Synergetic Control Strategies for Shipboard DC ... - Semantic Scholar
Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
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Synergetic Control Strategies for Shipboard DC Power Distribution Systems I. Kondratiev, Member, IEEE, and R. Dougal, Member, IEEE
Abstract— This paper presents a synergetic control design for a generalized component of modern ship DC distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design uses a state space averaged model of the system and overcomes the system’s nonlinearity, multi-connectivity, and high dynamic order, which are challenging features for control development. The design results in general analytical control strategies, conditions for counterweighing the system’s nonlinearity, and stability conditions for the closed-loop operation. The control introduces invariant manifolds into the state space of the system and suppresses errors in output voltage and current sharing. The coefficients of invariant manifolds allow us not only to counterweigh nonlinearity and allocate the type and ratings of current sharing for each converter, but also allow to change the number of operating converters. The control strategies are first applied to a two-converter system to demonstrate how the designed control dynamically allocates the current sharing within the system. Next, a two-converter system under synergetic control and feedback linearization control shows superiority of synergetic control performance. Finally, an eight-converter system presents a coordinative control strategy that optimizes the system’s efficiency by changing the number of operating converters.
U
I. INTRODUCTION
of DC power distribution systems that widely employ zonal structure and comprise paralleled converters has become a common practice in ship designs [1,2,3,4]. Among the numerous benefits such as a decrease of electrical and thermal stress in a component as well as overall system cost and weight, these DC distribution systems provide a capacity for nearly instantaneous reconfiguration [3]. Control design that fully utilizes this capacity could bring the ship power system operation to its highest possible invulnerability and efficiency. Unfortunately, such challenging properties of power systems as nonlinearity, multi-connectivity, and high-dimensionality prevent control design from exploiting reconfiguration fully and efficiently. However, a newly developed synergetic control theory provides the necessary tools for the design of analytical control capable of utilizing the reconfiguration capacity of these complex DC power systems at different control levels. A common element of modern DC zonal electrical
distribution systems is power converters connected in parallel and feeding service converter modules [3]. Control strategies for this element define reconfiguration features of the system. Ideally, at local control level these control strategies perform such technological tasks as maintaining allocated current sharing among power converters and stabilizing the output voltage. For the system control level, these control strategies deliver control channels that are able to change the allocation of current sharing and of the system structure to achieve such features of the system operation as highest efficiency or fastest response. Unfortunately, this element appears to be difficult not only for analysis of the system’s behavior but also for control design [1,2]. In practice, to simplify the study of the system, the influence of the service converter modules is approximated by constant power load nonlinearity. Such approximation reduces the dynamic order of the system, decouples the control design problem for the paralleled converter system and for the service converter modules [4]. The schematic of the simplified system based on paralleled buck converters is shown in Fig. 1. Different control design methods within the classical and advanced approaches have been applied to this system.
SE
This work was supported in part by ONR grants N00014-02-1-0623, N00014-03-1-0434. I.Kondratiev (ph: 803-777-8966; email:[email protected]) and R. Dougal (email:[email protected]) are with Electrical Engineering Department, University Of South Carolina, Columbia, SC 29208 USA
1-4244-0989-6/07/$25.00 ©2007 IEEE.
Fig. 1. System containing m parallel-connected buck converters
Traditionally, classical control theory attracts engineers due to the variety of control design approaches which were developed to use simple linear small-signal models [5,6]. These approaches design different control laws that stabilize the system; this includes proportional, integral-proportional etc. However, due to operating the system only in the vicinity of the operating point with a specified quality, these
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laws cannot control the system beyond the linear limits [7]. As a result, the inability of linear controls to predict dynamic behavior of constant power load nonlinearity could lead to chaotic behavior of the system and sometimes could result in a system collapse [8] because of the negative impedance instability [1]. On the other hand, the problem of control design for paralleled buck converters has been addressed by such advanced techniques as sliding control [7], feedback linearization control [9], fuzzy logic control [10], and synergetic control [11,12,13,14]. Almost all these approaches are focused on equal current sharing among system converters and do not consider system level optimization of the system behavior. In contrast, the results presented in [12,14] have addressed the problem of dynamic current sharing by introducing this capability of synergetic control on the basis of a two-buck converter system feeding resistive load and constant power load respectively. Synergetic control theory has made it possible to look at control design for the system at a more generalized level due to fully analytical procedure of control design. Our approach incorporates in control design not only channels for dynamic current sharing allocation but also control channels for changes of the system configuration. This procedure uses nonlinear model of the system and ensures global (or semiglobal) asymptotic stability of the closed-loop system. Moreover, synergetic control induces a self-organized motion in the system that takes control of the dynamic degrees of freedom of the system and as a result reduces the dynamic order of the controlled system and simplifies the analysis of the closed loop system behavior. The objectives of the research are to derive universal synergetic control laws that provide control channels for the dynamic allocation of types and ratings of current sharing among converters and control channels for a change of the number of converters operating in parallel; to study the closed loop stability of the system; and to evaluate the performance of the closed-loop system by simulation. This paper presents a design of generalized control strategies for the common element of DC zonal electric distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design results in general control strategies for the system, the condition for counterweighing the impact of constant power load, and stability conditions. Next, we apply the developed control strategies to three different systems. First on two-converter system, by varying coefficients of the macro-variables, we study the capability of the designed control for the dynamic allocation of type and rating of current sharing. We compare the performances of the two-converter system presented in [9] under synergetic control and feedback linearization control designed in [9]. Finally, we develop coordinative control that improves the efficiency of the eight-converter system by changing the number of operating converters.
The ultimate benefit of the research is that the designed control strategies enable more control of dynamic current sharing, improve the system efficiency, decrease stress on the system components, and promote simplification of control strategies. However, this digest covers only the most significant data such as the system model; general control strategies; general stability conditions; parameters of the used system; some simulation results, including impact of control law coefficients on system current sharing and transient response; performance comparison; and efficiency optimization. II. SYNERGETIC CONTROL DESIGN The system containing an arbitrary number of paralleled converters feeding constant power load is presented in Fig. 1. In model development, we limit the description level for each component by their lossless subset. Similarly, we limit the description level of the load by keeping only its capacitive, resistive, and constant power components, while sudden change of the system load is accounted for by disturbance M(t). Finally, we restrict the description level for the system model by using state space averaging technique [15] as a tool for model development. A. System Model The state space averaged model of the system presented in (1) is derived using the following assumptions: the system operates in a continuous conduction mode; switching occurs at a very-high switching frequency relative to the filter dynamics; parasitic effects are ignored; state variables are the averaged vc1 capacitor voltage and iLi inductance currents; constant power load is represented as dependent current source value of which equals to power consumed by the load (Pload) divided by output voltage (vc1); a piecewise constant current disturbance M(t) sufficiently approximates variations of the system load and any uncertainty of system parameters; the origin of the system model is at the specified steady state voltage(Vc1,ref); the zero order dynamics adequately represents the behavior of the sources Ei.
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⎧ Pload 1 ⎛ m ⎞ v + Vc1,ref M ⎜ ∑ iLj ⎟ − − + ⎪v ′ = ⎜ ⎟ ( ) C R C v V C C + j 1 = t ext t c ref t t 1 , ⎝ ⎠ ⎪ ⎪ 1 ⎪iL′ 1 = − ⋅ (v + Vc1,ref ) + u1 L1 ⎨ ⎪ ... ⎪ ⎪ 1 ′ =− ⋅ (v + Vc1,ref ) + u m ⎪iLm Lm ⎩ where v and iLi are state variables;
(1)
v = v c1 − Vc1,ref m
Ct = ∑ Ci − Cext is the total output capacitance of the system i =1
M(t) is a current disturbance influencing the system; dci is the desired switch duty of the ith converter;
ui = d ci ⋅ Ei / Li i = 1, m
In order to simplify control derivation results we
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represented system model (1) in the following form: ⎧⎪ X& = F1 I + F2 X + CF3 ( X ) ⎨& ⎪⎩ I = F4 ( X , I ) + U ( X , I )
(2)
where X=(M,v)T and I=(i1,…,im)T
B. Goals of the Control Design Introduced into the system model (1), disturbance M(t) represents the dependent current source, which allows the controller to take into account the change of the system load. Since in real life applications converter systems are usually influenced by sudden load change, the disturbance M(t) is approximated by piece-wise constant function (3). As a result of introduction of an additional dynamic coordinate M(t) into the system model (1), the system voltage becomes invariant to the system load resistance.
(v c 1 − V c 1 , ref dM = dt TI ⋅ Rd
)=
v TI ⋅ Rd
C. Control Synthesis Procedure From a mathematical standpoint, synergetic control design [16] is based on a new method for generating control laws, or feedback, that direct the system from arbitrary initial conditions into the vicinity of manifolds (4). The asymptotically stable motion along these manifolds toward the end attractors is then ensured by proper choice of coefficients in macro-variables. On these attractors the desired properties of the controlled system are guarantied. In short, the synergetic control design procedure is as follows. For system (1) the designer’s specifications are formulated as a set of macro-variables (5) based on the system state variables. The number of macro-variables in the set equals the number of control channels. ψ i (X, I ) = 0 (4) (5)
Assuming that the desired evolution of the macrovariables is as in (5) and substituting the system model (1) in functional equation (6) we get an asymptotically stable control, which ensures the desired dynamic properties. & + Ψ = 0 T ⋅Ψ T = E ⋅ (T 1 ...
Tm
)T
Ti > 0
E – is unity matrix dimension mxm
or Ψ = AI − B1 X − B 2 F3 ( X ) where
Ψ = (c1 ⋅ψ 1
(6)
...
c m ⋅ψ m )
T
ci={0,1}, i=1,m – is coefficient controlling the operation of ith converter
⎛ − a11 ⎛ a13 ... a1n ⎞ ⎟ ⎜ ⎜ A = ⎜ ... ... .. ⎟ B1 = ⎜ ... ⎟ ⎜− a ⎜a ⎝ m1 ⎝ m 3 ... amn ⎠
⎛ F3 ( X ) = ⎜ 0 ⎜ ⎝
(3)
The primary goal of the control is to maintain a specified output voltage Vc1,ref even while the system is affected by the disturbance M(t). Another important goal is to ensure proper sharing of current among the parallel-connected converters. At the design stage, we do not make any particular assumptions about current sharing except that the converter currents relate linearly to each other; that is, they each supply some constant fraction of the load current that correspond to active current sharing. All these control design goals were incorporated into the system model by the corresponding coordinate transformation. The resulting control laws must ensure asymptotic stability of the closed loop system.
ψ i = ψ i (X, I ), i = 1, m
Defining a macro-variable as shown in (7) and solving the system of functional equations (6) accounting for system model (2), yields control laws presented in (8). m ai ,n+1 ψ i = ai1 ⋅ M + ai 2 ⋅ v + ∑ ai , j+2iLj + ; i = 1, m ( v + V j =1 (7) c1,ref )Ct
1 (v + Vc1,ref
⎞ ⎟ ) ⎟⎠
− a12 ⎞ ⎛ 0 − a1n+1 ⎞ ⎟ ⎟ ⎜ ... ⎟ B2 = ⎜ ... ... ⎟ ⎜0 −a ⎟ − am 2 ⎟⎠ mn +1 ⎠ ⎝
T
U ( X , I ) = A−1 ⋅ ((B1 + B2 F4 )(F1I + F2 X + CF3 ( X )) − T −1Ψ ) + − F4 ( X , I )
(8)
where
⎛ 1 F4 = ⎜ 0 − ⎜ (v + Vc1,ref ⎝
)
2
⎞ ⎟ ⎟ ⎠
T
(9)
According to synergetic control theory [16], control laws (8) ensure asymptotically stable convergence towards the specified manifolds (4) within (3-5)Ti and, as shown later, results in current sharing specified by coefficients in (7). D. Behavior on Manifolds and Stability of the System It can be shown that under synergetic control the closed loop system has an attracting set. As a result, the representing point of the system reaches an attracting set, which is the intersection of manifolds. The study of the system behavior on the intersection of manifolds helps to understand system stability. The properties of the system motion on manifolds are defined by the following system of algebraic and dynamic equations: ⎧ X& = F1 I + F2 X + CF3 ( X ) ⎨ (10) Ψ=0 ⎩ or ⎧⎪ X& = F1 I + F2 X + CF3 ( X ) ⎨ ⎪⎩ I = − A−1 (B1 X + B2 F3 ( X ))
(11)
In order to define the properties of the system dynamics on manifolds, we eliminate dependent variables such as inductor currents and derive the following dynamic system: X& = (F2 − F1 A−1B1 )X + (C − F1 A−1B2 )F3 ( X ) (12) The equation (12) shows that the behavior of the closed loop system on manifolds is defined by two components: linear and nonlinear. It also can be seen that manifold coefficients provide flexibility in definition of both components. So, we define matrix B2 in such a way that the following conditions are true. C − F1 A −1 B 2 = 0 (13) Equation (13) represents the condition for counterweighing the nonlinear impact of a constant power load as well as for the linearization of the system (2). Accounting for condition (13), equation (12) can be rewritten as follows:
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(14) The equation (14) defines a linear dynamic system, properties of which depend on the choice of the coefficients of matrix A and B1. Coefficients of these matrixes need to be chosen to ensure stable motion of the system (14). Hence, synergetic control design results in analytical control laws that ensure asymptotic stability of the closed loop system and compensates for system nonlinearity by a particular choice of coefficients. The allocation of type and ratings of current sharing in the system is defined by coefficients of macro variables.
After the transients are passed, the representing point of the two-converter system inevitably reaches the intersection of the manifolds (4). Hence, the motion of the system on the manifolds is defined by the following equations: ⎛ a13 ⎜⎜ ⎝ a23
In this section we present a short description and some simulation results for three systems, which are used to study different features of the designed control strategies. First we study the properties of current sharing in a two-converter system. Next, we perform simulations of a two-converter system presented in [9] under synergetic control and feedback linearization control developed in [9]. Finally, for an eight-converter system we present a coordinative control strategy that optimizes the system’s performance by changing the number of converters operating in parallel.
TABLE II A FEW POSSIBLE SYNERGETIC CONTROL STRATEGIES FOR THE TWO-CONVERTER SYSTEM
TABLE I NOMINAL VALUE OF SYSTEM PARAMETERS
Resistance of inductors Filter inductance Filter capacitance Input voltage Output voltage Switching frequency Time constants Integrator coef. Law coef. Law coef. Load resistance Disturbance resistance Disturbance constant power
Symbol rL1=rL2=rLn L1=L2=Ln C1=C2=Cn E VC,ref=U0 T1=T2=Tn η= ηn a11=a11n a12=a12n Rext Rextd Pload
Control Strategy
Nominal Value 0.2 Ohm 6.1 mH 3000 μF 12 V 4.0 V 4.846 kHz 0.015 128 70 5 2 Ohm 2 Ohm 4W
A. Current Sharing in Two-Converter System To study the properties of current sharing we use a twoconverter system, the parameters of which are shown in Table I. In practice, master-slave [17], and democratic [18] current sharing strategies are widely used in paralleled converters systems. The major difference between masterslave and democratic current sharing is that master-slave sharing provides a fixed ratio of current sharing in transient and static regimes of operation, while democratic sharing has a variable ratio of current sharing during transients and a fixed ratio in the static regime of operation. This section shows important details of the impact of macro-variable coefficients on current sharing in the system and illustrates that synergetic control laws provide power systems with dynamic management of responsibilities, including the dynamic reconfiguration and the allocation of current sharing.
(15)
The study shows that: - coefficients a13, a14, a23, and a24 allocate the current sharing ratings, - coefficients a15, and a25 provide the ability to compensate the impact of the constant power load, - coefficient a22 defines the type of current sharing as shown in Table II, - coefficients a11 and a12 influence the system’s transient response as shown in Fig. 2 and Fig. 3 respectively, - time constants T1 and T2 the define speed of convergence of the macro-variables toward manifolds, - coefficients a12 and a22 define which converter is responsible for managing errors in the output voltage.
III. SYSTEM SIMULATION AND EXPERIMENTAL VERIFICATION
Parameter Name
a15 ⎛ ⎞ ⎜ − a M − a12 v − ⎟ a14 ⎞ ⎛ iL1 ⎞ ⎜ 11 ( v + Vc1, ref ) ⎟ ⎟⎟ ⋅ ⎜⎜ ⎟⎟ = ⎟ a25 a23 ⎠ ⎝ iL 2 ⎠ ⎜ − a M − a v − 22 ⎜ 21 ⎟ ( ) + v V c1, ref ⎠ ⎝
Manifold structure
Master-slave
⎛ a13 ⎜⎜ ⎝ a 23
Democratic
⎛ a 13 ⎜⎜ ⎝ a 23
a15 ⎛ ⎞ a14 ⎞ ⎜ − a M − a12 v − ⎟ ⎟⎟ ⋅ I = ⎜ 11 ( v + Vc1,ref ) ⎟ a 24 ⎠ ⎜ ⎟ 0 ⎝ ⎠ a 15 ⎞ ⎛ a 14 ⎞ ⎟ ⎜ − a M − a 12 v − ⎟⎟ ⋅ I = ⎜ 11 ( v + V c1, ref ) ⎟ a 24 ⎠ ⎟ ⎜ − a 22 v ⎠ ⎝
B. Performance Comparison As a base for comparison of synergetic control we use a two-converter system under feedback linearization control presented in [9]. Synergetic control parameters are tuned to satisfy the same design requirements as in [9]: equal current sharing, 750V output voltage, output voltage overshoot less than 6%, and the transient response are such that the settling time is less than 10 msec. The parameters of the twoconverter system are as follows: L1=1.35 mH, L2=1.25 mH, C1=2.6mF, C2=2.5mF, Vc1,ref=750V, and an input voltage are E1=E2=850V. The load is uncertain with the parameters falling in the following bounds: 2.81 Ω ≤ Rext ≤ 50 Ω; 5.11mF ≤ Ct ≤ 10mF; 0 kW ≤ Pload ≤ 200 kW. d = − A ⋅ (vc1 − Vc1.ref ) − B ⋅ (i L1 − i L 2 ) − D ⋅ idif − E ⋅ where
i dif vc21
− F ⋅M
(16)
idif = (iL1 + iL 2 ) − (io1 + io 2 )
Synergetic control strategies and feedback linearization control strategies are transformed into the same form as shown in (16).
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Corresponding coefficients of control laws (16) are presented in Table III.
a) output voltage error
nonzero values, the synergetic control law allows the system to nullify output voltage error through the introduced integral action. However, the value of F influences the speed of integration.
a) output voltage error a) Switch on transients of the output voltage transients during power up
b) inductor currents Fig. 2 Impact of a11 on transient response of the two-converter system (a11= 0.5a11n , a11n, and 2a11n)
b) inductor currents Fig. 3. Impact of a12 on transient response of the two-converter system (a12=0.5a12n, a12n, and 2a12n)
Next, the designed systems have been simulated using an averaged representation of the buck converters. The series resistance of each inductor is 0.5 mΩ. For each control, four simulations for extreme resistance and capacitance values have been performed. The combinations of load resistance and capacitance are as follows: Rext=2.81 Ω, Ct=5.11mF; Rext=50 Ω, Ct=5.11mF; Rext=2.81 Ω, Ct=10mF; Rext= 50 Ω, Ct=10mF. The system is initially energized with PLoad=0 kW and is allowed to operate in the steady state up to 0.09 s. At 0.09 s PLoad is stepped to 150 kW. At 0.12 s PLoad is stepped to 200 kW. At 0.15 s PLoad is stepped down to 0 kW. TABLE III CONTROL LAWS COEFFICIENTS Name
A B D E F
Coefficient Value Synergetic Converter 1 Converter 2 0.00608 0.00554 0.0002647 0.0002451 0.00224 0.00208 0.000454 0.0004207 0.0035 0.003244
Feedback Linearization Converter 1 Converter 2 0.00207 0.00124 0.000715 0.000662 0.00136 0.00126 34.26 31.72 0 0
b) Switch on transients of the output voltage when constant power load steps
c) Switch off transients of the output voltage when constant power load steps down from 100 % to 0 % Fig. 4. Simulation results for the closed loop two-converter system. (1- synergetic control, 2- feedback linearization control, and 3- intersection of transient responses)
C. Efficiency Optimization This section presents efficiency optimization for a power distribution system that contains eight paralleled subunits. Each subunit is a buck converter rated at 50 kW. The subunit parameters are presented in Table IV. The converter efficiency curve against power output is shown in Fig. 5. TABLE IV. BUCK CONVERTER PARAMETERS
It can be seen from Fig. 4 that the closed loop system with synergetic control shows solid, stable, and fast response and at the same time better accuracy of output voltage than the system with feedback linearization control. The solid, stable and fast response is a result of the dynamic order reduction of the closed loop system. In the feedback linearization case, the closed system is of the fourth order. The closed loop system with synergetic control law starts from the fourth order and, through the influence of the control, becomes a second order system. As a result, the closed loop system with synergetic control does not have significant delay during switching on and has a better transient response. In contrast to feedback linearization control under which errors of voltage regulation and of current sharing have finite
Parameter
Symbol
Value
Capacitance External capacitance Inductance Time constant Control Coef. Control Coef. Control Coef. Control Coef. Index
Ci
2.0mF
Cext
8.0mF
Li Ti ai1 ai2 ai3
0.7mH 0.0002 10 100 0.5 10000 1,..,8
ηi i
TABLE V. LOAD POWER THRESHOLD LEVELS Threshold number P1 P2 P3 P4 P5 P6 P7
Level, kW 45 84 120 150 180 210 240
Synergetic control parameters are tuned to ensure a stable operation of any number of paralleled converters (from 1 to 8), equal current sharing, and 200V output voltage. The control task is to optimize the system’s efficiency by proper choice of the coordinative control functions ci(Pouti)
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100
100
80
80
Efficiency
Efficiency
that put corresponding converter online if ci=1 and offline if ci=0. In other words, power levels thresholds Pouti (i=1,..,7) that optimize the system’s power efficiency are to be chosen to maximize the system efficiency. Due to flat part in efficiency curve in Fig. 5, the optimization task has multiple solutions. One possible solution is presented in Table V. Simulation results in Fig. 6 shows the efficiency of the zonal distribution system in both cases when the system operates without coordinative control and with coordinative control that uses switching threshold as shown in Table V. It can be seen in Fig. 6 that the coordination introduced into the system significantly improves the efficiency of the system operation under light loads.
60 40 20
design is that it provides scalable control strategies, stability conditions, and conditions for counterweighing the system nonlinearity for any number of paralleled converters. The use of the design will bring a new level of flexibility and reliability in DC zonal distribution systems. ACKNOWLEDGMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. REFERENCES [1] [2]
b)
60
a)
40
[3]
20
0 0
20 40 60 80 % of rated power
100
Fig. 5. Efficiency of the converter
[4]
0 0
200
400 600 Power, kW
800
Fig. 6. Efficiency of the zonal distribution system. ( a) without coordination b) with coordinative control)
[5] [6]
IV. CONCLUSION
[7]
This paper presents a synergetic control design for a generalized component of modern ship DC distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design overcomes the system’s nonlinearity, multi-connectivity, and high dynamic order, which are challenging features for control development. The design results in general analytical control strategies, conditions for counterweighing the system’s nonlinearity, and stability conditions for the closedloop operation. The control introduces invariant manifolds into the state space of the system and suppresses errors in output voltage and current sharing. The coefficients of invariant manifolds allow us not only to counterweigh nonlinearity and allocate the type and ratings of current sharing for each converter, but also allow to change the number of operating converters. The paper illustrates the application of the designed strategies on different systems such as two- and eight- converter systems. The results show that synergetic control provides an extended flexibility for allocating types and ratings of current sharing dynamically. A comparison of performances of a two-converter system under synergetic control and feedback linearization control shows a superior performance of the synergetic control, which provides a faster response and nullifies errors of current sharing and output voltage. Finally, an eightconverter system presents a coordinative control strategy that optimizes the system’s efficiency by changing the number of operating converters. The ultimate benefit of the presented synergetic control 4749
[8] [9] [10] [11] [12] [13] [14]
[15] [16] [17] [18]
S. D. Sudhoff, D.H. Schmucker, R.A. Youngs, and H.J. Hegner, “Stability analysis of DC distribution system using admittance space constraints”, SAE International, 04 1998, pp.21-35. J.G. Ciezki and R.W. Ashton, “Selection and Stability issues associated with a navy shipboard DC zonal electric distribution system” IEEE Tr. on P. Del., Vol. 15, April 2000, pp. 665-669. S.D.Pekarek and al. “Overview of a Naval Combat Survivability Program”, Proceedings of the 13th Ship Control Systems Symposium (SCSS 2003), Orlando, Florida, April 7-9, 2003 A.H. Rivetta et al. “Analysis and Control of a Buck DC-DC Converter Operating With Constant Power Load in Sea and Undersea Vehicles”, IEEE Trans on Ind. Appl., vol.42, no2., 2006 S.Sanders, J.Noworolski, X.Z. Liu, and G.C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electron., vol.6,pp.251-258,Apr.1991. D.M.Mitchell,DC-DC Switching Regulator Analysis, New York McGraw-Hill,1988 S. K. Mazunder, A. H. Nayfeh, and D. Borojevic, “Robust Control of Parallel DC-DC Buck Converters by Combining IntegralVariable-Structure and Multiple-Sliding-Surfaces Control Schemes,” IEEE Trans. on P. E., vol. 17, pp.428-437, May 2002. C.K. Tse, Complex Behavior of Switching Power Converters, Publisher: CRC Press, 2003. J. G. Ciezki and R. W. Ashton ,”The Design of Stabilizing Controls for Shipboard DC-to-DC Buck Choppers Using Feedback Linearization Techniques,” Proceed. Of PESC’98, pp.335-341 B.Tomescu and et all “Improved large-signal performance of paralleled dc-dc converters current sharing using fuzzy logic control,” IEEE Trans. P. E., vol.14, pp. 573-577, May 1999. I. Kondratiev, R. Dougal, E. Santi, and G. Veselov, “Synergetic control DC-DC buck converters with constant power load,” in IEEE Power Electron. Specialist Conf Rec., 2004, pp.3758-3764. I. Kondratiev, R. Dougal, E. Santi, and G. Veselov, “Synergetic control for m parallel-connected DC-DC Buck converters,” in IEEE Power Electron. Specialist Conf Rec., 2004, pp. 182- 188. I. Kondratiev, “A Synergetic Control for Parallel-Connected DCDC Buck Converters,”Ph.D, dissertation, USC, Columbia, SC, May 2005. I. Kondratiev et al, “General Synergetic Control Strategies for Arbitrary Number of Paralleled Buck Converters Feeding Constant Power Load: Implementation of Dynamic Current Sharing,” IEEE Int. Symp. on Ind. El. (ISIE 06) pp.257-261, Canada, July 2006. R. Ericson, S. Cuk, and R. Middlebrook, “Large-scale modeling and analysis of switching regulators,” IEEE PESC Rec., 1982, pp. 240-250. A.A. Kolesnikov, Synergetic Control Theory, Moscow – Taganrog: TSURE, 1994 V.J Throttuvelil, and G.C. Verghese, “Analysis and Control Design of Paralleled DC/DC Converters with Current Sharing”, IEEE Trans. on Power Electron., Vol. 13, pp.635-644, 1998. M.M. Jovanovic et al. “A Novel, Low-Cost Implementation of “Democratic” Load-Current Sharing of Paralleled Converter Modules”, IEEE Trans. on P. E. Vol.11, pp.604-611, 1996.
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Synergetic Control Strategies for Shipboard DC Power Distribution Systems I. Kondratiev, Member, IEEE, and R. Dougal, Member, IEEE
Abstract— This paper presents a synergetic control design for a generalized component of modern ship DC distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design uses a state space averaged model of the system and overcomes the system’s nonlinearity, multi-connectivity, and high dynamic order, which are challenging features for control development. The design results in general analytical control strategies, conditions for counterweighing the system’s nonlinearity, and stability conditions for the closed-loop operation. The control introduces invariant manifolds into the state space of the system and suppresses errors in output voltage and current sharing. The coefficients of invariant manifolds allow us not only to counterweigh nonlinearity and allocate the type and ratings of current sharing for each converter, but also allow to change the number of operating converters. The control strategies are first applied to a two-converter system to demonstrate how the designed control dynamically allocates the current sharing within the system. Next, a two-converter system under synergetic control and feedback linearization control shows superiority of synergetic control performance. Finally, an eight-converter system presents a coordinative control strategy that optimizes the system’s efficiency by changing the number of operating converters.
U
I. INTRODUCTION
of DC power distribution systems that widely employ zonal structure and comprise paralleled converters has become a common practice in ship designs [1,2,3,4]. Among the numerous benefits such as a decrease of electrical and thermal stress in a component as well as overall system cost and weight, these DC distribution systems provide a capacity for nearly instantaneous reconfiguration [3]. Control design that fully utilizes this capacity could bring the ship power system operation to its highest possible invulnerability and efficiency. Unfortunately, such challenging properties of power systems as nonlinearity, multi-connectivity, and high-dimensionality prevent control design from exploiting reconfiguration fully and efficiently. However, a newly developed synergetic control theory provides the necessary tools for the design of analytical control capable of utilizing the reconfiguration capacity of these complex DC power systems at different control levels. A common element of modern DC zonal electrical
distribution systems is power converters connected in parallel and feeding service converter modules [3]. Control strategies for this element define reconfiguration features of the system. Ideally, at local control level these control strategies perform such technological tasks as maintaining allocated current sharing among power converters and stabilizing the output voltage. For the system control level, these control strategies deliver control channels that are able to change the allocation of current sharing and of the system structure to achieve such features of the system operation as highest efficiency or fastest response. Unfortunately, this element appears to be difficult not only for analysis of the system’s behavior but also for control design [1,2]. In practice, to simplify the study of the system, the influence of the service converter modules is approximated by constant power load nonlinearity. Such approximation reduces the dynamic order of the system, decouples the control design problem for the paralleled converter system and for the service converter modules [4]. The schematic of the simplified system based on paralleled buck converters is shown in Fig. 1. Different control design methods within the classical and advanced approaches have been applied to this system.
SE
This work was supported in part by ONR grants N00014-02-1-0623, N00014-03-1-0434. I.Kondratiev (ph: 803-777-8966; email:[email protected]) and R. Dougal (email:[email protected]) are with Electrical Engineering Department, University Of South Carolina, Columbia, SC 29208 USA
1-4244-0989-6/07/$25.00 ©2007 IEEE.
Fig. 1. System containing m parallel-connected buck converters
Traditionally, classical control theory attracts engineers due to the variety of control design approaches which were developed to use simple linear small-signal models [5,6]. These approaches design different control laws that stabilize the system; this includes proportional, integral-proportional etc. However, due to operating the system only in the vicinity of the operating point with a specified quality, these
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laws cannot control the system beyond the linear limits [7]. As a result, the inability of linear controls to predict dynamic behavior of constant power load nonlinearity could lead to chaotic behavior of the system and sometimes could result in a system collapse [8] because of the negative impedance instability [1]. On the other hand, the problem of control design for paralleled buck converters has been addressed by such advanced techniques as sliding control [7], feedback linearization control [9], fuzzy logic control [10], and synergetic control [11,12,13,14]. Almost all these approaches are focused on equal current sharing among system converters and do not consider system level optimization of the system behavior. In contrast, the results presented in [12,14] have addressed the problem of dynamic current sharing by introducing this capability of synergetic control on the basis of a two-buck converter system feeding resistive load and constant power load respectively. Synergetic control theory has made it possible to look at control design for the system at a more generalized level due to fully analytical procedure of control design. Our approach incorporates in control design not only channels for dynamic current sharing allocation but also control channels for changes of the system configuration. This procedure uses nonlinear model of the system and ensures global (or semiglobal) asymptotic stability of the closed-loop system. Moreover, synergetic control induces a self-organized motion in the system that takes control of the dynamic degrees of freedom of the system and as a result reduces the dynamic order of the controlled system and simplifies the analysis of the closed loop system behavior. The objectives of the research are to derive universal synergetic control laws that provide control channels for the dynamic allocation of types and ratings of current sharing among converters and control channels for a change of the number of converters operating in parallel; to study the closed loop stability of the system; and to evaluate the performance of the closed-loop system by simulation. This paper presents a design of generalized control strategies for the common element of DC zonal electric distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design results in general control strategies for the system, the condition for counterweighing the impact of constant power load, and stability conditions. Next, we apply the developed control strategies to three different systems. First on two-converter system, by varying coefficients of the macro-variables, we study the capability of the designed control for the dynamic allocation of type and rating of current sharing. We compare the performances of the two-converter system presented in [9] under synergetic control and feedback linearization control designed in [9]. Finally, we develop coordinative control that improves the efficiency of the eight-converter system by changing the number of operating converters.
The ultimate benefit of the research is that the designed control strategies enable more control of dynamic current sharing, improve the system efficiency, decrease stress on the system components, and promote simplification of control strategies. However, this digest covers only the most significant data such as the system model; general control strategies; general stability conditions; parameters of the used system; some simulation results, including impact of control law coefficients on system current sharing and transient response; performance comparison; and efficiency optimization. II. SYNERGETIC CONTROL DESIGN The system containing an arbitrary number of paralleled converters feeding constant power load is presented in Fig. 1. In model development, we limit the description level for each component by their lossless subset. Similarly, we limit the description level of the load by keeping only its capacitive, resistive, and constant power components, while sudden change of the system load is accounted for by disturbance M(t). Finally, we restrict the description level for the system model by using state space averaging technique [15] as a tool for model development. A. System Model The state space averaged model of the system presented in (1) is derived using the following assumptions: the system operates in a continuous conduction mode; switching occurs at a very-high switching frequency relative to the filter dynamics; parasitic effects are ignored; state variables are the averaged vc1 capacitor voltage and iLi inductance currents; constant power load is represented as dependent current source value of which equals to power consumed by the load (Pload) divided by output voltage (vc1); a piecewise constant current disturbance M(t) sufficiently approximates variations of the system load and any uncertainty of system parameters; the origin of the system model is at the specified steady state voltage(Vc1,ref); the zero order dynamics adequately represents the behavior of the sources Ei.
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⎧ Pload 1 ⎛ m ⎞ v + Vc1,ref M ⎜ ∑ iLj ⎟ − − + ⎪v ′ = ⎜ ⎟ ( ) C R C v V C C + j 1 = t ext t c ref t t 1 , ⎝ ⎠ ⎪ ⎪ 1 ⎪iL′ 1 = − ⋅ (v + Vc1,ref ) + u1 L1 ⎨ ⎪ ... ⎪ ⎪ 1 ′ =− ⋅ (v + Vc1,ref ) + u m ⎪iLm Lm ⎩ where v and iLi are state variables;
(1)
v = v c1 − Vc1,ref m
Ct = ∑ Ci − Cext is the total output capacitance of the system i =1
M(t) is a current disturbance influencing the system; dci is the desired switch duty of the ith converter;
ui = d ci ⋅ Ei / Li i = 1, m
In order to simplify control derivation results we
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represented system model (1) in the following form: ⎧⎪ X& = F1 I + F2 X + CF3 ( X ) ⎨& ⎪⎩ I = F4 ( X , I ) + U ( X , I )
(2)
where X=(M,v)T and I=(i1,…,im)T
B. Goals of the Control Design Introduced into the system model (1), disturbance M(t) represents the dependent current source, which allows the controller to take into account the change of the system load. Since in real life applications converter systems are usually influenced by sudden load change, the disturbance M(t) is approximated by piece-wise constant function (3). As a result of introduction of an additional dynamic coordinate M(t) into the system model (1), the system voltage becomes invariant to the system load resistance.
(v c 1 − V c 1 , ref dM = dt TI ⋅ Rd
)=
v TI ⋅ Rd
C. Control Synthesis Procedure From a mathematical standpoint, synergetic control design [16] is based on a new method for generating control laws, or feedback, that direct the system from arbitrary initial conditions into the vicinity of manifolds (4). The asymptotically stable motion along these manifolds toward the end attractors is then ensured by proper choice of coefficients in macro-variables. On these attractors the desired properties of the controlled system are guarantied. In short, the synergetic control design procedure is as follows. For system (1) the designer’s specifications are formulated as a set of macro-variables (5) based on the system state variables. The number of macro-variables in the set equals the number of control channels. ψ i (X, I ) = 0 (4) (5)
Assuming that the desired evolution of the macrovariables is as in (5) and substituting the system model (1) in functional equation (6) we get an asymptotically stable control, which ensures the desired dynamic properties. & + Ψ = 0 T ⋅Ψ T = E ⋅ (T 1 ...
Tm
)T
Ti > 0
E – is unity matrix dimension mxm
or Ψ = AI − B1 X − B 2 F3 ( X ) where
Ψ = (c1 ⋅ψ 1
(6)
...
c m ⋅ψ m )
T
ci={0,1}, i=1,m – is coefficient controlling the operation of ith converter
⎛ − a11 ⎛ a13 ... a1n ⎞ ⎟ ⎜ ⎜ A = ⎜ ... ... .. ⎟ B1 = ⎜ ... ⎟ ⎜− a ⎜a ⎝ m1 ⎝ m 3 ... amn ⎠
⎛ F3 ( X ) = ⎜ 0 ⎜ ⎝
(3)
The primary goal of the control is to maintain a specified output voltage Vc1,ref even while the system is affected by the disturbance M(t). Another important goal is to ensure proper sharing of current among the parallel-connected converters. At the design stage, we do not make any particular assumptions about current sharing except that the converter currents relate linearly to each other; that is, they each supply some constant fraction of the load current that correspond to active current sharing. All these control design goals were incorporated into the system model by the corresponding coordinate transformation. The resulting control laws must ensure asymptotic stability of the closed loop system.
ψ i = ψ i (X, I ), i = 1, m
Defining a macro-variable as shown in (7) and solving the system of functional equations (6) accounting for system model (2), yields control laws presented in (8). m ai ,n+1 ψ i = ai1 ⋅ M + ai 2 ⋅ v + ∑ ai , j+2iLj + ; i = 1, m ( v + V j =1 (7) c1,ref )Ct
1 (v + Vc1,ref
⎞ ⎟ ) ⎟⎠
− a12 ⎞ ⎛ 0 − a1n+1 ⎞ ⎟ ⎟ ⎜ ... ⎟ B2 = ⎜ ... ... ⎟ ⎜0 −a ⎟ − am 2 ⎟⎠ mn +1 ⎠ ⎝
T
U ( X , I ) = A−1 ⋅ ((B1 + B2 F4 )(F1I + F2 X + CF3 ( X )) − T −1Ψ ) + − F4 ( X , I )
(8)
where
⎛ 1 F4 = ⎜ 0 − ⎜ (v + Vc1,ref ⎝
)
2
⎞ ⎟ ⎟ ⎠
T
(9)
According to synergetic control theory [16], control laws (8) ensure asymptotically stable convergence towards the specified manifolds (4) within (3-5)Ti and, as shown later, results in current sharing specified by coefficients in (7). D. Behavior on Manifolds and Stability of the System It can be shown that under synergetic control the closed loop system has an attracting set. As a result, the representing point of the system reaches an attracting set, which is the intersection of manifolds. The study of the system behavior on the intersection of manifolds helps to understand system stability. The properties of the system motion on manifolds are defined by the following system of algebraic and dynamic equations: ⎧ X& = F1 I + F2 X + CF3 ( X ) ⎨ (10) Ψ=0 ⎩ or ⎧⎪ X& = F1 I + F2 X + CF3 ( X ) ⎨ ⎪⎩ I = − A−1 (B1 X + B2 F3 ( X ))
(11)
In order to define the properties of the system dynamics on manifolds, we eliminate dependent variables such as inductor currents and derive the following dynamic system: X& = (F2 − F1 A−1B1 )X + (C − F1 A−1B2 )F3 ( X ) (12) The equation (12) shows that the behavior of the closed loop system on manifolds is defined by two components: linear and nonlinear. It also can be seen that manifold coefficients provide flexibility in definition of both components. So, we define matrix B2 in such a way that the following conditions are true. C − F1 A −1 B 2 = 0 (13) Equation (13) represents the condition for counterweighing the nonlinear impact of a constant power load as well as for the linearization of the system (2). Accounting for condition (13), equation (12) can be rewritten as follows:
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FrA16.3 X& = (F2 − F1 A −1 B1 )X
(14) The equation (14) defines a linear dynamic system, properties of which depend on the choice of the coefficients of matrix A and B1. Coefficients of these matrixes need to be chosen to ensure stable motion of the system (14). Hence, synergetic control design results in analytical control laws that ensure asymptotic stability of the closed loop system and compensates for system nonlinearity by a particular choice of coefficients. The allocation of type and ratings of current sharing in the system is defined by coefficients of macro variables.
After the transients are passed, the representing point of the two-converter system inevitably reaches the intersection of the manifolds (4). Hence, the motion of the system on the manifolds is defined by the following equations: ⎛ a13 ⎜⎜ ⎝ a23
In this section we present a short description and some simulation results for three systems, which are used to study different features of the designed control strategies. First we study the properties of current sharing in a two-converter system. Next, we perform simulations of a two-converter system presented in [9] under synergetic control and feedback linearization control developed in [9]. Finally, for an eight-converter system we present a coordinative control strategy that optimizes the system’s performance by changing the number of converters operating in parallel.
TABLE II A FEW POSSIBLE SYNERGETIC CONTROL STRATEGIES FOR THE TWO-CONVERTER SYSTEM
TABLE I NOMINAL VALUE OF SYSTEM PARAMETERS
Resistance of inductors Filter inductance Filter capacitance Input voltage Output voltage Switching frequency Time constants Integrator coef. Law coef. Law coef. Load resistance Disturbance resistance Disturbance constant power
Symbol rL1=rL2=rLn L1=L2=Ln C1=C2=Cn E VC,ref=U0 T1=T2=Tn η= ηn a11=a11n a12=a12n Rext Rextd Pload
Control Strategy
Nominal Value 0.2 Ohm 6.1 mH 3000 μF 12 V 4.0 V 4.846 kHz 0.015 128 70 5 2 Ohm 2 Ohm 4W
A. Current Sharing in Two-Converter System To study the properties of current sharing we use a twoconverter system, the parameters of which are shown in Table I. In practice, master-slave [17], and democratic [18] current sharing strategies are widely used in paralleled converters systems. The major difference between masterslave and democratic current sharing is that master-slave sharing provides a fixed ratio of current sharing in transient and static regimes of operation, while democratic sharing has a variable ratio of current sharing during transients and a fixed ratio in the static regime of operation. This section shows important details of the impact of macro-variable coefficients on current sharing in the system and illustrates that synergetic control laws provide power systems with dynamic management of responsibilities, including the dynamic reconfiguration and the allocation of current sharing.
(15)
The study shows that: - coefficients a13, a14, a23, and a24 allocate the current sharing ratings, - coefficients a15, and a25 provide the ability to compensate the impact of the constant power load, - coefficient a22 defines the type of current sharing as shown in Table II, - coefficients a11 and a12 influence the system’s transient response as shown in Fig. 2 and Fig. 3 respectively, - time constants T1 and T2 the define speed of convergence of the macro-variables toward manifolds, - coefficients a12 and a22 define which converter is responsible for managing errors in the output voltage.
III. SYSTEM SIMULATION AND EXPERIMENTAL VERIFICATION
Parameter Name
a15 ⎛ ⎞ ⎜ − a M − a12 v − ⎟ a14 ⎞ ⎛ iL1 ⎞ ⎜ 11 ( v + Vc1, ref ) ⎟ ⎟⎟ ⋅ ⎜⎜ ⎟⎟ = ⎟ a25 a23 ⎠ ⎝ iL 2 ⎠ ⎜ − a M − a v − 22 ⎜ 21 ⎟ ( ) + v V c1, ref ⎠ ⎝
Manifold structure
Master-slave
⎛ a13 ⎜⎜ ⎝ a 23
Democratic
⎛ a 13 ⎜⎜ ⎝ a 23
a15 ⎛ ⎞ a14 ⎞ ⎜ − a M − a12 v − ⎟ ⎟⎟ ⋅ I = ⎜ 11 ( v + Vc1,ref ) ⎟ a 24 ⎠ ⎜ ⎟ 0 ⎝ ⎠ a 15 ⎞ ⎛ a 14 ⎞ ⎟ ⎜ − a M − a 12 v − ⎟⎟ ⋅ I = ⎜ 11 ( v + V c1, ref ) ⎟ a 24 ⎠ ⎟ ⎜ − a 22 v ⎠ ⎝
B. Performance Comparison As a base for comparison of synergetic control we use a two-converter system under feedback linearization control presented in [9]. Synergetic control parameters are tuned to satisfy the same design requirements as in [9]: equal current sharing, 750V output voltage, output voltage overshoot less than 6%, and the transient response are such that the settling time is less than 10 msec. The parameters of the twoconverter system are as follows: L1=1.35 mH, L2=1.25 mH, C1=2.6mF, C2=2.5mF, Vc1,ref=750V, and an input voltage are E1=E2=850V. The load is uncertain with the parameters falling in the following bounds: 2.81 Ω ≤ Rext ≤ 50 Ω; 5.11mF ≤ Ct ≤ 10mF; 0 kW ≤ Pload ≤ 200 kW. d = − A ⋅ (vc1 − Vc1.ref ) − B ⋅ (i L1 − i L 2 ) − D ⋅ idif − E ⋅ where
i dif vc21
− F ⋅M
(16)
idif = (iL1 + iL 2 ) − (io1 + io 2 )
Synergetic control strategies and feedback linearization control strategies are transformed into the same form as shown in (16).
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Corresponding coefficients of control laws (16) are presented in Table III.
a) output voltage error
nonzero values, the synergetic control law allows the system to nullify output voltage error through the introduced integral action. However, the value of F influences the speed of integration.
a) output voltage error a) Switch on transients of the output voltage transients during power up
b) inductor currents Fig. 2 Impact of a11 on transient response of the two-converter system (a11= 0.5a11n , a11n, and 2a11n)
b) inductor currents Fig. 3. Impact of a12 on transient response of the two-converter system (a12=0.5a12n, a12n, and 2a12n)
Next, the designed systems have been simulated using an averaged representation of the buck converters. The series resistance of each inductor is 0.5 mΩ. For each control, four simulations for extreme resistance and capacitance values have been performed. The combinations of load resistance and capacitance are as follows: Rext=2.81 Ω, Ct=5.11mF; Rext=50 Ω, Ct=5.11mF; Rext=2.81 Ω, Ct=10mF; Rext= 50 Ω, Ct=10mF. The system is initially energized with PLoad=0 kW and is allowed to operate in the steady state up to 0.09 s. At 0.09 s PLoad is stepped to 150 kW. At 0.12 s PLoad is stepped to 200 kW. At 0.15 s PLoad is stepped down to 0 kW. TABLE III CONTROL LAWS COEFFICIENTS Name
A B D E F
Coefficient Value Synergetic Converter 1 Converter 2 0.00608 0.00554 0.0002647 0.0002451 0.00224 0.00208 0.000454 0.0004207 0.0035 0.003244
Feedback Linearization Converter 1 Converter 2 0.00207 0.00124 0.000715 0.000662 0.00136 0.00126 34.26 31.72 0 0
b) Switch on transients of the output voltage when constant power load steps
c) Switch off transients of the output voltage when constant power load steps down from 100 % to 0 % Fig. 4. Simulation results for the closed loop two-converter system. (1- synergetic control, 2- feedback linearization control, and 3- intersection of transient responses)
C. Efficiency Optimization This section presents efficiency optimization for a power distribution system that contains eight paralleled subunits. Each subunit is a buck converter rated at 50 kW. The subunit parameters are presented in Table IV. The converter efficiency curve against power output is shown in Fig. 5. TABLE IV. BUCK CONVERTER PARAMETERS
It can be seen from Fig. 4 that the closed loop system with synergetic control shows solid, stable, and fast response and at the same time better accuracy of output voltage than the system with feedback linearization control. The solid, stable and fast response is a result of the dynamic order reduction of the closed loop system. In the feedback linearization case, the closed system is of the fourth order. The closed loop system with synergetic control law starts from the fourth order and, through the influence of the control, becomes a second order system. As a result, the closed loop system with synergetic control does not have significant delay during switching on and has a better transient response. In contrast to feedback linearization control under which errors of voltage regulation and of current sharing have finite
Parameter
Symbol
Value
Capacitance External capacitance Inductance Time constant Control Coef. Control Coef. Control Coef. Control Coef. Index
Ci
2.0mF
Cext
8.0mF
Li Ti ai1 ai2 ai3
0.7mH 0.0002 10 100 0.5 10000 1,..,8
ηi i
TABLE V. LOAD POWER THRESHOLD LEVELS Threshold number P1 P2 P3 P4 P5 P6 P7
Level, kW 45 84 120 150 180 210 240
Synergetic control parameters are tuned to ensure a stable operation of any number of paralleled converters (from 1 to 8), equal current sharing, and 200V output voltage. The control task is to optimize the system’s efficiency by proper choice of the coordinative control functions ci(Pouti)
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100
100
80
80
Efficiency
Efficiency
that put corresponding converter online if ci=1 and offline if ci=0. In other words, power levels thresholds Pouti (i=1,..,7) that optimize the system’s power efficiency are to be chosen to maximize the system efficiency. Due to flat part in efficiency curve in Fig. 5, the optimization task has multiple solutions. One possible solution is presented in Table V. Simulation results in Fig. 6 shows the efficiency of the zonal distribution system in both cases when the system operates without coordinative control and with coordinative control that uses switching threshold as shown in Table V. It can be seen in Fig. 6 that the coordination introduced into the system significantly improves the efficiency of the system operation under light loads.
60 40 20
design is that it provides scalable control strategies, stability conditions, and conditions for counterweighing the system nonlinearity for any number of paralleled converters. The use of the design will bring a new level of flexibility and reliability in DC zonal distribution systems. ACKNOWLEDGMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. REFERENCES [1] [2]
b)
60
a)
40
[3]
20
0 0
20 40 60 80 % of rated power
100
Fig. 5. Efficiency of the converter
[4]
0 0
200
400 600 Power, kW
800
Fig. 6. Efficiency of the zonal distribution system. ( a) without coordination b) with coordinative control)
[5] [6]
IV. CONCLUSION
[7]
This paper presents a synergetic control design for a generalized component of modern ship DC distribution systems such as an arbitrary number of paralleled buck converters feeding a constant power load. The design overcomes the system’s nonlinearity, multi-connectivity, and high dynamic order, which are challenging features for control development. The design results in general analytical control strategies, conditions for counterweighing the system’s nonlinearity, and stability conditions for the closedloop operation. The control introduces invariant manifolds into the state space of the system and suppresses errors in output voltage and current sharing. The coefficients of invariant manifolds allow us not only to counterweigh nonlinearity and allocate the type and ratings of current sharing for each converter, but also allow to change the number of operating converters. The paper illustrates the application of the designed strategies on different systems such as two- and eight- converter systems. The results show that synergetic control provides an extended flexibility for allocating types and ratings of current sharing dynamically. A comparison of performances of a two-converter system under synergetic control and feedback linearization control shows a superior performance of the synergetic control, which provides a faster response and nullifies errors of current sharing and output voltage. Finally, an eightconverter system presents a coordinative control strategy that optimizes the system’s efficiency by changing the number of operating converters. The ultimate benefit of the presented synergetic control 4749
[8] [9] [10] [11] [12] [13] [14]
[15] [16] [17] [18]
S. D. Sudhoff, D.H. Schmucker, R.A. Youngs, and H.J. Hegner, “Stability analysis of DC distribution system using admittance space constraints”, SAE International, 04 1998, pp.21-35. J.G. Ciezki and R.W. Ashton, “Selection and Stability issues associated with a navy shipboard DC zonal electric distribution system” IEEE Tr. on P. Del., Vol. 15, April 2000, pp. 665-669. S.D.Pekarek and al. “Overview of a Naval Combat Survivability Program”, Proceedings of the 13th Ship Control Systems Symposium (SCSS 2003), Orlando, Florida, April 7-9, 2003 A.H. Rivetta et al. “Analysis and Control of a Buck DC-DC Converter Operating With Constant Power Load in Sea and Undersea Vehicles”, IEEE Trans on Ind. Appl., vol.42, no2., 2006 S.Sanders, J.Noworolski, X.Z. Liu, and G.C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electron., vol.6,pp.251-258,Apr.1991. D.M.Mitchell,DC-DC Switching Regulator Analysis, New York McGraw-Hill,1988 S. K. Mazunder, A. H. Nayfeh, and D. Borojevic, “Robust Control of Parallel DC-DC Buck Converters by Combining IntegralVariable-Structure and Multiple-Sliding-Surfaces Control Schemes,” IEEE Trans. on P. E., vol. 17, pp.428-437, May 2002. C.K. Tse, Complex Behavior of Switching Power Converters, Publisher: CRC Press, 2003. J. G. Ciezki and R. W. Ashton ,”The Design of Stabilizing Controls for Shipboard DC-to-DC Buck Choppers Using Feedback Linearization Techniques,” Proceed. Of PESC’98, pp.335-341 B.Tomescu and et all “Improved large-signal performance of paralleled dc-dc converters current sharing using fuzzy logic control,” IEEE Trans. P. E., vol.14, pp. 573-577, May 1999. I. Kondratiev, R. Dougal, E. Santi, and G. Veselov, “Synergetic control DC-DC buck converters with constant power load,” in IEEE Power Electron. Specialist Conf Rec., 2004, pp.3758-3764. I. Kondratiev, R. Dougal, E. Santi, and G. Veselov, “Synergetic control for m parallel-connected DC-DC Buck converters,” in IEEE Power Electron. Specialist Conf Rec., 2004, pp. 182- 188. I. Kondratiev, “A Synergetic Control for Parallel-Connected DCDC Buck Converters,”Ph.D, dissertation, USC, Columbia, SC, May 2005. I. Kondratiev et al, “General Synergetic Control Strategies for Arbitrary Number of Paralleled Buck Converters Feeding Constant Power Load: Implementation of Dynamic Current Sharing,” IEEE Int. Symp. on Ind. El. (ISIE 06) pp.257-261, Canada, July 2006. R. Ericson, S. Cuk, and R. Middlebrook, “Large-scale modeling and analysis of switching regulators,” IEEE PESC Rec., 1982, pp. 240-250. A.A. Kolesnikov, Synergetic Control Theory, Moscow – Taganrog: TSURE, 1994 V.J Throttuvelil, and G.C. Verghese, “Analysis and Control Design of Paralleled DC/DC Converters with Current Sharing”, IEEE Trans. on Power Electron., Vol. 13, pp.635-644, 1998. M.M. Jovanovic et al. “A Novel, Low-Cost Implementation of “Democratic” Load-Current Sharing of Paralleled Converter Modules”, IEEE Trans. on P. E. Vol.11, pp.604-611, 1996.
