2013 5th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)
Synergetics Approach to Multi-Machine Power System Hierarchical Nonlinear Control Andrey A. Kuz’menko Synergetics and Control Processes Department Southern Federal University Taganrog, Russia
[email protected] channels affect each other, cascade arrangements of PID controllers are used. The wide use of such controllers is explained by their structural simplicity, very simple software implementation of the control computation, which does not require that all the plant state variables are measured, and a relative simplicity of the controller fine tuning for a specific plant. A drawback of such controllers is not very high control quality, especially for complex plants that include nonlinear static and dynamic elements and delay blocks; furthermore, fine tuning of the controller coefficients is a “creative” procedure, which strongly depends on the practical experience of the service engineer. In addition, such systems guarantee the normal operation only in the vicinity of the nominal operation mode. Methods for designing linear control systems are described in [2]. However, MMPS are complex nonlinear dynamical interrelated systems whose elements exchange energy, matter, and information. Internal MMPS processes are multidimensional, multiply connected, and nonlinear. These features present insurmountable obstacles for conventional control theory methods. It is clear that proper nonlinear mathematical models and nonlinear control theory must be used to design effective control systems for MMPS processes. This theory is synergetic control theory. It was developed in [5]. If a linear MMPS model is used, adaptive control is typically designed using classical methods of linear adaptive control theory [1]-[3]. It was shown in [6] that the adaptivity of control systems is presently achieved by using conventional linear controllers and methods of fuzzy control systems or artificial neural networks. However, in the case of fuzzy systems, one faces the so called curse of dimensionality because the number of rules increases proportionally to a degree of the number of input variables. Furthermore, it was mentioned in [6] that the adjustment of linear controllers is mostly performed by experienced experts that use general concepts of the physics of the underlying processes or using the trial-and-error approach. This approach does not guarantee that optimal parameter values are found, and it strongly depends of the human factor. In the paper we consider synergetics approach to problem of nonlinear hierarchic control laws design for processes of power energy generating in MMPS. This approach based on methods and principles of synergetics conception of modern theory of control [5], [6].
Abstract— According to principle of hierarchy, any complex dynamics system consists of some aggregation of local subsystems. In turn, each of these systems includes powerhardware and informational (control) subsystems. These subsystems are in close interaction with each other. Hierarchy's high level control law is a supervise law for local control laws directing to actuators of control system. In the paper we consider synergetics approach to problem of nonlinear hierarchic control laws design for processes of power energy generating in multimachine power system (MMPS). MMPS is nonlinear, dynamics, multidimensional, and multilinked system. In this system extreme and chaotic operation modes can take place. Synergetics approach is based on methods and principals of nonlinear dynamics and synergetics conception of modern control theory. We explore the problem of MMPS control operating high power buses. Basing on proposed approach we have designed principally new nonlinear anti-chaotic regulator for MMPS. Keywords– multi-machine power system; synergetics approach; nonlinear control; hierarchical synthesis.
I.
INTRODUCTION
It is a fact, that in order to overcome the problem of over dimensioning in complex dynamics systems (DS) we may use a hierarchy approach. By this way we can explore any of complex DS as some aggregation of local subsystems. And each of these subsystems consists of power-matter and intelligent (control) subsystems closely interacting with each other. Hierarchy’s high level control law is a supervise law for local control laws directing to actuators of control system. So this law forms target values for local control subsystems. One of the main tasks of control MMPS is to maintain a unified frequency of the generated voltage. The quality and efficiency of electric energy generation in MMPS can be considerably improved by upgrading control algorithms for generation units. From this viewpoint, the most promising directions are nonlinear, intelligent, adaptive, robust, and predictive control [1]-[4]. Presently, mainly linear control systems with controllers, that are PID controllers class, are used to control MMPS. PID controllers also make it possible to implement simpler control laws by eliminating certain control components. These controllers are fine tuned using either linear models of plants or using overspeed characteristics. In the case of multiply connected plants, such as turbogenerator, whose control
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II.
Variations in the workload of the MMPS due to connection and disconnection of power consumers or emergency situations (such as short circuits or line break), electric power changes unpredictably, and this disturbance can be represented by a piecewise constant function. The statement of the control problem is as follows: design nonlinear hierarchical controls for the MMPS described by Eqs. (1)–(3) that accomplish the main technological task of the stabilizing of SG output voltage and the turbine output power; that is, we want to guarantee that technological invariants (4) and (5) are fulfilled. Furthermore, we must ensure the adaptability to changing loads; that is, there is the problem of suppressing the nonmeasured external disturbance. In linear control theory, this problem is solved using the procedure described in [1]. For the external disturbances, a model in the form of a system of homogeneous differential equations with known coefficients and unknown initial conditions is set up. The model of disturbances is combined with the model of the plant, and an observer is designed for the resultant extended system. The third equation of system (1) is a model of such external piecewise constant disturbance operating turbogenerator thru power system [5]. So, this equation is needed for providing suppression of this disturbance that corresponds with synergetic conception in control theory [5]. The proposed procedure of MMPS (1)–(3) hierarchical control system design consists of 4 steps.
STATEMENT OF THE PROBLEM
Consider a MMPS as a group of turbogenerators is core component of power system generating subsystem. Local subsystems for this group are turbine and synchronous generator (SG) control subsystems. And power system total frequency and output control subsystem is a top of system’s hierarchy. As an initial model of turbogenerator we consider the generally used model [7]. Introduce this model as following subsystems: dδ i = si ; S PS : dt ⎛ PТi − E qi2 y ii sin (α ii ) − ⎞ ds i ⎟; (1) = b1i ⎜ ⎜ ⎟ dt ( ) − − − − E E y w sin δ δ α qi qj ij i j ij i ⎝ ⎠ dwi = ξ i s i , j = 1,2, j ≠ i; dt dEqi S SG : = b2i (− Eqi + b3i ⋅ si sin (δ i − α ij ) + U1i ); (2) dt dP ST : Тi = b4 i (− PТi + qi ⋅ Ci ); dt dqi (3) = b6i (− γ i (qi ) − b5i si + hi ); dt dhi = b7 i (− hi + U 2i ), dt where S PS is high level subsystem; S SG , ST are low level subsystems (for SG and turbine correspondingly); U1i ,U 2i are local controls for low level subsystems; i = 1,2 is turbogenerator number; j = 1,2, j ≠ i ; yij = y ji , α ij = α ji are
III.
At the first step we form the first aggregation of local macrovariables for each SG subsystem and for i-th turbogenerator turbine separately:
mutual conductance of SG and its auxiliary angles; δ i is SG rotor’s turning angle about synchronous axis of rotation; si = (ω0 − ωi ) / ω0 is slip; ωi is rotation frequency of SG, ω0 is synchronous frequency of rotation, PТi is mechanical output of turbine; Eq is SG’s synchronous EMF; C = const is
ψ 1i = U G2 0 i − U Gi2 ; ψ 2 i = hi − ϕ1i ,
(6)
that might satisfy solution of functional equations: Tkiψ ki (t ) + ψ ki = 0.
respective vapors pressure before turbine; qi is consumption of vapor at turbine input; γ i (qi ) is function considering actuating motor motion limitation; hi is signal of turbine secondary speed controller; bij are MMPS constants, wi is
(7)
From now on let’s i = 1,2 . Let us find the analytical form for excitation U 1i control law providing stabilization of SG output voltage. Solve Eq. (7) with respect to U 1i considering macrovariables ψ 1i (6) with its total derivatives, and plant equations (1) and (2). Finally, we obtain:
variable that modeling external piecewise disturbance M i (t ) . Local control law design is performed for each turbogenerator individually, without taking into account its interaction. Let’s introduce following invariants (i.e. subtotality of local subsystem's targets) for each individual turbogenerator by the number of local control loops [4]: 1) stabilizing of SG output voltage (4) U G 0 i − U Gi = 0, where U G 0i is desired SG voltage; 2) stabilizing of turbine output power PTi0 − PTi = 0 ,
PROCEDURE OF HIERARCHICAL CONTROL LAW SYNTHESIS
U 1i = Eqi − b3i ⋅ si sin (δ i − α ij ) + +
b2i
(A E i
Δi + + Bi (δ i )) qi
ψ 1i 1 , T1i 2b2i (Ai Eqi + Bi (δ i ))
(8)
where Δ i = EqiU c yij xdi (sin (δ i − α ij ) − yii xdi sin (δ i − α ij + α ii ))si ;
Ai = 1 − 2 yii xdi cos(α ii ) + ( yii xdi ) ; 2
(5)
Bi (δ i ) = U c y12 xdi (cos(δ i − α12 ) − yii xdi cos( δ i −α12 +α ii
where PTi0 is output target value.
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2013 5th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)
Since ψ 1i = 0 , ψ 2i = 0 then by (6) the invariant (4) is satisfied. When the system image point comes into neighborhood of intersection of manifolds ψ 1i = 0 and
(5) is satisfied then PTi = PTi0 follows from (10). Substitute this expression into (1). Let us introduce the following macrovariable to define control PTi0 in obtained system of equations:
ψ 2i = 0 , a dynamical decomposition will take place.
ψ 5i = ξ i si + β i wi .
At the second step of synthesis procedure we might define the internal control ϕ1i (si , PTi , qi ) . So introduce the following macrovariable for decomposed system obtained at the first step
ψ 3i = qi − ϕ 2 i (PTi ) .
It must satisfy solution of equation (7). Finally we get coordinating control law: PTi0 = PTi0 (δ i , δ j , si , wi ) = Eqi2 (δ i ) yii sin (α ii ) +
+ Eqi (δ i )Eqj (δ j )y12 sin (δ i − δ j − α12 ) −
(9)
It must satisfy solution of equation (7). Similarly, at the third step we also have the loop of internal control ϕ 2 i (PTi ) . But now we can satisfy invariant (5). Therefore let the final macrovariable be:
ψ 4 i = PTi0 − PTi .
⎛ βi ⎞ ⎛ 1 β ⎞ − wi ⎜⎜ − 1⎟⎟ − si ⎜⎜ + i ⎟⎟, b T b T b ξ 1i ⎠ ⎝ i 1i 5i ⎠ ⎝ 1i 5 i Bi (δ i ) , j = 1,2, j ≠ i . Ai Output targets for turbine local control laws (12) are obtained by (14) in a way providing equality of mechanical and electrical output at turbogenerator’s shaft with taking into account an action of external piecewise constant disturbance. Note that local control laws (8), (12) consist of only state variables of local turbogenerator, but coordinating control (14) includes local variables δ i , si , wi and state variables δ j of where E qi (δ i ) = Bi2 (δ i ) − Ai (Di − U G2 0i ) −
(10)
It also must satisfy solution of equation (7). Finally we get
ϕ 2i (PTi ) =
PТi 1 (PTi0 − PTi ) . + Ci Ci b4iT4i
Since we now know expression for ϕ 2 i (PTi ) , therefore we
can define the internal control ϕ1i (si , PTi , qi ) . Let’s write the expression for derivative of macrovariable (9) provided by equations of corresponding decomposed system. Therefore ϕ 1i (s i , PTi , q i ) = γ i (q i ) + b5i s i +
neighboring turbogenerator as well. A condition of closedloop system (1)–(3), (8), (12), (14) stability has the form Tki > 0, i = 1,2; k = 1,5; β i > 0 .
. (11) ⎛ 1 ⎞ 1 ⎟b4i (− PТi + q i ⋅ C i ) − ⎜1 − ψ 3i ⎟ ⎜ b4 i T 4 i ⎠ b6i T3i ⎝ So, at the second step we can find the expression for turbine local control laws U 2i : combining equation (7), corresponding 1 + b6 i C i
IV.
MODELING
The external disturbance M i (t ) is formed as piecewise function: it reflects variations in the consumed electric power, takes constant values on certain time intervals, which corresponds to certain values of the load. Fig. 1–6 show modeling results for closed-loop system (1)–(3), (8), (12), (14) under action of piecewise constant disturbances (shown on fig. 7) at following control laws parameter values: U G 0i = 1.1 ;
macrovariable ψ 2 i in (6), and equations (1), (3) we obtain:
⎡ ∂ϕ1i dPTi ⎤ + ⎥ ⎢ ⎢ ∂PTi dt ⎥ 1 ⎢ ∂ϕ1i dqi ⎥ 1 +⎥ − (12) U 2i (si , PTi , qi , hi ) = hi + ψ 2i . ⎢+ b7 i ⎢ ∂qi dt ⎥ T2i b7i ⎢ ∂ϕ ds ⎥ ⎢+ 1i i ⎥ ⎢⎣ ∂si dt ⎥⎦ In order to get the final analytical expression for control law U 2i substitute corresponding partial derivations (11) in expression (12) with using MMPS equations (1), (3). The final control (12) will include output PTi0 target values. It’s formed by the high level system on the basis of satisfying the global engineering invariant. This invariant is a subtotality of high level system’s targets, such as turbogenerator ω i = ω 0 agreed rotation frequency. For model (1) this follows si = 0 .
(14)
ξ i = β i = 1 ; T1i = T2i = T4i = T5i = 0.25 ; T3i = 2 . Modeling results prove that designed hierarchical control laws provide satisfying the sub-totality of low level system targets (4), (5) (see fig. 1, 5) as well as the sub-totality of high level system targets (13) (see fig. 3). Moreover a piecewise constant disturbances are suppressed and chaotic modes are not arisen. We have built principally new algorithms for control of MMPS. Design of the high-efficiency control system for a MMPS was provided by dividing of the whole complex interrelated power system into high and low level subsystem, and by hierarchical synthesis of local and coordinated control laws. This control system is featured by: • Simplified implementation of control laws; • Providing a desired activity by turbogenerator; • Turbogenerator’s rotation synchronous frequency and constant output voltage;
(13)
So, the target P is a coordinating control for high level subsystem (1) with taking into account an electrical interrelation between SG. The fourth step is dedicated to synthesis of this control. Since at ψ 4i = 0 the local invariant 0 Ti
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Figure 1. Transients of ψ 4i (t )
Figure 2. Transients of mechanical output PTi (t ) and angles δ i (t )
Figure 3. Transients of slips
Figure 4. Transients of SG’ EMF E qi (t ) and hi (t )
Figure 5. Transients of output voltages U Gi (t )
Figure 6. Transients of controls (12)
and electrical powers PEi (t )
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simulation proves efficiency for designed system of group control. ACKNOWLEDGMENT Author thanks Russian Foundation for Basic Research (Grant 13-08-01008-a) and Southern Federal University Grants Program #213.01-24/2013-98 for financial and scientific support. REFERENCES [1] [2] [3] [4]
Figure 7. External disturbance
• • •
[5]
Asymptotical stability of closed-loop system in a whole; Coordinated control of frequency and output providing a balance of power at turbogenerators' shaft; Suppression of external piecewise constant disturbances.
[6]
[7]
Therefore basing on methods of synergetic control theory we built anti-chaotic control for complex power system using hierarchical principles of control law design. Computer
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Q. Lu, Y. Sun, S. Mei, Nonlinear Control Systems and Power System design. Kluwer Academic Publishers, 2001. P. Kundur, Power System Stability and Control. McGraw-Hill, 1994. P. M. Anderson, A. A. Fouad, Power System Control and Stability. New York: IEEE Press, 1994. H. Huerta, A. G. Loukianov, and J. M. Canedo, “Robust multi-machine power systems control via high order sliding modes”, Electric Power Systems Research, 2011. A. A. Kolesnikov, Synergetic Control Theory. Moscow: Energoatomizdat, 1994 [in Russian]. A. A. Kuz’menko, “Nonlinear Adaptive Control of a Turbogenerator”, Journal of Computer and Systems Sciences International, vol. 47, no. 1, pp. 103-110, 2008. V. N. Kozlov, V. N. Shashihin, “Synthesis of coordinated robust control of interlinked synchronous generators”, Electrichestvo magazine, no. 9, pp. 20-26, 2000 [in Russian].
