Discontinuous Controller for Power Systems ... - Semantic Scholar
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Discontinuous Controller for Power Systems: Sliding-Mode Block Control Approach Alexander G. Loukianov, José M. Cañedo, Member, IEEE, Vadim I. Utkin, Fellow, IEEE, and Javier Cabrera-Vázquez
Abstract—Based on the complete model of the plant, a sliding-mode stabilizing controller for synchronous generators is designed. The block control approach is used in order to derive a nonlinear sliding surface, on which the mechanical dynamics are linearized. This combined approach enables us to compensate the inherent nonlinearities of the generator and to reject high-level external disturbances. A nonlinear observer is designed for estimation of the rotor fluxes and mechanical torque. Index Terms—Nonlinear system, observer, power system control, sliding-mode control, synchronous generator.
I. INTRODUCTION
O
VER THE LAST 15 years, two major problems have been faced in the area of modern power system control: the problem of loss of synchronism or angle instability, and the problem of terminal voltage instability and collapse due to over loading on transmission line, reactive constraint and fault. Therefore, a fundamental problem in the design of feedback controllers for power systems is that of robust stabilizing both frequency and voltage magnitude, and achieving a specified transient performance. Robustness implies operation with adequate stability margins and admissible performance level in spite of plant parameters variations and in the presence of external disturbances. This paper deals with excitation control of a single synchronous machine connected to an infinite bus (Fig. 1). The control schemes of synchronous machines are commonly based on a reduced-order linearized model and classical control algorithms that ensure asymptotic stability of the equilibrium point under small perturbations [1], [2]. Recently, to overcome the limitation of linear control, attention has been focused on implementation of modern control technique, e.g., an adaptive linear control [3]–[6], passivity-based approach [7]–[10], intelligent control such as fuzzy logic [11]–[13] and neural networks [14], control based on direct Lyapunov method [15], [16], feedback linearization (FL) technique [17]–[21], and control based on adaptive FL [22], [23].
Manuscript received March 11, 2002; revised August 25, 2003. Abstract published on the Internet January 13, 2004. This work supported by CONACyT, Mexico, under Grant 36960A. A. G. Loukianov and J. M. Cañedo are with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara 44550, Mexico (e-mail: [email protected]). V. I. Utkin is with the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210-1272 USA. J. Cabrera-Vàzquez was with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara 44550, Mexico. He is now with the Electronic Engineering Department, University of Guadalajara, Guadalajara 44840, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/TIE.2004.825264
Fig. 1.
Syncronous machine connected to an infinite bus.
All of the mentioned controllers provide larger stability margins with respect to traditional ones. But these control schemes were designed for a reduced order plant. In [24], it has been shown however that the effects of unmodeled stator and rotor electrical dynamics cannot be neglected since they affect the electromechanical dynamics in the case of large perturbation. The detailed seventh-order model of synchronous machine (five equations for electrical dynamics and two ones for mechanical dynamics) has been considered and a nonlinear controller using this model and FL technique, has been designed in [25] to enhance transient stability. The proposed nonlinear control law is a function of all plant parameters and disturbances. In practice some of these parameters are subjected to variations as a result of a change in the system loading and/or in the system configuration. Since the detailed model is so involved, a direct use of the FL technique results in a computationally expensive control algorithm. Moreover, this control scheme does not take into account practical limitation on the magnitude of the excitation voltage, and an observer design problem was not solved. On the other hand, a fruitful and relatively simple approach, especially when dealing with nonlinear plants subjected to perturbations, is based on the use of variable-structure control (VSC) with sliding mode [26]. First and foremost, this enables high accuracy and robustness to disturbances and plant parameter variations to be obtained. Second, the control variables of the basic sliding mode control law rapidly switch between extreme limits, which is ideal for the direct operation of the switched mode power converters of synchronous generators. Sliding-mode controllers for power systems have been designed in [27]–[35], however, for reduced-order plants only, to the best of our knowledge. Application of these controllers to a full-order plant would cause undesirable chattering and impair accuracy, since unmodeled dynamics can be excited. In this paper, based on the full-order plant model, we shall resort to the block control [36] and sliding-mode techniques [26], the combination of which overcomes most of the mentioned problems: they are simple then [25], with low computational demands, and take into account structural constraints of the controller. The block control technique is applied to design a nonlinear sliding surface in such a way that the sliding-mode
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LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
dynamics are represented by a linear system with desired eigenvalues. For a given bound on the control signal a discontinuous control law ensuring stability of the sliding mode is proposed. The main feature of the proposed control scheme is robustness with respect to disturbances and plant parameter variations [37]. This paper is organized as follows. Section II reviews the detailed eighth-order model of the synchronous machine. In Section III, the block control technique is extended for design a discontinuous control strategy for a class of nonlinear systems presented in so-called Block Controllable Form with unmatched perturbations, and internal dynamics. This class describes dynamics of electromechanical plants, in particular, power systems. In Section IV, a nonlinear observer design is presented. Based on these results, in Section V, a sliding mode observerbased excitation controller for a single synchronous machine connected to an infinite bus, is derived. Section V discusses simulation results.
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with
where and are the direct-axis and quadrature-axis self-inis the rotor self-inductance; and are ductances; the direct-axis and quadrature-axis damper windings self-inducand are the direct-axis and quadrature-axis tances; magnetizing inductances. Subtracting from (2), and gives the following transformation: (3)
II. PLANT MODEL The complete mathematical model of the single machine infinite-bus system consists of electrical and mechanical dynamics and load constraints. The electrical dynamics comprising the stator and rotor damper windings, with the currents as the state variables, after Park’s transformation, can be expressed as follows [24]:
which reduces the system (1) to the form (4)
(1) where
. The complete where mathematical description includes the swing equation [24] (5) (6)
and are the the field flux and current, respectively; , , , , , and are the direct-axis and quadrature-axis damper windings fluxes and currents, respectively; , , and are the direct-axis and quadrature-axis stator fluxes and is the excurrents, respectively; is the angular velocity and and are the direct-axis and quadracitation control input; ture-axis terminal voltages;
is the rated synchronous speed, where is the power angle; is the inertia constant; is the mechanical torque applied is the electrical torque, expressed in terms to the shaft; and of the currents as follows: (7) The mechanical torque function of time. Thus,
is assumed to be a slowly varying (8)
and are the stator and field resistances; , , and are the damper windings resistances. Analysis of relationship between fluxes and currents shows that the sensitivity of the fluxes with respect to parameters variations is less then the sensitivity of the currents. Therefore, it is more suitable the representation of the electrical dynamics in and the stator current . This kind of terms of the rotor flux model can be obtained from (1) using the following transformation between fluxes and currents: (2)
The equilibrium equation for the external network of the synchronous machine connected to an infinite bus is (9) where
is the value of the infinite-bus voltage; and are the transformer plus transmission line resistance and inductance. Here, all the state variables as well as the parameters of the model (1)–(9) are expressed in per unit. Combining (1)–(9), the
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complete model of the generator is presented in the state-space form
must be chosen such that the matrix rank, and the sliding-mode equation
,(
) has full (16)
(10) where where is the control,
,
, and
has the desired properties, including stability as a minimum requirement. Secondly, a discontinuous control if if
(17)
is designed to guarantee convergence of the motion projection on the subspace , governed by is defined by the equation shown at the bottom of the page, , ,( ), , , and are , , , , positive constant parameters depending on , , , , , , , , and .
where and , are smooth functions to be selected. If the disturbance satisfies the so-called matching consuch that dition [38], i.e., there exists a matrix
III. CONTROL METHOD In this section, the method and underlying ideas are described in generic terms to show the generality of our approach. A. Formulation of the Problem
and then reduces to simply
. In this case, (16)
(19)
Consider the following system subject to disturbance: (11) (12) is the state vector, where control vector to be bounded by
(18)
is the (13)
, is a vector of external disturbances generated by an external system described by (14) and the columns of and The vector field are smooth and bounded mappings of class . It is assumed , , , . The that standard sliding mode design procedure is comprised of the two subproblems. First, the nonlinear sliding manifold in the state space of the system (15)
), however, it is This equation has the reduced order ( still nonlinear and nonautonomous. One possible approach to ensuring stability of system (19) in the autonomous case is connected with the input–output linearization techniques [39]. Another approach is the ”backstepping” [40] that is based on the use of step-by-step Lyapunov functions. In this paper, the decomposition block control method is adopted to design a nonwhich stabilizes the linear time-varying sliding function sliding-mode equation. Another important aim is to provide inin cases where it does variance with respect to disturbance not satisfy (18). B. Block Control for a Class of Nonlinear Systems Consider the following decomposed system: (20) (21)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
where
is an
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1 new variables vector,
,
(22) (23) (24) where the vector
is decomposed as , , , is a 1 vector, and the indexes ( ) define the structure of the system and satisfy the following relation:
and is the identity matrix. Proposition 3: Suppose that the assumptions H1) and H2) are satisfied. Then, the transformation (29) reduces the system (20)–(23) to the following desired form:
(25) Definition 1: The system (20)–(23) is referred to as Nonlinear Block Controllable Form with disturbance and internal , multiplying the fictitious control dynamics if the matrix in each th block of (20)–(23), has full rank (26) Remark 2: The model of the synchronous generator has the form (20)–(24) with the load angle as output. or , therefore, The relation (25) means consider the plant with the structure (27)
(30) , is a bounded function, , . Proof: In order to prove the proposition the method of in, , as a duction will be used, considering fictitious control vector in each th block of (20)–(22). Step 1: Define the tracking error as where
(31) . Therefore, the inverse As follows from (27), we have under condition (26) exists. Let the fictitious control matrix in the first block rewritten as
that includes both cases. Suppose that the output should follow produced by given external system the reference signal (28) The control design procedure for the system (20)–(24) consists of three parts. First, using block control technique a transformation of the system (20)–(22) to a desired form is derived. As result a desired sliding manifold will be derived. Secondly, a discontinuous control strategy which stabilizes sliding mode is proposed, and finally, the stability of a sliding-mode equation with zero dynamics is analyzed. In order to reduce the system (20)–(22) to a desired form the following assumptions are introduced. , and H1): The elements of are continuously differentiable functions of th order, , with respect to all arguments in interval , and all derivatives are bounded. H2): The condition (26) is satisfied. Based on these assumptions and taking into account the structure (27), the following recursive transformation is introduced:
(32) with
, be selected as (33)
where is an 1 new variables vector, . The transand , with input (33) formed first block in new variables . has the desired form (30), i.e., Step 2: Taking the derivative of (29) along the trajectories of (20)–(21), results in (34) where
and , . From relation (27) it follows . Therefore, the matrix is not square, in (34) is chosen as and the fictitious control (35) 1 new variables vector, , denotes where is a pseudoinverse matrix of , . Thus, (34) with (35) takes the form (30), that is, . Now, the following assumption is introduced. can be orH3): Assume that the elements of the matrix dered such that the square matrix
(29)
, has
with .
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Based on this assumption, the variable using (35), of the form (29), that is,
can be obtained
where ,
Step : At this stage it is possible to show that if we have, ) steps, the transformed blocks of the system after ( (for (20)–(22) with new variables (27)) of the form
, and , . Thus, transformation (29) reduces the system (20)–(23) to (30). Remark 4: Note that the assumption Hi permits us to use with constant parameters, which simplifies the the matrix transformation. If this assumption is not satisfied, then it is poswhich orthogonal sible to use, for instance, the matrix , instead of . to Now, in order to generate sliding mode in (30), a natural choice of the sliding manifold using transformation (29) is (41) Then, the following discontinuous control is proposed:
(36)
(42)
(37)
Proposition 5: The control law (42) guaranteies the convergence of the closed-loop system motion to manifold (41) in a finite time defined as
with
then on the th step of the transformation procedure, we will have the transformed the th block with new state vector similar to (36). Then, the equation for (37) with (20) and (21) is of the form
Proof: Taking the derivative of trajectories of (40)–(42) gives
along the
(38) Using the following relations , we have:
where , , , the vector (35), of the form
and
, . For the case in (38) can be selected similar to In the domain (39)
where
is pseudo inverse matrix of , . Thus, (38) with (39) takes the same form as (36), that is, . For this step, the assumption H3 is generalized as follows. can be ordered such that Hi): The elements of the matrix
(43) , is negative the derivative definite, that guarantees convergence of the state vector to the (41). In order to demonstrate this convergence manifold , and using is finite, first we assume that , we have . Therefore, , or (44)
Under this assumption, the recursive transformation can be obtained as
Using the comparison lemma [41], a solution of (44) can be . Thus, . Therefore, vanishes in , and sliding mode some finite time, after this time. starts on the manifold Now, a sliding mode equation of reduced, th order can be derived of the form estimated as
Step : On the last step, calculating the time derivative of (29) gives the last block
(45)
(40)
(46)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
, , and is a solution of the equation (40) . In order to analyze the calculated as stability of a solution of (45) and (46), first the variables of this and . system are redefined for simplicity as Then, the system is represented as
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where
(47) (48) with
, and
.. . .. .
.. . .. .
.. .. .
. ..
.
.. . .. .
.. . .. .
(56) where , , , and are positive constants. Proposition 6: Suppose assumptions A1) and A2) hold, and (57) where . Then the solution of the system (47)–(48) is uniformly ultimately bounded, and the converges exponentially to zero. tracking error along the traProof: Calculate the derivative of . As jectories of (48), follows from (55) and (56),
In the domain defined as for
where . The behavior of (47) and (48) will be ana: lyzed in the region (49) (50) in which the sliding-mode existence condition (43) holds. Since the matrix is Hurwitz, the solution of subsystem (47) is exponentially stable, that is, (51) where
is a positive-definite solution of the Lyapunov equation for . Then, the dynamics of the system (47)–(48) on the invariant subspace are governed by the system
the derivative tive definite, then
(58)
with
is nega, and (59)
, . Under condition (57), we where . That have means the solution belongs to region (50) for . In order to estimate an ultimate bound of the all of (47)–(48), first is defined from (58) solution . using (51) as Then taking into account condition (57), gives . Therefore, application of Corollary 5.3 [41] gives or . tends to zero, solution is uniformly ultimately Since . Thus, taking into account (59), the bounded by solution can be finally estimated as
(52) that presents the zero dynamics. This system is subject to nonvanishing perturbation. Therefore, we introduce the following assumptions. A1): A solution of zero dynamics (52) converges exponentially to a compact set (53) A2): The mapping in time and locally Lipshitz in
is continuous and
such that (54)
where is a Lipshitz constant. By assumption A1) and the converse theorem [41], there exin a Lyapunov ists for all function , with the following properties:
(55)
In addition, since converges exponentially to zero, we . have The results obtained in Propositions 3–6, can be formulated in the following theorem. Theorem 7: If the assumptions H1)–Hr), A1) and A2), and the conditions (57) are satisfied, then a solution of the closed-loop system (20)–(23) with (42)–(41) is uniformly ultimately bounded, and the tracking error (31) converges exponentially to zero. IV. NONLINEAR OBSERVER In this section, we deal with an observer problem for a class of nonlinear perturbed systems governed by (60) (61) (62) where and the measured state vectors, and
are is the
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state vector to be estimated, and are appropriately linear and bilinear functions with respect to , that is,
by assumption B2 such that . The solution of (69)–(70) with as
, can be expressed
(63) (71) (72)
(64) , and are matrices with constant elements, and a perturbation produced by an given external system
is , and , , . The stability of this solution is analyzed by application of the Lyapunov’s characteristic indexes (LCI) technique [42]. First, (72) defined as the LCI of the solution
with (65)
The following is assumed. B1): The system (60)–(62) is input-to-state locally stable. is observable.
B2): The pair
is Hurwitz. B3): The matrix Note that models of electrical systems, including generators and motors, can be represented in the form (60)–(64) presents mechanical variables, the speed where vector presents the stator currents, vector and angle; vector presents the rotor fluxes, and presents the load torque (for motors) or mechanical torque (for generators). In the case of , , synchronous generator, we have , and . In order to estimate and , the following nonlinear observer with linear correction terms is proposed:
is negative since matrix by assumption B3 is Hurwitz. Furthermore, by assumption B1 the state of (60)–(62) is ultimately bounded for a bounded control input . Hence, the elements of and, consequently, are bounded. That results in . From the LCI properties [42] it follows
Hence, the LCI of
(66) where , , and are estimates of vectors , and , respectively; and are observer matrices. Using (60)–(66), the following observer error system is derived: (67) (68) , , , and . Thus, error system (67) and (68) is considered as a linear system with time-varying parameters. The stability of this system is investigated in the following theorem. Theorem 8: If the assumptions B1)–B3) are met then there and such that a solution of (67) and (68) is asympare totically stable. Proof: The system (67) and (68) by the nonsingular transwhere
and
formation
is reduced as (69) (70)
(71)
is negative, too. This means that the solution of (69) and (70) and, consequently, (67) and (68) is asymptotically stable, that is, , . Hence, , and . V. GENERATOR CONTROL DESIGN In this section, based on the above results, a sliding-mode observer-based excitation controller for a single synchronous machine connected to an infinite bus is derived. The control oband terminal voltage stajectives are rotor angle , speed bility enhancement. First, a mechanics stabilizing control design is studied. Secondly, a voltage control design is outlined, and finally, a control switching logic is proposed. A. Mechanics Stabilizing Control The sliding mode controller design procedure will be divided into two steps. First, exploring the block control technique presented in Section III, a sliding surface will be formed. Then, a discontinuous control law will be designed to make attractive this surface. The first subsystem of (10) is presented in the Block Controllable Form (20)–(22) which consists of three blocks (73)
where , , are eigenvalues of
, , and , . We can choose the matrices
, and
(74) (75)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
while the second subsystem of (10) presents the internal dynamics (23). The perturbation is produced by the exosystem , and control to be bounded by (76) is the maximum value of the excitation voltage. The where control goal is to make the angle be equal to a reference signal , and the speed be equal to the rated synchronous speed . In accordance with the block control technique, we define as the control error (77) Then, selection of fictitious control , where gives
in the first block as and is a new variable,
(78) Using (77) and (78), (73) and (74) in terms of new variables and may be written as (79) (80) where
In the second step, the selection of fictitious control , gives the switching function
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guarantees The control (85) under the condition in a finite time the convergence of the state to surface interval. Once this is achieved, the sliding motion is governed in the full state space by (86) (87) Therefore, the sliding-mode control (85) yields invariant , in the subspace state space of the closed-loop system. The dynamics on this invariant subspace are zero dynamics. In of in (87) is reorder to derive these dynamics, vector , placed by : where mapping , is defined by (77), (78), and (81). , , and Then, is zeroed, that means , thus, (88) Note that sliding-mode dynamics (86) and (87) can be considered as a particular case of the system (47) and (48) or (45) and (46), while zero dynamics (88) is a particular case of (52). Since the mapping in (88) is smooth, assumptionA2) (see Section III) in this case is met. Now, we will show that assumption A1 is also satisfied. The right-hand side of (88) consists of linear and nonlinear parts, thus, with
in (80) as ,
(81) is described such that a sliding-mode motion on surface ) by the following linear system: in the state space ( (82) ) and ( ). This system correwith desired eigenvalues ( sponds to the linearized mechanical dynamics of the closed-loop and means and system. Note that . Therefore, the control goal requires be a constant. Thus, and will be put in (78) and (81) to zero. The switching function design has been outlined. Now, a control law will be investigated. Projection of the closed-loop system motion on subspace can be described by (83) , , and is a positive function for . Then, the equivalent control is (83) of the form calculated as a solution of where
For these dynamics, the following assumptions are introduced. Hurwitz. C1): The matrix is , . C2): By assumption C1) there is a Lyapunov function , with the following properties: and where is a positive-definite solution of the Lyapunov equation for , and . calculated as Using the above relations, the derivative for is negative in the region . In this region a solution of (88) estimated as
with
(84) Now, taking into account (76), the control is proposed as
and (85)
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exponentially converges to the compact set (53) defined as
is the equivalent control calculated from (93) where , guarantees the converas gence of the control error to zero in a finite time. C. Control Switching Logic
that is, assumption A1) (see Section III), is met. Hence, by Proposition 6 a solution of (86) and (87) is ultimately bounded, and the control error converges exponentially to zero.
Having just one control input for two switching surfaces and , and taking into account (85) and (94), a control strategy can be proposed, finally, of the form if if
B. Generator Voltage Control The control objective is the control of terminal voltage, defined as (89) Using (10), (9) can be represented as (90) where
and . Because parameter is very small, the control term in (90) can be neglected, and then we have
if if
(96) (97)
with . Therefore, we are proposing a hierarchical control action through the proposed logic (96) and (97). Since the mechanical dynamics are slower than the electrical ones, we spend reaches the control resources, at first, to stabilize . When the boundary layer with width , the control resources will be spending to stabilize , that is, to regulate terminal voltage . , the control action After convergence of such that to , ( ). reduces the boundary layer width from Thus, the controller maintains the value of and within desired accuracy and . In order to avoid oscillations in steady state, the load angle must correspond to the terminal voltage refreference value mechanical torque value , such that to have erence value in steady state the following quality:
Using (1)–(7), after some routine calculations, a steady-state approximate expression is derived as
(91) where The solution of this equation for estimation of the reference (0.03% of error) can be found as (98), shown at the bottom of the next page, where , the coefficients are given in the Appendix. and is the positive function for all of the form switching function
. Define the D. Observer Design (92)
where have
is a voltage reference. Then, from (91) and (92) we
, speed , and stator currents We consider angle and as measured signals. Based on this information we consider the generator model (10) with (8) as a partic, , ular case of system (60)–(65) where and . Thus,
(93) where
. The control (94)
under the following condition: (95)
(99)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
The fluxes , and mechanical torque , can be estimated by means of the nonlinear observer (66) which takes in this case the following form:
(100) where , and are the estimated variables; and are observer gains. The convergence of observer (100) may now be analyzed by examining the following, derived from (99) and (100), error dynamics equation: (101) where
, , , . Thus, the nonlinear observer (100) can and be seen as a linear system with time-varying parameters when and are assumed known functions. The variables generator dynamics is input-to-output locally stable. The pair is observable, and additionally, matrix is Hurwitz since its eigenvalues
are real and negative (see Section VI). Therefore, the assumptions B1)–B3) (see Section IV) in this case are met, hence, a solution of (101) by Theorem 2 is asymptotically stable. The and are employed in the resulting estimates , control law (96).
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VI. SIMULATION RESULTS This section presents the simulation results, emphasizing the effectiveness of the previously designed sliding-mode controller. The performance of the proposed controller were tested on the complete eighth-order model of the generator connected to an infinite bus through a transmission line (Fig. 1). The parameters of the synchronous machine and transmission , system are (all in p.u., except where indicated): , , , , , , , , , , s, rad s , , , and . For these parameters we obtain the parameters of mathematical model (10) (nominal values and experienced a variation of ): variations after , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . Setand , the steady state is comting , , puted as , , , , , and . and The controller gains were adjusted to and the observer gains were chosen as and , of matrix (101). resulting in the eigenvalues The remaining observer eigenvalues are , , and . The first set of simulations (Figs. 2–9) depicts results under s, a three-phase short circuit two different events: 1) at for a period of 150 ms is simulated at the transformer terminals s, experienced a pulse 0.4 for 5 s (exand 2) at ternal perturbations). The solid line is the response of the proposed sliding-mode controller applied to the nominal plant and the dotted line is the case of plant parameter perturbations: the experiences variations of introvalue of parameter , , , , , , and ducing variations of parameters in (10). Figs. 2–9 reveal some important aspects. 1) State variables hastily reach a steady-state condition after small and large disturbances, exhibiting the stability of the closed-loop system. 2) The estimated signals are closely related to the actual ones, exhibiting a robust performance of the observer. 3) The terminal voltage recovers their steady-state value after the short circuit.
(98)
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Fig. 5.
q -axis(1) damper winding flux x
Fig. 6.
d-axis damper winding flux x
Fig. 7.
q -axis(2) damper winding flux x
(p.u.) and its estimate.
Fig. 2. Angle x (rad).
(p.u.) and its estimate.
Fig. 3. Speed x (rad/s) and its estimate.
Fig. 4. Excitation flux x (p.u.) and its estimate.
4) In order to maintain in steady state , the ref, produced by (98), is changed when the meerence experienced a pulse. chanical torque 5) In the case of the parameter variations, we can observe that the estimated variables converge to a steady state de, but the steady state of the fined by the new value of is invariant with respect to the outputs, namely, and parameter variations. Multiple simulations were carried out considering different operation points, and good performance for both the controller and the observer were obtained.
Fig. 8. Terminal voltage V (p.u.).
(p.u.) and its estimate.
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
Fig. 9.
Fig. 10.
Mechanical torque T
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(p.u.) and its estimate.
Fig. 12.
Terminal voltage V (p.u.).
Fig. 13.
Mechanical torque T
Angle x (rad).
(p.u.) and its estimate.
compared with the standard IEEE-type DC1 linear controller s, a three-phase (dashed line). The events are: 1) at short circuit for a period of 150 ms occurs at the transformer s, experienced a pulse 0.05 for terminals and 2) at 5 s (Fig. 13). The performance comparison of two controllers exhibits the clear advantage of the proposed sliding-mode nonlinear controller. VII. CONCLUSION
Fig. 11.
Speed x (rad/s).
The second set of simulations, Figs. 10–12, shows the performance of the nonlinear sliding-mode controller (solid line)
A nonlinear controller based on the combination of the block control linearization and sliding-mode control techniques was proposed. A model used for control is fully detailed nonlinear, and this model takes in account all interactions between the electrical and mechanical dynamics and load constraints. A nonlinear observer for estimation of the excitation and rotor fluxes, and the mechanical torque, was designed. A new controller was tested through simulation under two most important perturbations in the power systems: variation of the mechanical
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torque, large fault (a 150-ms short circuit), and variation of parameters. The simulation results show that the observer-based sliding-mode controller with the proposed logic is able to achieve the mechanical dynamics and the generator terminal voltage robust stability under small and large disturbances, as well as uncertainty in the value of parameters. APPENDIX
REFERENCES [1] F. R. Schleif, H. D. Hunkins, G. E. Martin, and E. E. Hattan, “Ieee excitation control to improve powerline stability,” IEEE Trans. Power App. Syst., vol. PAS-87, pp. 189–202, June 1968. [2] F. P. de Mello and C. Concordia, “Concepts of synchronous machine stability as affected by excitation control,” IEEE Trans. Power App. Syst., vol. PAS-88, pp. 316–329, Apr. 1969. [3] A. Ghandakly and P. Idowu, “Design of a model reference adaptive stabilizer for the exciter and governor loops of power generators,” IEEE Trans. Power Syst., vol. 5, pp. 887–893, May 1990. [4] C. Mao, O. Malik, G. Hope, and J. Fan, “An adaptive generator excitation controller based on linear control,” IEEE Trans. Energy Conversion, vol. 5, pp. 673–678, Dec. 1990. [5] A. Ghandakly and J. Dai, “An adaptive synchronous generator stabilizer design by generalized multivariable pole shifting (GMPS) technique,” IEEE Trans. Power Syst., vol. 7, pp. 1239–1244, Aug. 1992. [6] K. M. Son and J. K. Park, “On the robust LQR control of TCSC for damping power system oscillations,” IEEE Trans. Power Syst., vol. 15, pp. 1306–1312, Aug. 2000. [7] A. Y. Pogromsky, A. L. Fradkov, and D. J. Hill, “Passivity based damping of power system oscillations,” in Proc. IEEE CDC, Kobe, Japan, 1996, pp. 512–517. [8] R. Ortega, A. Stankovic, and P. Stefanov, “A passivation approach to power systems stabilization,” in Proc. IFAC NOLCOS, Enshede, Netherlands, 1998, pp. 320–325. [9] Z. Xi and D. Cheng, “Passivity based stabilization and control of hamiltonian systems with dissipation and its application to power systems,” Int. J. Control, vol. 73, no. 18, pp. 1686–1691, 2000. [10] M. Galaz, R. Ortega, A. Bazanella, and A. Stankovic, “An energy-shaping approach to excitation control of synchronous generator,” in Pros. ACC, Arlington, VA, June 2001, pp. 817–822. [11] C. Lim and T. Hiyama, “Comparison study between a fuzzy logic stabilizer and self tuning stabilizer,” Comput. Ind., vol. 21, pp. 199–215, 1993. [12] P. Dash, S. Mishra, and A. Liew, “Design of a fuzzy PI controller for power system applications,” J. Intell. Fuzzy Syst., vol. 3, pp. 155–163, 1995. [13] P. Hoang and K. Tomsovic, “Design and analysis of an adaptive fuzzy power system stabilizer,” in Proc. IEEE PES Winter Meeting, 1996, pp. 7–12. [14] Y. Hsu and L. Chen, “Tuning of power system stabilizers using an artificial neural network,” IEEE Trans. Energy Conversion, vol. 6, pp. 612–619, Dec. 1991. [15] A. S. Bazanella, A. S. Silva, and P. Kokotovic, “Lyapunov design of excitation control for synchronous machine,” in Proc. 36th IEEE CDC, San Diego, CA, 1997, pp. 211–216. [16] J. Machowski, S. Robak, J. W. Bialek, J. R. Bumby, and N. Abi-Samra, “Decentralized stability-enhancing control of synchronous generator,” IEEE Trans. Power Syst., vol. 15, pp. 1336–1345, Nov. 2000.
[17] R. Marino, “An example of a nonlinear regulator,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 276–279, Feb. 1984. [18] J. W. Chapman, M. D. Ilic, C. A. King, and H. Kaufman, “Stabilizing a multimachine power system via decentralized feedback linearizing excitation control,” IEEE Trans. Power Syst., vol. 8, pp. 830–839, May 1993. [19] Y. Wang, D. J. Hill, R. H. Middleton, and L. Gao, “Transient stability, enhancement and voltage regulation of power systems,” IEEE Trans. Power Syst., vol. 8, pp. 620–627, May 1993. [20] W. Mielczarski and A. M. Zajaczowski, “Multivariable nonlinear controller for synchronous generator,” Opt. Control Applicat. Methods, vol. 15, pp. 49–65, 1994. [21] T. Landhiri and A. T. Alouani, “Design of a nonlinear excitation controller for synchronous generator using the concept of feedback linearization,” in Proc. ACC, Albuquerque, NM, 1997, pp. 545–551. [22] S. Jain, F. Khorrami, and Fardanesh, “Adaptive nonlinear excitation control of power systems with unknown interconnections,” IEEE Trans.Contr. Syst. Technol., vol. 2, pp. 1832–1839, Dec. 1994. [23] T. Lahdhiri and A. T. Alouani, “On the robust control of synchronous generator,” in Proc. ACC, Philadelphia, PA, June 1998, pp. 3798–3801. [24] P. M. Anderson and A. Fouad, Power System Control and Stability. New York: IEEE Press, 1994. [25] O. Akhkrif, F. Okou, L. Dessaint, and R. Champagne, “Application of multivariable feedback linearization scheme for rotor angle stability and voltage regulation of power system,” IEEE Trans. Power Syst., vol. 14, pp. 620–628, May 1999. [26] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992. [27] A. Y. Sivaramakrishnan, M. V. Hariharan, and M. C. Srisailam, “Design of variable-structure load-frequency controller using pole assignment technique,” Int. J. Control, vol. 40, no. 3, pp. 487–498, 1984. [28] K. Panicker, S. Asno, and C. Bhatia, “The transient stabilization of a synchronous machine using variable structure systems theory,” Int. J. Control, vol. 42, pp. 715–724, 1985. [29] G. P. Matthews, R. A. De Carlo, P. Hawley, and S. Lfebre, “Toward a feasible variable structure control design for a synchronous machine connected to an infinite bus,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 1918–1924, Dec. 1986. [30] Y. Wang et al., “Variable structure facts controllers for power system transient stability,” IEEE Trans. Power Syst., vol. 7, pp. 305–313, Feb. 1992. [31] M. Kothari, J. Nanda, and K. Bhattacharya, “Design of variable structure power system stabilizers with desired eigenvalues in the sliding mode,” Proc. Inst. Elect. Eng., pt. C, vol. 140, pp. 263–268, 1993. [32] R. El-Khazali, G. Heydt, and R. DeCarlo, “Stabilization of power system using variable structure output feedback control,” in Proc. ACC, 1994, pp. 1183–1187. [33] M. E. Aggoune, F. Boudjeman, A. Bensenouci, A. Hellal, M. R. Elmesai, and S. V. Vadari, “Design of variable structure voltage regulator using pole assignment technique,” IEEE Trans. Automat. Contr., vol. 39, pp. 1674–1679, Oct. 1994. [34] P. Dash, N. Sahoo, S. Elangovan, and A. Liew, “Sliding mode control of a static controller for synchronous generator stabilization,” Elect. Power Energy Syst., vol. 18, pp. 55–64, 1996. [35] Y. Park and W. Kim, “Discrete-time adaptive sliding mode power system stabilizer with only input/output measurements,” Elect. Power Energy Syst., vol. 18, pp. 509–517, 1996. [36] A. G. Loukianov, “Nonlinear block control with sliding mode,” Automat. Remote Control, vol. 59, no. 7, pp. 916–933, 1998. [37] A. G. Loukianov, V. I. Utkin, J. M. Cañedo, J. M. Ramirez, and J. Cabrera-Vázquez, “Control of the system synchronous generator-exciter via VSC,” in Proc. IEEE CDC, Sydney, Australia, Dec. 2000, pp. 3057–3062. [38] B. Drazenovic, “The invariance conditions for variable structure systems,” Automatica, vol. 5, pp. 287–295, 1969. [39] A. Isidori, Nonlinear Control Systems. Berlin, Germany: Springer-Verlag, 1989. [40] I. Kanellakopulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, July 1991. [41] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996. [42] B. P. Demidovich, Lectures on Mathematical Theory of Stability. Moscow, U.S.S.R.: Nauka, 1981.
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Alexander G. Loukianov was born in Moscow, Russia, in 1946. He graduated from the Polytechnic Institute, Moscow, Russia, in 1975, and received the Ph.D. degree in automatic control from the Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia, in 1985. In 1978, he joined the Institute of Control Sciences, where he was Head of the Discontinuous Control Systems Laboratory from 1994 to 1995. In 1995–1997, he held a visiting position at the University of East London, U.K. Since April 1997, he has been with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, as a Professor of electrical engineering graduate programs. His research interests include nonlinear systems robust control and variable-structure systems with sliding modes as applied to dynamical plants with delay, electric drives and power systems control, robotics, Spice, and automotive control. During 1992–1995, he was in charge of an industrial project between his Institute and the largest Russian car plant, in addition to several international projects supported by INTAS and INCO-COPERNICUS, Brussels. He has authored more than 70 technical papers published in international journal and conferences, and has served as reviewer for different international journals and conferences.
Vadim I. Utkin (SM’98–F’03) was born in Moscow, U.S.S.R., in 1937. He graduated in electrical engineering from the Moscow Power Institute, Moscow, Russia, and received the Ph.D. degree in control system engineering from the Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia, in 1964. In 1960, he joined the Institute of Control Sciences, where he was Head of the Discontinuous Control Systems Laboratory from 1973 to 1994. Currently, he is the Ford Chair of Electromechanical Systems at The Ohio State University, Columbus, where he teaches in the System Dynamics, Measurements and Control program. He has held visiting positions at universities in the U.S., Japan, Italy, and Germany. He is one of the originators of the concepts of variable-structure systems and sliding-mode control. He has authored five books and more than 250 technical papers. His current research interests are the control of infinite-dimensional systems (including flexible manipulators), sliding modes in discrete-time systems, and microprocessor implementation of sliding-mode control of electric drives and alternators, robotics, engines, vehicles, and industrial processes. Prof. Utkin is an Honorary Doctor of the University of Sarajevo. He is an Associate Editor of the International Journal of Control.
José M. Cañedo (M’93) was born in Mazatlán, Mexico, in 1950. He graduated in electrical engineering–power systems from the National Polytechnic Institute of México, México City, México, in 1980, and received the Ph.D. degree from the Moscow Power Institute, Moscow, U.S.S.R., in 1985. From 1988 to 1994, he was a Researcher with the Federal Power Commission of México. Since 1997, he has been with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, as a Professor of electrical engineering graduate programs. His research interests include nonlinear robust control of power systems and electric machines (motors and generators).
Javier Cabrera-Vázquez was born in Mexico City, Mexico, in 1950. He received the Engineer’s degree in Communications and Electronics from the Polytechnic National Institute, México City, México, in 1991, the Master’s degree in sciences from the University of Guadalajara, Guadalajara, Mexico, in 1998, and the Ph.D. degree in electrical engineering from the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, in 2002. Currently, he is a Professor in the Electronic Engineering Department, University of Guadalajara. His research interests include parametric identification and nonlinear systems control with sliding modes as applied to electric drives and power systems control.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004
Discontinuous Controller for Power Systems: Sliding-Mode Block Control Approach Alexander G. Loukianov, José M. Cañedo, Member, IEEE, Vadim I. Utkin, Fellow, IEEE, and Javier Cabrera-Vázquez
Abstract—Based on the complete model of the plant, a sliding-mode stabilizing controller for synchronous generators is designed. The block control approach is used in order to derive a nonlinear sliding surface, on which the mechanical dynamics are linearized. This combined approach enables us to compensate the inherent nonlinearities of the generator and to reject high-level external disturbances. A nonlinear observer is designed for estimation of the rotor fluxes and mechanical torque. Index Terms—Nonlinear system, observer, power system control, sliding-mode control, synchronous generator.
I. INTRODUCTION
O
VER THE LAST 15 years, two major problems have been faced in the area of modern power system control: the problem of loss of synchronism or angle instability, and the problem of terminal voltage instability and collapse due to over loading on transmission line, reactive constraint and fault. Therefore, a fundamental problem in the design of feedback controllers for power systems is that of robust stabilizing both frequency and voltage magnitude, and achieving a specified transient performance. Robustness implies operation with adequate stability margins and admissible performance level in spite of plant parameters variations and in the presence of external disturbances. This paper deals with excitation control of a single synchronous machine connected to an infinite bus (Fig. 1). The control schemes of synchronous machines are commonly based on a reduced-order linearized model and classical control algorithms that ensure asymptotic stability of the equilibrium point under small perturbations [1], [2]. Recently, to overcome the limitation of linear control, attention has been focused on implementation of modern control technique, e.g., an adaptive linear control [3]–[6], passivity-based approach [7]–[10], intelligent control such as fuzzy logic [11]–[13] and neural networks [14], control based on direct Lyapunov method [15], [16], feedback linearization (FL) technique [17]–[21], and control based on adaptive FL [22], [23].
Manuscript received March 11, 2002; revised August 25, 2003. Abstract published on the Internet January 13, 2004. This work supported by CONACyT, Mexico, under Grant 36960A. A. G. Loukianov and J. M. Cañedo are with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara 44550, Mexico (e-mail: [email protected]). V. I. Utkin is with the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210-1272 USA. J. Cabrera-Vàzquez was with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara 44550, Mexico. He is now with the Electronic Engineering Department, University of Guadalajara, Guadalajara 44840, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/TIE.2004.825264
Fig. 1.
Syncronous machine connected to an infinite bus.
All of the mentioned controllers provide larger stability margins with respect to traditional ones. But these control schemes were designed for a reduced order plant. In [24], it has been shown however that the effects of unmodeled stator and rotor electrical dynamics cannot be neglected since they affect the electromechanical dynamics in the case of large perturbation. The detailed seventh-order model of synchronous machine (five equations for electrical dynamics and two ones for mechanical dynamics) has been considered and a nonlinear controller using this model and FL technique, has been designed in [25] to enhance transient stability. The proposed nonlinear control law is a function of all plant parameters and disturbances. In practice some of these parameters are subjected to variations as a result of a change in the system loading and/or in the system configuration. Since the detailed model is so involved, a direct use of the FL technique results in a computationally expensive control algorithm. Moreover, this control scheme does not take into account practical limitation on the magnitude of the excitation voltage, and an observer design problem was not solved. On the other hand, a fruitful and relatively simple approach, especially when dealing with nonlinear plants subjected to perturbations, is based on the use of variable-structure control (VSC) with sliding mode [26]. First and foremost, this enables high accuracy and robustness to disturbances and plant parameter variations to be obtained. Second, the control variables of the basic sliding mode control law rapidly switch between extreme limits, which is ideal for the direct operation of the switched mode power converters of synchronous generators. Sliding-mode controllers for power systems have been designed in [27]–[35], however, for reduced-order plants only, to the best of our knowledge. Application of these controllers to a full-order plant would cause undesirable chattering and impair accuracy, since unmodeled dynamics can be excited. In this paper, based on the full-order plant model, we shall resort to the block control [36] and sliding-mode techniques [26], the combination of which overcomes most of the mentioned problems: they are simple then [25], with low computational demands, and take into account structural constraints of the controller. The block control technique is applied to design a nonlinear sliding surface in such a way that the sliding-mode
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dynamics are represented by a linear system with desired eigenvalues. For a given bound on the control signal a discontinuous control law ensuring stability of the sliding mode is proposed. The main feature of the proposed control scheme is robustness with respect to disturbances and plant parameter variations [37]. This paper is organized as follows. Section II reviews the detailed eighth-order model of the synchronous machine. In Section III, the block control technique is extended for design a discontinuous control strategy for a class of nonlinear systems presented in so-called Block Controllable Form with unmatched perturbations, and internal dynamics. This class describes dynamics of electromechanical plants, in particular, power systems. In Section IV, a nonlinear observer design is presented. Based on these results, in Section V, a sliding mode observerbased excitation controller for a single synchronous machine connected to an infinite bus, is derived. Section V discusses simulation results.
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with
where and are the direct-axis and quadrature-axis self-inis the rotor self-inductance; and are ductances; the direct-axis and quadrature-axis damper windings self-inducand are the direct-axis and quadrature-axis tances; magnetizing inductances. Subtracting from (2), and gives the following transformation: (3)
II. PLANT MODEL The complete mathematical model of the single machine infinite-bus system consists of electrical and mechanical dynamics and load constraints. The electrical dynamics comprising the stator and rotor damper windings, with the currents as the state variables, after Park’s transformation, can be expressed as follows [24]:
which reduces the system (1) to the form (4)
(1) where
. The complete where mathematical description includes the swing equation [24] (5) (6)
and are the the field flux and current, respectively; , , , , , and are the direct-axis and quadrature-axis damper windings fluxes and currents, respectively; , , and are the direct-axis and quadrature-axis stator fluxes and is the excurrents, respectively; is the angular velocity and and are the direct-axis and quadracitation control input; ture-axis terminal voltages;
is the rated synchronous speed, where is the power angle; is the inertia constant; is the mechanical torque applied is the electrical torque, expressed in terms to the shaft; and of the currents as follows: (7) The mechanical torque function of time. Thus,
is assumed to be a slowly varying (8)
and are the stator and field resistances; , , and are the damper windings resistances. Analysis of relationship between fluxes and currents shows that the sensitivity of the fluxes with respect to parameters variations is less then the sensitivity of the currents. Therefore, it is more suitable the representation of the electrical dynamics in and the stator current . This kind of terms of the rotor flux model can be obtained from (1) using the following transformation between fluxes and currents: (2)
The equilibrium equation for the external network of the synchronous machine connected to an infinite bus is (9) where
is the value of the infinite-bus voltage; and are the transformer plus transmission line resistance and inductance. Here, all the state variables as well as the parameters of the model (1)–(9) are expressed in per unit. Combining (1)–(9), the
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complete model of the generator is presented in the state-space form
must be chosen such that the matrix rank, and the sliding-mode equation
,(
) has full (16)
(10) where where is the control,
,
, and
has the desired properties, including stability as a minimum requirement. Secondly, a discontinuous control if if
(17)
is designed to guarantee convergence of the motion projection on the subspace , governed by is defined by the equation shown at the bottom of the page, , ,( ), , , and are , , , , positive constant parameters depending on , , , , , , , , and .
where and , are smooth functions to be selected. If the disturbance satisfies the so-called matching consuch that dition [38], i.e., there exists a matrix
III. CONTROL METHOD In this section, the method and underlying ideas are described in generic terms to show the generality of our approach. A. Formulation of the Problem
and then reduces to simply
. In this case, (16)
(19)
Consider the following system subject to disturbance: (11) (12) is the state vector, where control vector to be bounded by
(18)
is the (13)
, is a vector of external disturbances generated by an external system described by (14) and the columns of and The vector field are smooth and bounded mappings of class . It is assumed , , , . The that standard sliding mode design procedure is comprised of the two subproblems. First, the nonlinear sliding manifold in the state space of the system (15)
), however, it is This equation has the reduced order ( still nonlinear and nonautonomous. One possible approach to ensuring stability of system (19) in the autonomous case is connected with the input–output linearization techniques [39]. Another approach is the ”backstepping” [40] that is based on the use of step-by-step Lyapunov functions. In this paper, the decomposition block control method is adopted to design a nonwhich stabilizes the linear time-varying sliding function sliding-mode equation. Another important aim is to provide inin cases where it does variance with respect to disturbance not satisfy (18). B. Block Control for a Class of Nonlinear Systems Consider the following decomposed system: (20) (21)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
where
is an
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1 new variables vector,
,
(22) (23) (24) where the vector
is decomposed as , , , is a 1 vector, and the indexes ( ) define the structure of the system and satisfy the following relation:
and is the identity matrix. Proposition 3: Suppose that the assumptions H1) and H2) are satisfied. Then, the transformation (29) reduces the system (20)–(23) to the following desired form:
(25) Definition 1: The system (20)–(23) is referred to as Nonlinear Block Controllable Form with disturbance and internal , multiplying the fictitious control dynamics if the matrix in each th block of (20)–(23), has full rank (26) Remark 2: The model of the synchronous generator has the form (20)–(24) with the load angle as output. or , therefore, The relation (25) means consider the plant with the structure (27)
(30) , is a bounded function, , . Proof: In order to prove the proposition the method of in, , as a duction will be used, considering fictitious control vector in each th block of (20)–(22). Step 1: Define the tracking error as where
(31) . Therefore, the inverse As follows from (27), we have under condition (26) exists. Let the fictitious control matrix in the first block rewritten as
that includes both cases. Suppose that the output should follow produced by given external system the reference signal (28) The control design procedure for the system (20)–(24) consists of three parts. First, using block control technique a transformation of the system (20)–(22) to a desired form is derived. As result a desired sliding manifold will be derived. Secondly, a discontinuous control strategy which stabilizes sliding mode is proposed, and finally, the stability of a sliding-mode equation with zero dynamics is analyzed. In order to reduce the system (20)–(22) to a desired form the following assumptions are introduced. , and H1): The elements of are continuously differentiable functions of th order, , with respect to all arguments in interval , and all derivatives are bounded. H2): The condition (26) is satisfied. Based on these assumptions and taking into account the structure (27), the following recursive transformation is introduced:
(32) with
, be selected as (33)
where is an 1 new variables vector, . The transand , with input (33) formed first block in new variables . has the desired form (30), i.e., Step 2: Taking the derivative of (29) along the trajectories of (20)–(21), results in (34) where
and , . From relation (27) it follows . Therefore, the matrix is not square, in (34) is chosen as and the fictitious control (35) 1 new variables vector, , denotes where is a pseudoinverse matrix of , . Thus, (34) with (35) takes the form (30), that is, . Now, the following assumption is introduced. can be orH3): Assume that the elements of the matrix dered such that the square matrix
(29)
, has
with .
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Based on this assumption, the variable using (35), of the form (29), that is,
can be obtained
where ,
Step : At this stage it is possible to show that if we have, ) steps, the transformed blocks of the system after ( (for (20)–(22) with new variables (27)) of the form
, and , . Thus, transformation (29) reduces the system (20)–(23) to (30). Remark 4: Note that the assumption Hi permits us to use with constant parameters, which simplifies the the matrix transformation. If this assumption is not satisfied, then it is poswhich orthogonal sible to use, for instance, the matrix , instead of . to Now, in order to generate sliding mode in (30), a natural choice of the sliding manifold using transformation (29) is (41) Then, the following discontinuous control is proposed:
(36)
(42)
(37)
Proposition 5: The control law (42) guaranteies the convergence of the closed-loop system motion to manifold (41) in a finite time defined as
with
then on the th step of the transformation procedure, we will have the transformed the th block with new state vector similar to (36). Then, the equation for (37) with (20) and (21) is of the form
Proof: Taking the derivative of trajectories of (40)–(42) gives
along the
(38) Using the following relations , we have:
where , , , the vector (35), of the form
and
, . For the case in (38) can be selected similar to In the domain (39)
where
is pseudo inverse matrix of , . Thus, (38) with (39) takes the same form as (36), that is, . For this step, the assumption H3 is generalized as follows. can be ordered such that Hi): The elements of the matrix
(43) , is negative the derivative definite, that guarantees convergence of the state vector to the (41). In order to demonstrate this convergence manifold , and using is finite, first we assume that , we have . Therefore, , or (44)
Under this assumption, the recursive transformation can be obtained as
Using the comparison lemma [41], a solution of (44) can be . Thus, . Therefore, vanishes in , and sliding mode some finite time, after this time. starts on the manifold Now, a sliding mode equation of reduced, th order can be derived of the form estimated as
Step : On the last step, calculating the time derivative of (29) gives the last block
(45)
(40)
(46)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
, , and is a solution of the equation (40) . In order to analyze the calculated as stability of a solution of (45) and (46), first the variables of this and . system are redefined for simplicity as Then, the system is represented as
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where
(47) (48) with
, and
.. . .. .
.. . .. .
.. .. .
. ..
.
.. . .. .
.. . .. .
(56) where , , , and are positive constants. Proposition 6: Suppose assumptions A1) and A2) hold, and (57) where . Then the solution of the system (47)–(48) is uniformly ultimately bounded, and the converges exponentially to zero. tracking error along the traProof: Calculate the derivative of . As jectories of (48), follows from (55) and (56),
In the domain defined as for
where . The behavior of (47) and (48) will be ana: lyzed in the region (49) (50) in which the sliding-mode existence condition (43) holds. Since the matrix is Hurwitz, the solution of subsystem (47) is exponentially stable, that is, (51) where
is a positive-definite solution of the Lyapunov equation for . Then, the dynamics of the system (47)–(48) on the invariant subspace are governed by the system
the derivative tive definite, then
(58)
with
is nega, and (59)
, . Under condition (57), we where . That have means the solution belongs to region (50) for . In order to estimate an ultimate bound of the all of (47)–(48), first is defined from (58) solution . using (51) as Then taking into account condition (57), gives . Therefore, application of Corollary 5.3 [41] gives or . tends to zero, solution is uniformly ultimately Since . Thus, taking into account (59), the bounded by solution can be finally estimated as
(52) that presents the zero dynamics. This system is subject to nonvanishing perturbation. Therefore, we introduce the following assumptions. A1): A solution of zero dynamics (52) converges exponentially to a compact set (53) A2): The mapping in time and locally Lipshitz in
is continuous and
such that (54)
where is a Lipshitz constant. By assumption A1) and the converse theorem [41], there exin a Lyapunov ists for all function , with the following properties:
(55)
In addition, since converges exponentially to zero, we . have The results obtained in Propositions 3–6, can be formulated in the following theorem. Theorem 7: If the assumptions H1)–Hr), A1) and A2), and the conditions (57) are satisfied, then a solution of the closed-loop system (20)–(23) with (42)–(41) is uniformly ultimately bounded, and the tracking error (31) converges exponentially to zero. IV. NONLINEAR OBSERVER In this section, we deal with an observer problem for a class of nonlinear perturbed systems governed by (60) (61) (62) where and the measured state vectors, and
are is the
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state vector to be estimated, and are appropriately linear and bilinear functions with respect to , that is,
by assumption B2 such that . The solution of (69)–(70) with as
, can be expressed
(63) (71) (72)
(64) , and are matrices with constant elements, and a perturbation produced by an given external system
is , and , , . The stability of this solution is analyzed by application of the Lyapunov’s characteristic indexes (LCI) technique [42]. First, (72) defined as the LCI of the solution
with (65)
The following is assumed. B1): The system (60)–(62) is input-to-state locally stable. is observable.
B2): The pair
is Hurwitz. B3): The matrix Note that models of electrical systems, including generators and motors, can be represented in the form (60)–(64) presents mechanical variables, the speed where vector presents the stator currents, vector and angle; vector presents the rotor fluxes, and presents the load torque (for motors) or mechanical torque (for generators). In the case of , , synchronous generator, we have , and . In order to estimate and , the following nonlinear observer with linear correction terms is proposed:
is negative since matrix by assumption B3 is Hurwitz. Furthermore, by assumption B1 the state of (60)–(62) is ultimately bounded for a bounded control input . Hence, the elements of and, consequently, are bounded. That results in . From the LCI properties [42] it follows
Hence, the LCI of
(66) where , , and are estimates of vectors , and , respectively; and are observer matrices. Using (60)–(66), the following observer error system is derived: (67) (68) , , , and . Thus, error system (67) and (68) is considered as a linear system with time-varying parameters. The stability of this system is investigated in the following theorem. Theorem 8: If the assumptions B1)–B3) are met then there and such that a solution of (67) and (68) is asympare totically stable. Proof: The system (67) and (68) by the nonsingular transwhere
and
formation
is reduced as (69) (70)
(71)
is negative, too. This means that the solution of (69) and (70) and, consequently, (67) and (68) is asymptotically stable, that is, , . Hence, , and . V. GENERATOR CONTROL DESIGN In this section, based on the above results, a sliding-mode observer-based excitation controller for a single synchronous machine connected to an infinite bus is derived. The control oband terminal voltage stajectives are rotor angle , speed bility enhancement. First, a mechanics stabilizing control design is studied. Secondly, a voltage control design is outlined, and finally, a control switching logic is proposed. A. Mechanics Stabilizing Control The sliding mode controller design procedure will be divided into two steps. First, exploring the block control technique presented in Section III, a sliding surface will be formed. Then, a discontinuous control law will be designed to make attractive this surface. The first subsystem of (10) is presented in the Block Controllable Form (20)–(22) which consists of three blocks (73)
where , , are eigenvalues of
, , and , . We can choose the matrices
, and
(74) (75)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
while the second subsystem of (10) presents the internal dynamics (23). The perturbation is produced by the exosystem , and control to be bounded by (76) is the maximum value of the excitation voltage. The where control goal is to make the angle be equal to a reference signal , and the speed be equal to the rated synchronous speed . In accordance with the block control technique, we define as the control error (77) Then, selection of fictitious control , where gives
in the first block as and is a new variable,
(78) Using (77) and (78), (73) and (74) in terms of new variables and may be written as (79) (80) where
In the second step, the selection of fictitious control , gives the switching function
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guarantees The control (85) under the condition in a finite time the convergence of the state to surface interval. Once this is achieved, the sliding motion is governed in the full state space by (86) (87) Therefore, the sliding-mode control (85) yields invariant , in the subspace state space of the closed-loop system. The dynamics on this invariant subspace are zero dynamics. In of in (87) is reorder to derive these dynamics, vector , placed by : where mapping , is defined by (77), (78), and (81). , , and Then, is zeroed, that means , thus, (88) Note that sliding-mode dynamics (86) and (87) can be considered as a particular case of the system (47) and (48) or (45) and (46), while zero dynamics (88) is a particular case of (52). Since the mapping in (88) is smooth, assumptionA2) (see Section III) in this case is met. Now, we will show that assumption A1 is also satisfied. The right-hand side of (88) consists of linear and nonlinear parts, thus, with
in (80) as ,
(81) is described such that a sliding-mode motion on surface ) by the following linear system: in the state space ( (82) ) and ( ). This system correwith desired eigenvalues ( sponds to the linearized mechanical dynamics of the closed-loop and means and system. Note that . Therefore, the control goal requires be a constant. Thus, and will be put in (78) and (81) to zero. The switching function design has been outlined. Now, a control law will be investigated. Projection of the closed-loop system motion on subspace can be described by (83) , , and is a positive function for . Then, the equivalent control is (83) of the form calculated as a solution of where
For these dynamics, the following assumptions are introduced. Hurwitz. C1): The matrix is , . C2): By assumption C1) there is a Lyapunov function , with the following properties: and where is a positive-definite solution of the Lyapunov equation for , and . calculated as Using the above relations, the derivative for is negative in the region . In this region a solution of (88) estimated as
with
(84) Now, taking into account (76), the control is proposed as
and (85)
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exponentially converges to the compact set (53) defined as
is the equivalent control calculated from (93) where , guarantees the converas gence of the control error to zero in a finite time. C. Control Switching Logic
that is, assumption A1) (see Section III), is met. Hence, by Proposition 6 a solution of (86) and (87) is ultimately bounded, and the control error converges exponentially to zero.
Having just one control input for two switching surfaces and , and taking into account (85) and (94), a control strategy can be proposed, finally, of the form if if
B. Generator Voltage Control The control objective is the control of terminal voltage, defined as (89) Using (10), (9) can be represented as (90) where
and . Because parameter is very small, the control term in (90) can be neglected, and then we have
if if
(96) (97)
with . Therefore, we are proposing a hierarchical control action through the proposed logic (96) and (97). Since the mechanical dynamics are slower than the electrical ones, we spend reaches the control resources, at first, to stabilize . When the boundary layer with width , the control resources will be spending to stabilize , that is, to regulate terminal voltage . , the control action After convergence of such that to , ( ). reduces the boundary layer width from Thus, the controller maintains the value of and within desired accuracy and . In order to avoid oscillations in steady state, the load angle must correspond to the terminal voltage refreference value mechanical torque value , such that to have erence value in steady state the following quality:
Using (1)–(7), after some routine calculations, a steady-state approximate expression is derived as
(91) where The solution of this equation for estimation of the reference (0.03% of error) can be found as (98), shown at the bottom of the next page, where , the coefficients are given in the Appendix. and is the positive function for all of the form switching function
. Define the D. Observer Design (92)
where have
is a voltage reference. Then, from (91) and (92) we
, speed , and stator currents We consider angle and as measured signals. Based on this information we consider the generator model (10) with (8) as a partic, , ular case of system (60)–(65) where and . Thus,
(93) where
. The control (94)
under the following condition: (95)
(99)
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
The fluxes , and mechanical torque , can be estimated by means of the nonlinear observer (66) which takes in this case the following form:
(100) where , and are the estimated variables; and are observer gains. The convergence of observer (100) may now be analyzed by examining the following, derived from (99) and (100), error dynamics equation: (101) where
, , , . Thus, the nonlinear observer (100) can and be seen as a linear system with time-varying parameters when and are assumed known functions. The variables generator dynamics is input-to-output locally stable. The pair is observable, and additionally, matrix is Hurwitz since its eigenvalues
are real and negative (see Section VI). Therefore, the assumptions B1)–B3) (see Section IV) in this case are met, hence, a solution of (101) by Theorem 2 is asymptotically stable. The and are employed in the resulting estimates , control law (96).
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VI. SIMULATION RESULTS This section presents the simulation results, emphasizing the effectiveness of the previously designed sliding-mode controller. The performance of the proposed controller were tested on the complete eighth-order model of the generator connected to an infinite bus through a transmission line (Fig. 1). The parameters of the synchronous machine and transmission , system are (all in p.u., except where indicated): , , , , , , , , , , s, rad s , , , and . For these parameters we obtain the parameters of mathematical model (10) (nominal values and experienced a variation of ): variations after , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . Setand , the steady state is comting , , puted as , , , , , and . and The controller gains were adjusted to and the observer gains were chosen as and , of matrix (101). resulting in the eigenvalues The remaining observer eigenvalues are , , and . The first set of simulations (Figs. 2–9) depicts results under s, a three-phase short circuit two different events: 1) at for a period of 150 ms is simulated at the transformer terminals s, experienced a pulse 0.4 for 5 s (exand 2) at ternal perturbations). The solid line is the response of the proposed sliding-mode controller applied to the nominal plant and the dotted line is the case of plant parameter perturbations: the experiences variations of introvalue of parameter , , , , , , and ducing variations of parameters in (10). Figs. 2–9 reveal some important aspects. 1) State variables hastily reach a steady-state condition after small and large disturbances, exhibiting the stability of the closed-loop system. 2) The estimated signals are closely related to the actual ones, exhibiting a robust performance of the observer. 3) The terminal voltage recovers their steady-state value after the short circuit.
(98)
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Fig. 5.
q -axis(1) damper winding flux x
Fig. 6.
d-axis damper winding flux x
Fig. 7.
q -axis(2) damper winding flux x
(p.u.) and its estimate.
Fig. 2. Angle x (rad).
(p.u.) and its estimate.
Fig. 3. Speed x (rad/s) and its estimate.
Fig. 4. Excitation flux x (p.u.) and its estimate.
4) In order to maintain in steady state , the ref, produced by (98), is changed when the meerence experienced a pulse. chanical torque 5) In the case of the parameter variations, we can observe that the estimated variables converge to a steady state de, but the steady state of the fined by the new value of is invariant with respect to the outputs, namely, and parameter variations. Multiple simulations were carried out considering different operation points, and good performance for both the controller and the observer were obtained.
Fig. 8. Terminal voltage V (p.u.).
(p.u.) and its estimate.
LOUKIANOV et al.: DISCONTINUOUS CONTROLLER FOR POWER SYSTEMS: SLIDING-MODE BLOCK CONTROL APPROACH
Fig. 9.
Fig. 10.
Mechanical torque T
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(p.u.) and its estimate.
Fig. 12.
Terminal voltage V (p.u.).
Fig. 13.
Mechanical torque T
Angle x (rad).
(p.u.) and its estimate.
compared with the standard IEEE-type DC1 linear controller s, a three-phase (dashed line). The events are: 1) at short circuit for a period of 150 ms occurs at the transformer s, experienced a pulse 0.05 for terminals and 2) at 5 s (Fig. 13). The performance comparison of two controllers exhibits the clear advantage of the proposed sliding-mode nonlinear controller. VII. CONCLUSION
Fig. 11.
Speed x (rad/s).
The second set of simulations, Figs. 10–12, shows the performance of the nonlinear sliding-mode controller (solid line)
A nonlinear controller based on the combination of the block control linearization and sliding-mode control techniques was proposed. A model used for control is fully detailed nonlinear, and this model takes in account all interactions between the electrical and mechanical dynamics and load constraints. A nonlinear observer for estimation of the excitation and rotor fluxes, and the mechanical torque, was designed. A new controller was tested through simulation under two most important perturbations in the power systems: variation of the mechanical
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torque, large fault (a 150-ms short circuit), and variation of parameters. The simulation results show that the observer-based sliding-mode controller with the proposed logic is able to achieve the mechanical dynamics and the generator terminal voltage robust stability under small and large disturbances, as well as uncertainty in the value of parameters. APPENDIX
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Alexander G. Loukianov was born in Moscow, Russia, in 1946. He graduated from the Polytechnic Institute, Moscow, Russia, in 1975, and received the Ph.D. degree in automatic control from the Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia, in 1985. In 1978, he joined the Institute of Control Sciences, where he was Head of the Discontinuous Control Systems Laboratory from 1994 to 1995. In 1995–1997, he held a visiting position at the University of East London, U.K. Since April 1997, he has been with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, as a Professor of electrical engineering graduate programs. His research interests include nonlinear systems robust control and variable-structure systems with sliding modes as applied to dynamical plants with delay, electric drives and power systems control, robotics, Spice, and automotive control. During 1992–1995, he was in charge of an industrial project between his Institute and the largest Russian car plant, in addition to several international projects supported by INTAS and INCO-COPERNICUS, Brussels. He has authored more than 70 technical papers published in international journal and conferences, and has served as reviewer for different international journals and conferences.
Vadim I. Utkin (SM’98–F’03) was born in Moscow, U.S.S.R., in 1937. He graduated in electrical engineering from the Moscow Power Institute, Moscow, Russia, and received the Ph.D. degree in control system engineering from the Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia, in 1964. In 1960, he joined the Institute of Control Sciences, where he was Head of the Discontinuous Control Systems Laboratory from 1973 to 1994. Currently, he is the Ford Chair of Electromechanical Systems at The Ohio State University, Columbus, where he teaches in the System Dynamics, Measurements and Control program. He has held visiting positions at universities in the U.S., Japan, Italy, and Germany. He is one of the originators of the concepts of variable-structure systems and sliding-mode control. He has authored five books and more than 250 technical papers. His current research interests are the control of infinite-dimensional systems (including flexible manipulators), sliding modes in discrete-time systems, and microprocessor implementation of sliding-mode control of electric drives and alternators, robotics, engines, vehicles, and industrial processes. Prof. Utkin is an Honorary Doctor of the University of Sarajevo. He is an Associate Editor of the International Journal of Control.
José M. Cañedo (M’93) was born in Mazatlán, Mexico, in 1950. He graduated in electrical engineering–power systems from the National Polytechnic Institute of México, México City, México, in 1980, and received the Ph.D. degree from the Moscow Power Institute, Moscow, U.S.S.R., in 1985. From 1988 to 1994, he was a Researcher with the Federal Power Commission of México. Since 1997, he has been with the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, as a Professor of electrical engineering graduate programs. His research interests include nonlinear robust control of power systems and electric machines (motors and generators).
Javier Cabrera-Vázquez was born in Mexico City, Mexico, in 1950. He received the Engineer’s degree in Communications and Electronics from the Polytechnic National Institute, México City, México, in 1991, the Master’s degree in sciences from the University of Guadalajara, Guadalajara, Mexico, in 1998, and the Ph.D. degree in electrical engineering from the Centro de Investigacion y de Estudios Avanzados del IPN, Guadalajara, Mexico, in 2002. Currently, he is a Professor in the Electronic Engineering Department, University of Guadalajara. His research interests include parametric identification and nonlinear systems control with sliding modes as applied to electric drives and power systems control.
