POWER ELECTRONIC SYSTEMS. Anatoly Kolesnikov, Gennady Veselov, Andy. Popov, Alex Kolesnikov,. Andy Kuzmenko. Department of Automatic Controls.
A SYNERGETIC APPROACH TO THE M ODELING OF POWER ELECTRONIC SYSTEMS Anatoly Kolesnikov, Gennady Veselov, Andy Popov, Alex Kolesnikov, Andy Kuzmenko Department of Automatic Controls Taganrog St. Univ of Radio Engineering Taganrog, Russia
ABSTRACT We introduce the Synergetic approach, which brings great improvements in computer modeling and simulation of power electronic systems. Essentially, the process involves synthesis of control laws that result in collapse of (at least two) initially high-dimension models into a single model of lower-dimension. The technique requires creation of state space attractors, artificial manifolds, that reflect the desirable operating regimes of the dynamic system We explain the process and give example results.
Roger A. Dougal, Igor Kondratiev Dept. of Electrical Engineering Univ. of South Carolina Columbia, SC 29208 The necessity of including the control laws into the power system object models leads to new tasks, which include the following: •
Developing methods for understanding the parametric and structural robustness (insensitivity) of the objects and the control laws. As part of the modeling, it is necessary to address the question of how the simultaneous drift of either the object’s and/or the regulator's parameters influence the dynamic stability and the dynamic qualities of the power system.
•
Developing methods to determine the connective stability of the power system in general and of its parts (groups, zones, subsystems). As a result of modeling, we have to answer the question of how spontaneous commutations and new links influence the stability of the power system.
INTRODUCTION The synergetic approach, used in the Virtual Test Bed (VTB) project as a basis for synthesizing control laws for power systems, brings great improvements in the computer modeling of dynamic systems. Synergetic control theory helps in the modeling of power electronic systems in the following way: Properly synthesized synergetic control strategies inevitably decompose a system that is initially described by several high dimensional models into a hierarchical succession of asymptotically stable lower dimensional models. As a result, it is possible, for instance, to collapse several complicated components of a system model into a single component described by much simpler equations. In the modeling of controlled power electronic systems there are tasks and features that are essentially different from those handled by traditional electric circuit modeling and simulation tools. These differences arise from the fact that controlled systems are described not only by the systems of differential equations of the circuit elements, but also by the differential equations that describe the control laws of the system. Such control laws reflect the corresponding control strategies and directly influence the behavior of the collective system.
PRINCIPLES OF SYNERGETIC CONTROL The Synergetic approach gives rise to a new ideological principle: even for a power system, control is a directed process of converting power, matter and information so as to ensure optimal functionality. This principle reveals the target orientation of control laws and deeply changes the emphasis of modeling by focusing attention on the result more than on the process. Control, as a measure of purposeful influence of a person on a power system and as a reflection of his aims and requirements, should weave together all the aspects of the system’s work as a whole. The same is true for its separate elements. In the VTB project, based on the synergetic concept of dynamic interaction among power, matter, and information, we have developed an applied theory for synthesizing vector regulators for nonlinear power systems. Synergetic Control Theory is a new direction
in control science based on the principles of directed self-organization and the use of the natural nonlinear qualities of dynamic objects. The basic principles of synergetic control theory are as follows.
test problem for control theory methods − from classical linear methods based on PID regulators to the modern ones based on Fuzzy Neural Networks:
x1′ = x2 ; x′2 = sinx1 + x3 ; x′ = u. 3
(1)
1. Artificial attractors – invariant manifolds – are formed in the state space of the object. On Assume that our target is the following attractor: these attractors, we ensure organization of the ψ = x1 + x2 + x3 = 0. (2) desired dynamic and static qualities of the controlled objects. Formation of the attractors The synergetic control law (3) that directs our is the reflection of a directed self-organization process. system onto the attractor (2) is: 2. The dominant principle of synergetic synthesis methods is the principle of compressiondecompression of the phase flow of the controllable systems.
where T is the time constant
3. The developer’s requirements are presented in the form of a system of invariants (technical, power, electromagnetic, etc.) which describe the desired operating modes of the controlled objects.
After the time needed to accomplish all the transient processes in the system, (3-5)⋅T, the object state point arrives on the target invariant manifold, equation (2) becomes true, and we can use this equation to simplify the equations of system (1), to:
Attractors created in the state space of the object simplify the modeling process by ensuring a radical lowering of the dimensionality of the comprehensive nonlinear model that describes the dynamic characteristic s of the controlled subsystem and of the power system in general. As a result, it is possible to model the entire system though a single subsystem defined by the decomposed system of equations, the dimension of which can be found according to the following:
dim A = n − k * m , where:dim A - dimension of the decomposed system; n - dimension of the initial system; m - dimension of the control vector; and k - number of sequentially used attractors.
EXAMPLE APPLICATIONS Inverted Pendulum As a first example, consider the following equations that describe an inverted pendulum. Due to its distinctive features this model has became a sort of
u= −
x1 + x2 ⋅ (T + 1) + x3 ⋅ (T + 1) + T ⋅ sin x1 (3) T
x1′ = x2 ; x′2 = sinx1 − x1 − x2 .
(4)
Equation (4) defines the behavior of the closed-loop system after the transient processes in the system have been accomplished.
Asynchronous motor Another interesting problem is that of maintaining a constant rotation speed of a motor while it is affected by an unmeasured load ml . Assume that we also require constant armature flux linkage to ensure good energy characteristic s for the drive. Consider the complete nonlinear model of an asynchronous motor, written in a rotating coordinate system as follows
3 2 kr 1 p x2 x3 − ml , 2 J J x& 2 ( t) = −α r x2 + kr rr x4 , r xx k 1 x&3 (t) = − e x3 − x1x4 − kr rr 3 4 − r x1x2 + u1 , le x2 le le r x x kα 1 x& 4 ( t) = − e x4 − x1 x3 − kr rr 3 3 − r r x2 + u2 , le x2 le le (5) x&1 (t) =
where x1 - rotation frequency; x2 − rotor’s flux linkage; x3 , x4 − damper current projections on the axes y and x of the rectangular domain; u1 , u2 − projections of the damper voltage on the axes y and x; J – inertia moment; ml – load moment;
kr =
le =
Lm r ; αr = r ; Lr Lr
expedient to select v4 so that the armature flux linkage is constant in the stabilized regime. According to the law of M.E.Kostenko, this allows the optimization of the energy processes in the drive. Let
x2st =ψ = const
(9)
and select v4 as follows
2 m
2 m r
v4 = −λ2 x2 + A1
2 r s
Ls Lr − L L r +Lr ; re = ; Lr L2r
Substituting v4 (10) into (8) yields
x& 2 ( t) = −α r x2 − kr rr λ2 x2 + kr rr A1
Here, Ls andLr − inductance of the stator and rotor;
rs and rr resistance of the stator and rotor; p − number of pole pairs; Lm − mutual inductance. Since the controls u1 and u2 directly influence the state variables x3 and x4 , we call these variables
ψ 1 = β11 ( x3 − v3 ) + β12 ( x4 − v4 ), ψ 2 = β 21 ( x3 − v3 ) + β 22 ( x4 − v4 ),
(10)
(6)
(11)
From (11) we see that in the stabilized regime
x2st =
A1
αr + λ2 kr rr
=ψ
(12)
In (12) the coefficients A1 and λ2 are selected so that they ensure the desired stabilized quantity. Let’s consider the first equation of the system (5) on the intersection of the manifolds. Assume ml = const .
where β11 ,K, β 22 - coefficients are selected in such a
Now since we are attempting to stabilize the frequency
way that equations ψ 1 = 0 and ψ 2 = 0 are linearly independent.
(13)
Assume that we require the manifolds to satisfy the following equations
ψ& 1(t) + α 1ψ 1 = 0, ψ& 2 (t) + α 2ψ 2 = 0,
(7)
where the arbitrary positive coefficients α 1 and α 2 determine the transient time to reach the manifolds ψ 1 = 0 and ψ 2 = 0 . According to (7) the duration of these processes is ( 3K4)
1
α max
). After the transients
are finished, the conditions ψ 1 = 0 and ψ 2 = 0 are satisfied, and also according to (6): x 3 = v 3 and
x4 = v4 . Therefore on manifolds ψ 1 = 0 and ψ 2 = 0 the trajectory of the system is described only by the first two equations, i.e. 3 2 kr 1 p x2v3 − ml , 2 J J x& 2(t) = −α r x2 + kr rr v4. x&1(t) =
(8)
Now selecting v3 and v4 , we are going to consider the second equation first. It would be most
x1 = ω Introducing the difference variable
y1 = ω − x1 ⇒ x1 = ω − y1 ,
(14)
we get
y& 1 = −
3 2 kr 1 p x 2 v3 + ml 2 J J
(15)
Introduction of an integrating unit into the control law suppresses the ml . We add the following equation in order to do this.
x&5 (t) = y1 = ω − x1 ,
(16)
Selecting v3 =
2J (µ x + µ1 y1) 3 p2kr x2 0 5
(17)
This results in
x&5 (t) = y1 , 1 y& 1 (t) = − µ 0 x5 − µ1 y1 + J ml .
(18)
Substituting the expression for x&5 ( t ) from the first equation into the second one, we get
&y&1 (t) + µ1 y&1 (t) + µ 0 y1 = 0
(19)
Selecting µ 0 > 0 , µ1 > 0 we ensure the suppression of the constant moment ml . So we selected v3 and v4 the following way:
2J ( µ 0 x5 + µ1 y1 ) 2 3 p kr x2 v4 = −λ2 x2 + A1 v3 =
(20)
Now find the control actions that move the system to the intersection of the manifolds. A joint solution of (5), (6), (7) accounting for (9), (13)-(20) gives the expressions for the controls:
α 1α 2l e u1 = ⋅ β1 β 4 − β 2 β 3
µ 0 (ω − x1 ) − k 3 1 µ p 2 r x x − m x2 2 3 l 1 2 J J 2J = (µ 0 x5 + µ 1 (ω − x1 )) ⋅ 3 p 2 k r x 22 (− α r x 2 + k r rr x 4 ) v& 4 ( t) = −λ2 (− α r x2 + kr rr x4 )
− (25)
2
β4 α 2 l e −
CONCLUSION
β1 β2 (bu − v&4 ) − ψ 1 − (v′3 − au ) − α1 α 1
β2 β3 (a u − v&3 ) − β 4 (bu − v& 4 ) − ψ 2 − α 1l e α 2 α1
(21)
u2 =
α 1α l ⋅ β1β4 − β 2β 3
β1 α 1 le −
β3 α 2 le
2 2 e
β3 β4 − α (a u − v& 3 ) − α (b u − v& 4 ) − ψ 2 − 2 2 β1 β2 − α (a u − v& 3 ) − α (bu − v& 4 ) − ψ 1 , 1 1 (22)
where
re x3 − x1 x4 − kr rr le r bu = − e x4 + x1x3 + kr rr le au = −
x3 x4 kr − x1x2 x2 le x3 x3 krα r − x2 x2 le
In formulas (21) and (22), v& 3 and according to the equations of the object:
v& 3 (t ) =
2J 3 p 2 k r x22
(23)
v& 4 are calculated
(µ 0 x& 5 − µ 1 x&1 )x 2 − (µ x + µ (ω − x ))x& = 0 5 1 1 2 (24)
These examples illustrate the method of simplifying the description of a system object by means of Synergetic control theory. The Synergetic approach can help not only to decrease the dimension of a modeled system, but also to ensure the following additional properties of the closed-loop system: • It lowers the importance or eliminates a number of standard problems typically encountered in the modeling and simulation process, including, for example, the problems of sparse matrices, problem stiffness, or selection of integration step size. • It allows the possibility of determining the parametric robustness (insensitivity) of the object, local subsystem, and power system in general. • It allows the possibility of determining the fullscale connective stability of the power system in general and of its parts. For all of the reasons discussed in this paper, the Synergetic approach can be very helpful in modeling large-scale power electronic systems.
ACKNOWLEDGEMENTS This work was supported by the US Office of Naval Research under Grant N00014-00-1-0131.
