and disadvantages, the damping injection loop is often designed to operate on a number (usually two) of these input signals [1,12]. In particular, the angle d is rarely available from measurements. Let CR(s) and CS(s) be the transfer functions of the regulating and stabilizing loops, respectively. Then, DuðsÞ Z CR ðsÞeV ðsÞ C CS ðsÞvðsÞ Z CR ðsÞ½eV ðsÞ C CR ðsÞK1 CS ðsÞvðsÞ
1
2
3
4
washout filter ffl{ stabilizing gain zfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflffl
z}|{ K
stabilizing and regulating loops stabilizing loop
0
1
2
3
4
90 80 70 60 50 40 30
when the shaft velocity deviation v is the signal used for control input computation. From Eq. (24), one has that
1
2
3
(37)
It is noted that (37) contains an ideal integrator 1/s and an ideal differentiator s. Controllers in industrial power systems has been traditionally implemented as analog controllers. The implementation of the ideal integrator and the differentiator on an analog framework led commonly to expensive devices. Frequently, a cheaper static electric network consisting of resistors and capacitors sufficiently approximates the differentiator s and the integrator 1/s. This is made as follows [19] 1 Ta s C 1 z ; Ta / Tb s Tb s C 1
0
lead lag blocks
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ Tw s T1 s C 1 T3 s C 1 Tw s C 1 T2 s C 1 T4 s C 1 (36)
DuS ðsÞ 1 Z K1;e ð3Þ C K2;e ð3Þ C K3;e ð3Þs vðsÞ s
16000 14000 12000 10000 8000 6000 4000
(35)
Secondly, let us show that KCR(s)-1CS(s), displays the following structure
KCR ðsÞK1 CS ðsÞZ
0
delta (DEG)
4. Controller structure
(38)
4
Time (s)
Fig. 6. SGIB system during a short circuit event under the proposed two-loop controller.
and sz
Tc s C 1 ; Tc [ T d Td s C 1
(39)
J. Alvarez-Ramirez et al. / Electrical Power and Energy Systems 27 (2005) 556–566
where Ta to Td are suitable time-constants. In this way, the practical damping injection controller is given by DuS ðsÞ T s C1 T s C1 Z K1;e ð3Þ a C K2;e ð3Þ C K3;e ð3Þ c vðsÞ Tb s C 1 Td s C 1
1.1
1.0
After some algebraic manipulations, the following practical transfer function is obtained:
0.9
(41)
where
Voltage, V (p.u.)
(40)
uS ðsÞ b1 s2 C b2 s C 1 Z CS ðsÞ Z KS vðsÞ ðTb s C 1ÞðTd s C 1Þ
565
Lead-Lag Exact
0.8
0.7
KS Z K1;e ð3Þ C K2;e ð3Þ C K2;e ð3Þ 1 b1 Z KK S ½Ta Td K1;e ð3Þ C Tb Td K2;e ð3Þ C Tb Tc K3;e ð3Þ 1 b2 Z KK S ½ðTa
(42)
C Td ÞK1;e ð3Þ C ðTb C Td ÞK2;e ð3Þ
C ðTb C Tc ÞK3;e ð3Þ
0.5 0
Notice that b1O0 and b2O0. If b22 K4b1 O 0, the numerator in (41) can be written as (TesC1)(TfsC1), 1 K1 where KTK e ! 0 and KTf ! 0 are the roots of the 2 polynomial b1 s C b2 sC 1Z 0. That is, the transfer function CS(s) can be written as the composition of a stabilizing gain Kin and two lead-lag blocks as follows: CS ðsÞ Z KS
0.6
Te s C 1 Tf s C 1 Tb s C 1 Td s C 1
On the other hand, the transfer function of the PI compensator (31)–(33) can be written as TuV s C 1 CR ðsÞ Z KP (43) TuV s In this way, one obtains KCR ðsÞK1 CS ðsÞ TuV s Te s C 1 Tf s C 1 ZKKP KS TuV s C 1 Tb s C 1 Td s C 1
(44)
Summarizing, it has been shown that the proposed twoloop excitation control configuration has structures of traditional control of power systems (i.e. wash-out filters, lead-lag blocks). 5. Implementation issues The practical two-loop controller is given by the stabilizing loop controller (24) and the regulator (31). The implementation of this practical excitation controller depends on: (a) the relative speed of the generator vZuKus and the terminal voltage VT, which are always available from measurements, (b) an estimate d^s;e of the operating power angle d^s , which is also available in practice, (c) an estimate ge of the high-frequency gain g, which can be estimated from standard synchronous machine modeling, and d) an estimate of the steady-state gain KuV and the timeconstant TuV, which can be estimated from plant data or system step response. In this way, the implementation of the proposed two-loop excitation controller does not require the exact
3
6
9
12
15
Time, t (s) Fig. 7. Performance of the two-loop configuration under lead-lag approximation when a short circuit occurs.
knowledge of the operating point or measurement of the power angle d(t). In this way, the implementation of the twoloop excitation controller can be made on the basis of available practical information of the SGIB system. The tuning of the two-loop excitation controller is particularly easy to carry out. The damping injection tuning depends only on the single parameter 3O0, which can be increased to enhance the damping of the controlled SGIB system. Numerical simulations suggest that values of 3 about 3–5 sK1 give good performance with acceptable damping rate. The regulating loop tuning depends only on the single timeconstant TuV, which can be reduced to accelerate the terminal voltage convergence. Numerical simulations in this paper suggest that values of TuV of the order of 0.5–1.0 s provide fast voltage convergence. These features facilitate the possible practical implementation of the two-loop controller. Example 1 (continued). Let us illustrate the performance of the two-loop excitation control configuration under the lead-lag approximations (40) and (41). The controller was tuned as follows, TbZ20 s and TdZ1.4 s were chosen [1,15]. Moreover, TaZ0.5 and TcZ20 s were chosen to meet the constraints Ta/Tb and Tc[Td. Fig. 7 presents the dynamics of the terminal voltage with exact and approximate integrator and differentiator. It is noted that the lead-lag filters (40) and (41) induce a small delay in the convergence of the SGIB system to the operating conditions, although the two-loop control performance is still acceptable. This simulations show that the practical approximations (40) and (41) do not degrade significantly the performance of damping injection controller (15). 6. Conclusions A two-loop excitation controller for synchronous machines has been proposed in this work. The stabilizing loop is a simple
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linear state-feedback excitation controller that increases the damping and enlarges the potential region of attraction of the operating point by moving away the unstable equilibrium point. The voltage regulator is composed basically by a PI compensator, which provides asymptotic regulation of the terminal voltage while maintaining ultimate boundedness of all trajectories in the domain of operation. Since the damping injection controller only requires measurements of the speed deviation and an estimate d^e of the operating power angle, it can be implemented in practice. The results in this paper show that it is possible to propose simple excitation control schemes by departing from modern state-space methods. Maybe one of the main advantages of this approach is that the tuning of the controller parameters is very systematic and straightforward. Specifically, the tuning of the damping action depends on the parameter 3 and an estimate of the high-frequency gain g, which is always available in practice. On the other hand, the tuning of the voltage controller requires estimates of a timeconstant and a steady-state gain, which can be obtained from plant data (e.g. step response). References [1] Kundur P. Power system stability and control. McGraw Hill; 1994. [2] Law KT, Hill DJ, Godfrey NR. Robust co-ordinated PSS-AVR design. IEEE Trans Power Syst 1994;9:1218–25. [3] Machowski J, Bialek JW, Robak S, Bumby JR. Excitation control system for use with synchronous generators. IEE Proc-Gener Transm Distrib 1998;145:537–46. [4] Bazanella AS, Silva AS, Kokotovic PV. Lyapunov design of excitation control for synchronous machines. Proceedings of 1997 conference on decision and control, San Diego California, USA 1997. [5] Sepulcre R, Jankovic M, Kokotovic PV. Constructive nonlinear control. Springer; 1996.
[6] Wang Y, Xie L, Hill DJ, Middleton RH. Robust nonlinear controler design for transient stability enhacement of power systems Proceedings of 1992 conference on decision and control, Tucson Arisona, USA 1992. [7] Galaz M, Ortega R, Bazanella A, Stankovic A. An energy-shaping approach to excitation control of synchronous generators Proceedings of 2001 American control conference, Arlington, Virginia, USA 2001. [8] de Leon Jesus, Gerardo Espinosa-Perez, Ivan Macias. Observer-based Control of a synchronous generator: a Hamiltonian approach. Int J Electric Power Energy Syst 2002;24(8):655–63. [9] Pogromsky AY, Fradkov AL, Hill DJ. Passivity based damping of power system oscillations Proceedings of 1996 conference on decision and control, Kobe Japan 1996. [10] Middlebrook RD. Topics in multi-loop regulators and current-mode programming. IEEE Trans Power Electron 1987;2:109–24. [11] Luyben WL. Parallel cascade control. Ind Eng Chem Fundam 1973;463: 1973. [12] Machowski J, Bialek JW, Bumby JR. Power system dynamics and stability. Wiley; 1997. [13] Wiggins S, Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer; 1990. [14] Bergan AR, Bergan AR. Power systems analysis. New Jersey: PrenticeHall; 1986. [15] Bourles H, Peres S, Margotin T, Houry MP. Analysis and design of a robust coordinated AVR/PSS. IEEE Trans Power Syst 1998;13(2): 568–75. [16] Escarela-Perez R, Arjona-Lopez MA, Melgoza-Vazquez E, CamperoLittlewood E, Aviles-Cruz C. A comprehensive finite-element model of a turbine-generator infinite-busbar system. Int J Finite Elem Anal Des 2004; 40(5):485–509. [17] Desoer ChA, Lin ChA. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans Autom Control 1985; 30(9):861–7. [18] Morari M, Zafiriou E. Robust process control. New Jersey: Prectice-Hall; 1989. [19] D’Azzo JJ, Houpis CH. Feedback control systems analysis and synthesis. 2nd ed. Tokyo: McGraw Hill; 1966.