Robust Nonlinear Synergetic Control for m-Parallel-Connected DC ...
Robust Nonlinear Synergetic Control for m-Parallel-Connected DC-DC Boost Converters Igor Kondratiev, Enrico Santi and Roger Dougal University of South Carolina Electrical Engineering Department, Columbia, USA
Abstract—This paper presents synergetic control design for an m-paralleled boost converter system under active current sharing. The presented design overcomes such problems of the system as multi-connectivity, nonlinearity, and high dimensionality. The set of macro-variables introduced during control design defines a set of invariant manifolds that accommodate different current sharing regimes such as masterslave and democratic current sharing simply by changing coefficients in the macro-variables. Invariant manifold formation contracts the system state space and makes it possible to perform stability analysis for an arbitrary number of paralleled boost converters. The stability analysis and current sharing analysis presented in this paper provide criteria for the choice of control law coefficients. The simulation results for a two-converter system validate the theory and show that the closed-loop system is characterized by stable operation, fast response, good voltage regulation, ability to nullify steady state error not only of output voltage but also of current sharing, capability to control the current drawn from each converter, ability to change current sharing during operation, and, most importantly, capability to withstand variation of the system parameters by400%. Index Terms — DC converters, synergetic control, nonlinear control, robust control.
I. I NTRODUCTION The fast growing interest in control strategies for parallel-connected converters has been caused by several factors such as growing consumer power demands, the increasing importance of dynamic power management, growing requirements for system reliability, and a design to decrease overall system cost [1]. However, in spite of the apparent simplicity of DC systems, the design of efficient control for autonomous DC power system is made difficult by nonlinearity which may result in chaotic behavior [2], high dimensionality, and multiconnectivity of the controlled system. As a result, most classical control design techniques are not capable of efficient control design for these systems. Such techniques as feedback linearization [3], sliding [4], and synergetic control [5] have addressed the above-mentioned problems for DC system control design using active current sharing [6], [7] for parallel-connected buck converter systems. Even though systems containing parallel-connected boost converters with active current sharing [8] can complement the operation of the buck converters when a boost of the source voltage is needed, highly-nonlinear behavior, multi-connectivity, and non-minimal phase properties of the system represent a particular challenge for control design. As a result, systems with parallel-connected boost converters are limited in use and serve as examples to illustrate nonlinear behavior [9]. However, the application
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of the synergetic control theory overcomes these challenging properties of the system and ensures asymptotic stability of the closed-loop system, robustness against parameter variation, dynamic regulation of current sharing characteristics within the system, nullifies errors of current sharing and voltage regulation and therefore can help the designer to extend the range of the system utilization. In this paper we consider a general case of an mparalleled boost converter system feeding a resistive load. For control design, we use synergetic control theory [10] which is based on ideas of self-organization. The theory allows designers to derive analytical control laws for nonlinear, high-dimensional, and multi-connected systems. The paper presents a derivation of the nonlinear PI control algorithms for m parallel-connected DC boost converters under active current sharing that nullifies steady state error of current sharing and of voltage regulation (for details see section III), stability assessment of the closed loop system (III-C1), study of robustness of the control algorithm against parameter variation (IV-A1), criteria for control coefficients selection (III-C), and illustration of the capabilities of the designed control for dynamic power sharing (III-C2,IV-A2). The design is applied to a twoconverter system to illustrate particular properties of the closed-loop system such as robustness against parameter variation (IV-A1), and dynamics of the current sharing and of the voltage response (IV-A2). In the paper, we use a two-converter system as an example to compare theoretical and simulation results. The comparison of theory and simulation shows a good agreement (III-C2,IV-A2). II. C ONCEPT OF S YNERGETIC C ONTROL T HEORY Synergetic control theory [11] is based on the ideas of modern mathematics, optimal control, and synergetics and exploits the capability of open systems to self-organize. Synergetic control introduces a holistic philosophy of the dynamically controlled interaction of energy, matter, and information within a system by generating positive and negative feedback. The philosophy of control design is based on the principle of dynamic expansioncontraction of the system state space. The expansion of the system state space enriches system dynamics by providing additional differential equations that describe the interaction of the system with the environment (for example load disturbances), which is important from the designers’ perspective. The purpose of the control is to eliminate unwanted dynamics of the system or to reduce excessive degrees of freedom. At the control design stage,
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the unwanted dynamics is eliminated in accordance with the introduced dynamic constrains which are presented as invariant manifolds in the state space of the system. A. Main Equation of Synergetic Control Theory For a nonlinear system, the model of which is represented in (1), synergetic control theory provides design procedure that is based on a new methodology for generating control u(ψ) = u(x) or feedback as a function of the specified macro-variables ψ(x) which operate as parameters of order in synergetic systems and are also called aggregated variables. x˙ i (t) = fi (x1 , ..., xn ) + ai+1 xi+1 , i = 1, n − 1 x˙ n (t) = fn (x1 , ..., xn ) + u
(1)
where: xi are the system state variables; fi (x1 , ..., xn ) are nonlinear functions defining the dynamic evolution of the system; u is an input variable. For solving a control design problem, the theory uses the following functional, which is now called Kolesnikov’s Functional [12]: Z ∞ Z ∞ ˙ JΣ = F (ψ, ψ)dt = (m2 φ2 (ψ)+c2 ψ˙ 2 (t))dt (2) 0
0
˙ is a continuous positive definite function; where: F (ψ, ψ) ψ(x1 , ...xn ) is an aggregated variable or a macro-variable that represents an arbitrary continuous (or piece wise continuous) function of the state vector x, and φ is a smooth function differentiable with respect to its arguments. Moreover, function φ(ψ) is chosen to be a) differentiable and invertible; b) φ(0) = 0; and c) φ(ψ)ψ > 0 ∀ ψ 6= 0. The resulting synergetic control (3) is found as a joint solution of a so-called functional equation (4) and system model (1). · ¸ ∂ψ −1 ∂ψ 1 n u=− Σk=1 (fk + ai+1 xi+1 ) + φ(ψ) ∂xn ∂xk T (3) ˙ + φ(ψ) = 0 T ψ(t) (4) where: an+1 = 0, T = c/m = 1/λ; moreover, from the synergetics point of view, λ is a parameter characterizing the job performed by the self-organizing forces. Equation (4) defines the evolution of the system’s macrovariable ψ(x) into manifold (5). ψ(x) = 0
(5)
Under control (3), manifold (5) becomes an attracting set (or attractor) in the state space of the closed-loop system. As a result, from an admissible arbitrary initial location in the state space, the system moves into a vicinity of the manifold and along the manifold to the equilibrium point. The stability of these two motions is analyzed independently. The stability of the system motion towards the manifold can be determined using Lyapunov function (6), a derivative of which (7) defines the basin of attraction of the attractor ψ(x) = 0 introduced into the system. V = 0.5ψ 2 > 0
(6)
1 V˙ (t) = − φ(ψ)ψ < 0 if T > 0 T
(7)
Equation (7) shows that the stability of the closed-loop system motion towards the attractor is not compromised by nonlinear properties of the system model (1) and the system motion is stable as long as the parameter T is greater than zero. Once the system reaches the vicinity of the invariant manifold ψ(t) = 0, synergetic control law (3) will keep it on the manifold. As a result, in the vicinity of the manifold ψ(x) = 0, the system behavior is described by the reduced order model (8) which is obtained by substituting a manifold equation (5) into the system (1) instead of a dynamic equation with control (the last one in our case). x˙ i (t) = fi (x1 , ..., xn ) + ai+1 xi+1 , i = 1, n − 1 ψ(x) = 0
(8)
It can be seen from (8), the application of synergetic control decreases the order of the system by 1; and the order equals to n − 1. Thus, the stability analysis of the closed loop system can be significantly simplified by a sequential application of the control design process. To continue the application of the design, it is assumed that now a state variable xn becomes an internal control. With similar assumptions, the design process can be recursively repeated for a required number of times. Moreover, the direct path of the recursion gives a set of decomposed models and the return path of the recursion defines the control and manifold structures. As a result of the synergetic recursive design for one control channel system, one invariant manifold is created in the system state space. Under synergetic control the system goes into a vicinity of the manifold, first, and along this manifold to the specified equilibrium state. Moreover, the control ensures asymptotically stable motion towards and along the manifold to the desired equilibrium state. As long as the system remains on this attractor, the specified properties of the controlled system are guaranteed. Without loss of generality the same approach can be applied for control design for multi-control channel systems. III. S YNERGETIC C ONTROL D ESIGN A. System Model A prototype system for this research is presented in Fig.1. The state space averaged model of the system presented in (9) is derived using the following assumptions: the system operates in a continuous conduction mode, switching occurs at a very-high switching frequency, parasitic effects are ignored, state variables are the averaged vc1 capacitor voltage, iLi inductor currents. Disturbance M (t) in the model (9) takes into account the impact of the load and parameters’ changes on the system. In addition, coordinate transformation was applied to the model (9). This transformation moved the system voltage equilibrium point of the resulting system (9) into the origin of the coordinate (v = uc − v0 ) and substituted duty ratio with
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Converter 1
i1
where: £ ¤T I = i1 ... im £ ¤T X = v M a1,1 ... a1,m ... ... A1 = ... a ... am,m · m,1 ¸T a1,m+1 ... am,m+1 A2 = a1,m+2 ... am,m+2
Load
d1
L1 E1
uc C
Rext
M(t)
Converter 2 Converter m
im
dm
Lm
ai,j , i, j=1, m : ∀γ 6= 0
Em
m X
λk γk 6= 0
k=1
Fig. 1.
where: λl =
System of m paralleled boost converters with resistive load
control ui = (1 − di ) for every converter. m dv 1 X v+v0 M = uj ij − − dt C R C ext C j=1 di1 E1 (v+v0 ) u1 dt = L − L 1 1 ... dim Em (v+v0 ) = − um dt Lm Lm dM =ηv dt B. Control Design
ψi (i1 , i2 , ..., im , v) = 0 Ti ψ˙ i + ψi = 0, Ti > 0, i = 1, m
(9)
m X
ai,j ij +ai,m+1 v+ai,m+2 M (t) ; i = i, m
j=1
Ψ = A1 I + A2 X
al,n in ; l=1, m
For control design we select macro-variables of general form as shown in (13) where coefficients ai,j are selected to ensure linear independence of vectors λl in (15), which provides necessary and sufficient conditions for existence of the manifolds intersection. Solving jointly the system of the functional equations (12) with the system model (9) and with the set of the macro-variables (13), the synergetic control for the system (9) is found as shown in (16). U = −Z −1 [ΛΨ − A1 E − A3 (v, M )] ,
(16)
where: det Z 6= 0 T
U = (u1 , u2 , ..., um ) z1,1 ... z1,m ... ... Z = ... zm,1 ... zm,m a zi,j |i6=j = i,m+1 ij C zi,j |i=j =
(10) (11)
(13)
ai,m+1 ij C
·
+
(v+v0 ) Li
m P
ai,k
k=1
¸T E1 Em T E= , ..., A3 = [a31 , ..., a3m ] L1 Lm µ ¶ ν+v0 M (t) a3i =ai,m+1 − + −ai,m+2 η · v Rext C C i, j = 1, m. (17)
(12)
From a mathematical standpoint, synergetic control design [10] is based on a new method for generating control laws ψi = ψi (i1 , i2 , ..., im , v) or feedbacks that direct the system from arbitrary initial conditions into a vicinity of manifolds ψi (i1 , i2 , ..., im , v) = 0 and then ensure asymptotically stable motion along these manifolds toward the end attractors. On these attractors the desired properties of the controlled system are guaranteed. In short, the synergetic control design procedure is as follows. For system (9) the designer’s specifications are formulated as a set of macro-variables (10) based on the system state variables. The number of macro-variables in the set equals the number of control channels. Controls are found as a joint solution of the system of functional equations (12) and system model (9). ψi =
(15)
n=1
Control design uses synergetic control theory [10] and implements active current sharing [8] under which power distribution among converters is based on adjusting of the reference voltage [7]. ψi = ψi (i1 , i2 , ..., im , v) ; i = 1, m
m X
(14)
As a result of synergetic control design procedure, we get a general nonlinear PI control law. As it follows from the synergetic control theory, for any Ti > 0 and as long as vectors λi satisfy conditions for linear independency (15) for any i = 1, m, synergetic control law (16) moves the system representing point, which describes the system motion in its state space, onto the intersection of the manifold (11), where the system becomes of reduced order. C. Closed-loop Behavior There are two important components in the behavior of a closed-loop system: the stability of its behavior and the way parameters of the control law are to be tuned to satisfy designers specifications.
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1) Stability: The closed-loop system goes first into the vicinity of the manifolds and along the manifolds towards the equilibrium point. The stability analysis for these two parts of the system motion is done separately. a) Motion Towards Manifolds: We construct Lyapunov function (18) using (6). Equation (19) shows the derivative of the Lyapunov function (18) and summarizes the stability conditions. X ψi V = 0.5ψi2 > 0 and Ti ψ˙i = − (18) Ti V˙ = −
m X ψ2 i
i=1
Ti
< 0 if Ti > 0
i=1: ψ1 = i1 + a1,m+1 v + a1,m+2 M (t) (v + v0 ) ai,m+1 z1,1 = − + i1 Li C ψi = i1 − ii Ei = E1 ; Li = L1 ai,m+1 z1,i = ii C (v + v0 ) (v + v0 ) zi,1 = − ; zi,i = L1 Li zi,j = 0 ; ∀j = 2, m ; j 6= i
(20)
Moreover, since all ψi = 0 and inductances Li are equal as well as feeding voltages Ei , it can be shown that on manifolds controls and currents are equal as shown in (21). uiψ = u1ψ , iiψ = i1ψ ; i = 2, m
(21)
Using all the above (20,21) simplification we get the following expression for the control that directs the closedloop system on the manifold intersection towards the equilibrium:
u1ψ = ³
a1,m+1 C
Mψ vψ +v0 C − Rext C
where: f (vψ , Mψ ) =
d2 vψ dvψ + b1 + b2 vψ = 0, dt2 dt where: mE1 a21,m+1 mL1 a31,m+1 + Rext C v0 C Rext C 2 2 ηmL1 a1,m+1 a1,m+2 ηmE1 a1,m+1 a1,m+2 η b2 = − − . Rext C 2 v0 C C (24)
(a1,m+1 vψ +a1,m+2 Mψ ) m+
(vψ +v0 ) L1
´.
(22)
1
b1 =
−
In accordance with Hurwitz stability criterion [13], the system (24) is asymptotically stable if coefficients b1 and b2 are greater than zero as shown in (25). mE1 a21,m+1 mL1 a31,m+1 + >0 Rext C v0 C Rext C 2 2 ηmL1 a1,m+1 a1,m+3 ηmE1 a1,m+1 a1,m+2 η b2 = − − >0 Rext C 2 v0 C C (25)
1
−
Conditions shown in (25) represent limiting constrains for selecting control law coefficients to ensure a stable motion of the closed-loop system along the intersection of manifolds. 2) Current Sharing: Above, to simplify analysis, we considered the case with equal current sharing and masterslave operation, and selected macro-variable coefficients accordingly. However, in this section, on an example of a two-converter system, we examine how the selection of coefficients in macro-variables influences the current sharing in the system. We also show that synergetic control laws provide power systems with a dynamic management of responsibilities, including dynamic reconfiguration and allocation of current sharing. For a two-converter system the general macro-variable structure is as follows: ψ1 = a1,1 i1 + a1,2 i2 + a1,3 v + a1,4 M ψ2 = a2,1 i1 + a2,2 i2 + a2,3 v + a2,4 M
´ + a1,m+2 ηvψ
(vψ +v0 ) u1ψ L1
Linearization of (23) in the vicinity of the equilibrium point where v and M approximately equal to zero gives us the following second order system:
b1 =
i = 2, m :
E1 L1 +a1,m+1
E 1 +a1,m+2 η · vψ −f (vψ , Mψ )+ dvψ L1 µ ¶ = Mψ (t) vψ +v0 dt (1+ai,m+1 ) − C Rext C (23) dMψ (t) = η · vψ dt
(19)
It can be seen from (19) that the motion of the closedloop system towards manifolds (10) is stable as long as the time constants Ti are greater than zero. b) Motion Along the Manifolds: During the second part of the motion, the system is moving along the intersection of manifolds (10). The resulting motion can be described by a reduced order dynamic model as shown in (23). To simplify the derivation of the reduced order model for an m-paralleled boost converter system, we consider the most common case of the system operation, when paralleled converters are fed from the same source, inductors have the same values of their inductances, and an equal current sharing among converters is enforced. All these assumptions lead us to the following simplifications:
³
The resulting closed-loop behavior is defined by the equation (23) where control u1ψ is as shown in (22).
(26)
After the transients are finished, the representing point of the two-converter system inevitably reaches the intersection of the manifolds (27). The equations (27) define
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the steady state part of the closed-loop system motion along the manifolds towards the equilibrium point. ψ1 = a1,1 i1ψ + a1,2 i2ψ + a1,3 vψ + a1,4 Mψ = 0 ψ2 = a2,1 i1ψ + a2,2 i2ψ + a2,3 vψ + a2,4 Mψ = 0
(27)
Thus, the system currents that we can calculate from (27) are functions of the output voltage error v and an integral of the voltage error M as shown in the following: a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = −a2,3 vψ − a2,4 Mψ or A · I = B · X.
(a) varying inductance
(28)
Moreover, the coefficients of matrices A and B in (28) define the properties of current sharing between the two converters on the manifolds. In systems with two parallel-connected converters, the only possible basic current sharing regimes are masterslave and democratic. When the system is on the manifolds, either of these can be achieved by a particular choice of the coefficients in (28). In particular, master-slave current sharing occurs when the master converter maintains the output voltage and the slave converter supplies current according to the rating assigned to it [6], which is usually defined for the slave converter by the master. On the other hand, democratic current sharing occurs when each of the two converters provides current in proportion to its current rating (21) and each converter independently accounts for the output voltage error [19]. a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = 0
(29)
a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = −a2,3 vψ
(30)
Equations (29) and (30) show the manifolds’ structure for master-slave and democratic current sharing respectively, and give important insights not only for the current sharing properties but also for the responsibility of each converter. For example, coefficients a11 , a12 , a21 , and a22 define what fraction of the current is allocated to each of the converters, and if a2,1 = −a2,2 equal current sharing is assigned between the converters. Moreover, coefficients a13 and a23 define which converter is responsible for managing errors of the output voltage. Equations (29) and (30) demonstrate that the difference between mater-slave and democratic current sharing corresponds to coefficient a23 . In the case of democratic sharing, this coefficient is not zero; hence, during the transition to the equilibrium state, coefficient a23 influences the rate of change of converters’ currents. Since the output voltage error equals zero at equilibrium, it does not affect static current sharing in the system.
(b) varying capacitance
Fig. 2. Output voltage error transients under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
(a) varying inductance
(b) varying capacitance
Fig. 3. State portraits (i1 ,v plain) of the system under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
IV. S IMULATION R ESULTS AND D ISCUSSION For simulation of a two-converter system, parameters of which are presented in (31), we performed control design using the algorithm proposed in (16). The resulting control laws can be easily reconstructed using (16) and are not presented here to save space. The control law parameters are presented in (32). L1 = L2 = Ln = 0.001H C = Cn = 0.001F E1 = E2 = En = 12V v0 = 15V Rext = 10Ω
(31)
a1,1 = 1 a1,2 = 1 a1,3 = 5 a1,4 = 10 a2,1 = 1 a2,2 = −1 a2,3 = 0 a2,4 = 0 T1 = T2 = Tn = 0.001s
(32)
A. Simulation Results Using Maple, we performed a number of simulations, in which we studied the ability of the control to withstand system parameters’ variation and flexibility of current sharing. 1) Robustness Against Parameter Variation: In this simulations we varied independently by two times each of the following parameters: inductance L1 = (0.5, 1, 2)Ln and capacitance C = (0.5, 1, 2)Cn , while the rest of the parameters were kept constant. Simulation results are presented in Figs. 2-5 which accordingly are output voltage transients, state portraits in state plain (i1 ,v), inductor current transients, and state portraits in state plain (i1 ,i2 ).
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(a) varying inductance
(b) varying capacitance
Fig. 4. Inductor current transients under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
(a) varying inductance
(b) varying capacitance
Fig. 5. State portraits (i1 ,i2 plain) of the system under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
a) Voltage and Current Transients: It can be seen from figures (2, 3 and 4) that on the one hand, the variation of the converter inductance does not impact the closedloop system voltage and current transients. On the other hand, the variation of the load capacitance directly and proportionally influences the settling time of the voltage and current transients. b) Current Sharing: The state portraits in state plane (i1 ,i2 ) presented in Fig.6 illustrate the current sharing between two converters. As it was predicted by the theory, the figure shows that the system’s representing point first goes into vicinity of the manifold i1 = i2 and along this manifold goes to the equilibrium point. The variation of the inductance results in insignificant impact on current sharing, and the capacitance variation does not affect the current sharing specified during the control design. 2) Current Sharing: State portraits in state plain (i1 ,i2 ) presented in Fig.6 illustrate operation of different current sharing strategies. Fig.6(a) shows master-slave current sharing with different current distribution among converter: case (1) illustrates equal current sharing (dash line) and in case (2) the current of the first converter i1 is twice the current of the second converter i2 (solid line). Fig.6(b) presents democratic current sharing. As it was predicted in section III-C2, in democratic current sharing the current sharing ratings during the transition depend on the output voltage error, and in steady state operation it becomes equal as specified in control design in our case. Fig.6(c) shows the state portrait for the system
(a) state portrait of the system (b) state portrait of the system under master-slave current sharing under democratic current sharing with different current ratings (MS1 (i1 = i2 , a23 = 3) i1 = i2 and MS2 i1 = 2i2 )
(c) state portrait of the system un- (d) transient behavior of the output der dynamic current sharing (a21 = voltage under different current shar0, a22 = 1, a23 = 3) ing Fig. 6. Transient behavior of the closed-loop system under different current sharing strategies
with dynamic current sharing. It can be seen that current withdrawn from the second converter in steady state equals zero. However, during transients in the system the second converter supplies an additional amount of current to the load. Thus, the second converter behaves as a capacitor by supplying the energy to the system load dynamically. Fig.6(d) presents voltage transients of the closed-loop system under the above mentioned four different current sharing strategies. As it can be seen from the figure, the deviation of the voltage transient responses from each other is insignificant. Thus, as it was predicted in section III-C2, the structure of macro-variables defines the properties of current sharing in the system. For example, structure of macro-variables presented in (29) results in master-slave current sharing and structure (30) leads to democratic current sharing. In particular, coefficients a11 , a12 , a21 , and a22 define what fraction of the current is allocated to each of the converters, and coefficients a13 and a23 define which converter is responsible for managing the error of the output voltage. During transients, coefficient a23 influences the rate of change of the converters’ currents. Thus, synergetic control proposed in this paper provides flexible means for selecting and adjusting energy flow within the system. B. Discussion As it was predicted by the theory, the simulations show that the system’s representing point first goes into vicinity
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of the manifolds intersection and along this intersection goes to the equilibrium point. Moreover, the manifold created in the closed-loop system state space shows very robust properties under 400% variation of the system parameters. Thus, synergetic control designed in the paper provides a good insensitivity for the closed-loop system to the converters parameters’ variations. Moreover, control law coefficients provide extraordinary flexibility not only for the allocation of current sharing and tuning the system transients but also for selecting the type of current sharing properties and responsibilities of each converter. The resulting flexibility in current sharing provided by synergetic control laws can benefit reconfigurable power systems by allocating current sharing and managing responsibilities dynamically. V. C ONCLUSION This paper presented synergetic control design for an m-paralleled boost converter system under active current sharing. The presented design overcame such problems of the system as multi-connectivity, nonlinearity, and high dimensionality. Multi-connectivity and nonlinearity of the system were taken into account by the presented set of manifolds, which define laws of interaction among converters. Moreover, the introduced set of macro-variables accommodates different current sharing regimes such as master-slave and democratic current sharing simply by changing coefficients in the macro-variables. Contraction of the system state space under synergetic control made it possible to draw a stability analysis for an arbitrary number of paralleled boost converters. The stability analysis and current sharing analysis presented in this paper explained the choice of control law coefficients. The simulation results for a two-converter system validated the theory and showed that the closed loop systems are characterized by stable operation; fast response; good voltage regulation; ability to nullify steady state error not only of output voltage but also of current sharing; capability to control the current withdrawn from each converter; and ability to change current sharing during operation, and more importantly, capability to withstand the variation of the system parameters by 400%.
[5] A. Kolesnikov, G. Veselov, A. Monti, F. Ponci, E. Santi, and R. Dougal, “Synergetic synthesis of dc-dc boost converter controllers: theory and experimental analysis,” in Applied Power Electronics Conference and Exposition, 2002. APEC 2002. Seventeenth Annual IEEE, vol. 1, 2002, pp. 409–415 vol.1. [6] J. Rajagopalan, K. Xing, Y. Guo, F. Lee, and B. Manners, “Modeling and dynamic analysis of paralleled dc/dc converters with master-slave current sharing control,” in Applied Power Electronics Conference and Exposition, 1996. APEC ’96. Conference Proceedings 1996., Eleventh Annual, vol. 2, 1996, pp. 678–684 vol.2. [7] V. Thottuvelil and G. Verghese, “Stability analysis of paralleled dc/dc converters with active current sharing,” in Power Electronics Specialists Conference, 1996. PESC ’96 Record., 27th Annual IEEE, vol. 2, 1996, pp. 1080–1086 vol.2. [8] I. Batarseh, K. Siri, and H. Lee, “Investigation of the output droop characteristics of parallel-connnected dc-dc converters,” in Power Electronics Specialists Conference, PESC ’94 Record., 25th Annual IEEE, 1994, pp. 1342–1351 vol.2. [9] M. di Bernardo and F. Vasca, “Discrete-time maps for the analysis of bifurcations and chaos in dc/dc converters,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits and Systems I: Regular Papers, IEEE Transactions on], vol. 47, pp. 130–143, 2000. [10] ed. by A.A. Kolesnikov, Sovremennaya Prikladnaya Teoria Uprablenia: Chasty I Sinergeticheskii podhod v teorii upravlenia. Moscow - Taganrog: TRTU, 2000, vol. 2. [11] A. Kolesnikov, Synergetic Control Theory, 1st ed. Moscow– Taganrog: TSURE, 1994, vol. 1. [12] G. Kondratiev, Geometrical Synthesis Theory Of Optimal Stationary Smooth Control Systems. Moscow: Fizmatlit, 2003. [13] R. Clark, “The Routh-Hurwitz stability criterion, revisited,” Control Systems Magazine, IEEE, vol. 12, pp. 119–120, 1992.
ACKNOWLEDGEMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. R EFERENCES [1] J. G. Ciezki and R. W. Ashton, “Selection and stability issues associated with a navy shipboard dc zonal distribution system,” IEEE Tr. on Power Delivery, vol. 15, pp. 665–669, April 2000. [2] C. Tse, Complex Behavior of Switching Power Converters, 1st ed. CRC Press, 2003, vol. 1. [3] J. G. Ciezki and R. W. Ashton, “The design of stabilizing controls for shipboard dc-to-dc buck choppers using feedback linearization techniques,” in Proc. PESC’98, pp. 335–341, 1998. [4] S. K. Mazunder, A. H. Nayfeh, and D. Borojevic, “Robust control of parallel dc-dc buck converters by combining integral-variablestructure and multiple-sliding-surfaces control schemes,” IEEE Tr. on Power Electron., vol. 17, pp. 428–437, May 2002.
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Abstract—This paper presents synergetic control design for an m-paralleled boost converter system under active current sharing. The presented design overcomes such problems of the system as multi-connectivity, nonlinearity, and high dimensionality. The set of macro-variables introduced during control design defines a set of invariant manifolds that accommodate different current sharing regimes such as masterslave and democratic current sharing simply by changing coefficients in the macro-variables. Invariant manifold formation contracts the system state space and makes it possible to perform stability analysis for an arbitrary number of paralleled boost converters. The stability analysis and current sharing analysis presented in this paper provide criteria for the choice of control law coefficients. The simulation results for a two-converter system validate the theory and show that the closed-loop system is characterized by stable operation, fast response, good voltage regulation, ability to nullify steady state error not only of output voltage but also of current sharing, capability to control the current drawn from each converter, ability to change current sharing during operation, and, most importantly, capability to withstand variation of the system parameters by400%. Index Terms — DC converters, synergetic control, nonlinear control, robust control.
I. I NTRODUCTION The fast growing interest in control strategies for parallel-connected converters has been caused by several factors such as growing consumer power demands, the increasing importance of dynamic power management, growing requirements for system reliability, and a design to decrease overall system cost [1]. However, in spite of the apparent simplicity of DC systems, the design of efficient control for autonomous DC power system is made difficult by nonlinearity which may result in chaotic behavior [2], high dimensionality, and multiconnectivity of the controlled system. As a result, most classical control design techniques are not capable of efficient control design for these systems. Such techniques as feedback linearization [3], sliding [4], and synergetic control [5] have addressed the above-mentioned problems for DC system control design using active current sharing [6], [7] for parallel-connected buck converter systems. Even though systems containing parallel-connected boost converters with active current sharing [8] can complement the operation of the buck converters when a boost of the source voltage is needed, highly-nonlinear behavior, multi-connectivity, and non-minimal phase properties of the system represent a particular challenge for control design. As a result, systems with parallel-connected boost converters are limited in use and serve as examples to illustrate nonlinear behavior [9]. However, the application
978-1-4244-1668-4/08/$25.00 ©2008 IEEE
of the synergetic control theory overcomes these challenging properties of the system and ensures asymptotic stability of the closed-loop system, robustness against parameter variation, dynamic regulation of current sharing characteristics within the system, nullifies errors of current sharing and voltage regulation and therefore can help the designer to extend the range of the system utilization. In this paper we consider a general case of an mparalleled boost converter system feeding a resistive load. For control design, we use synergetic control theory [10] which is based on ideas of self-organization. The theory allows designers to derive analytical control laws for nonlinear, high-dimensional, and multi-connected systems. The paper presents a derivation of the nonlinear PI control algorithms for m parallel-connected DC boost converters under active current sharing that nullifies steady state error of current sharing and of voltage regulation (for details see section III), stability assessment of the closed loop system (III-C1), study of robustness of the control algorithm against parameter variation (IV-A1), criteria for control coefficients selection (III-C), and illustration of the capabilities of the designed control for dynamic power sharing (III-C2,IV-A2). The design is applied to a twoconverter system to illustrate particular properties of the closed-loop system such as robustness against parameter variation (IV-A1), and dynamics of the current sharing and of the voltage response (IV-A2). In the paper, we use a two-converter system as an example to compare theoretical and simulation results. The comparison of theory and simulation shows a good agreement (III-C2,IV-A2). II. C ONCEPT OF S YNERGETIC C ONTROL T HEORY Synergetic control theory [11] is based on the ideas of modern mathematics, optimal control, and synergetics and exploits the capability of open systems to self-organize. Synergetic control introduces a holistic philosophy of the dynamically controlled interaction of energy, matter, and information within a system by generating positive and negative feedback. The philosophy of control design is based on the principle of dynamic expansioncontraction of the system state space. The expansion of the system state space enriches system dynamics by providing additional differential equations that describe the interaction of the system with the environment (for example load disturbances), which is important from the designers’ perspective. The purpose of the control is to eliminate unwanted dynamics of the system or to reduce excessive degrees of freedom. At the control design stage,
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the unwanted dynamics is eliminated in accordance with the introduced dynamic constrains which are presented as invariant manifolds in the state space of the system. A. Main Equation of Synergetic Control Theory For a nonlinear system, the model of which is represented in (1), synergetic control theory provides design procedure that is based on a new methodology for generating control u(ψ) = u(x) or feedback as a function of the specified macro-variables ψ(x) which operate as parameters of order in synergetic systems and are also called aggregated variables. x˙ i (t) = fi (x1 , ..., xn ) + ai+1 xi+1 , i = 1, n − 1 x˙ n (t) = fn (x1 , ..., xn ) + u
(1)
where: xi are the system state variables; fi (x1 , ..., xn ) are nonlinear functions defining the dynamic evolution of the system; u is an input variable. For solving a control design problem, the theory uses the following functional, which is now called Kolesnikov’s Functional [12]: Z ∞ Z ∞ ˙ JΣ = F (ψ, ψ)dt = (m2 φ2 (ψ)+c2 ψ˙ 2 (t))dt (2) 0
0
˙ is a continuous positive definite function; where: F (ψ, ψ) ψ(x1 , ...xn ) is an aggregated variable or a macro-variable that represents an arbitrary continuous (or piece wise continuous) function of the state vector x, and φ is a smooth function differentiable with respect to its arguments. Moreover, function φ(ψ) is chosen to be a) differentiable and invertible; b) φ(0) = 0; and c) φ(ψ)ψ > 0 ∀ ψ 6= 0. The resulting synergetic control (3) is found as a joint solution of a so-called functional equation (4) and system model (1). · ¸ ∂ψ −1 ∂ψ 1 n u=− Σk=1 (fk + ai+1 xi+1 ) + φ(ψ) ∂xn ∂xk T (3) ˙ + φ(ψ) = 0 T ψ(t) (4) where: an+1 = 0, T = c/m = 1/λ; moreover, from the synergetics point of view, λ is a parameter characterizing the job performed by the self-organizing forces. Equation (4) defines the evolution of the system’s macrovariable ψ(x) into manifold (5). ψ(x) = 0
(5)
Under control (3), manifold (5) becomes an attracting set (or attractor) in the state space of the closed-loop system. As a result, from an admissible arbitrary initial location in the state space, the system moves into a vicinity of the manifold and along the manifold to the equilibrium point. The stability of these two motions is analyzed independently. The stability of the system motion towards the manifold can be determined using Lyapunov function (6), a derivative of which (7) defines the basin of attraction of the attractor ψ(x) = 0 introduced into the system. V = 0.5ψ 2 > 0
(6)
1 V˙ (t) = − φ(ψ)ψ < 0 if T > 0 T
(7)
Equation (7) shows that the stability of the closed-loop system motion towards the attractor is not compromised by nonlinear properties of the system model (1) and the system motion is stable as long as the parameter T is greater than zero. Once the system reaches the vicinity of the invariant manifold ψ(t) = 0, synergetic control law (3) will keep it on the manifold. As a result, in the vicinity of the manifold ψ(x) = 0, the system behavior is described by the reduced order model (8) which is obtained by substituting a manifold equation (5) into the system (1) instead of a dynamic equation with control (the last one in our case). x˙ i (t) = fi (x1 , ..., xn ) + ai+1 xi+1 , i = 1, n − 1 ψ(x) = 0
(8)
It can be seen from (8), the application of synergetic control decreases the order of the system by 1; and the order equals to n − 1. Thus, the stability analysis of the closed loop system can be significantly simplified by a sequential application of the control design process. To continue the application of the design, it is assumed that now a state variable xn becomes an internal control. With similar assumptions, the design process can be recursively repeated for a required number of times. Moreover, the direct path of the recursion gives a set of decomposed models and the return path of the recursion defines the control and manifold structures. As a result of the synergetic recursive design for one control channel system, one invariant manifold is created in the system state space. Under synergetic control the system goes into a vicinity of the manifold, first, and along this manifold to the specified equilibrium state. Moreover, the control ensures asymptotically stable motion towards and along the manifold to the desired equilibrium state. As long as the system remains on this attractor, the specified properties of the controlled system are guaranteed. Without loss of generality the same approach can be applied for control design for multi-control channel systems. III. S YNERGETIC C ONTROL D ESIGN A. System Model A prototype system for this research is presented in Fig.1. The state space averaged model of the system presented in (9) is derived using the following assumptions: the system operates in a continuous conduction mode, switching occurs at a very-high switching frequency, parasitic effects are ignored, state variables are the averaged vc1 capacitor voltage, iLi inductor currents. Disturbance M (t) in the model (9) takes into account the impact of the load and parameters’ changes on the system. In addition, coordinate transformation was applied to the model (9). This transformation moved the system voltage equilibrium point of the resulting system (9) into the origin of the coordinate (v = uc − v0 ) and substituted duty ratio with
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Converter 1
i1
where: £ ¤T I = i1 ... im £ ¤T X = v M a1,1 ... a1,m ... ... A1 = ... a ... am,m · m,1 ¸T a1,m+1 ... am,m+1 A2 = a1,m+2 ... am,m+2
Load
d1
L1 E1
uc C
Rext
M(t)
Converter 2 Converter m
im
dm
Lm
ai,j , i, j=1, m : ∀γ 6= 0
Em
m X
λk γk 6= 0
k=1
Fig. 1.
where: λl =
System of m paralleled boost converters with resistive load
control ui = (1 − di ) for every converter. m dv 1 X v+v0 M = uj ij − − dt C R C ext C j=1 di1 E1 (v+v0 ) u1 dt = L − L 1 1 ... dim Em (v+v0 ) = − um dt Lm Lm dM =ηv dt B. Control Design
ψi (i1 , i2 , ..., im , v) = 0 Ti ψ˙ i + ψi = 0, Ti > 0, i = 1, m
(9)
m X
ai,j ij +ai,m+1 v+ai,m+2 M (t) ; i = i, m
j=1
Ψ = A1 I + A2 X
al,n in ; l=1, m
For control design we select macro-variables of general form as shown in (13) where coefficients ai,j are selected to ensure linear independence of vectors λl in (15), which provides necessary and sufficient conditions for existence of the manifolds intersection. Solving jointly the system of the functional equations (12) with the system model (9) and with the set of the macro-variables (13), the synergetic control for the system (9) is found as shown in (16). U = −Z −1 [ΛΨ − A1 E − A3 (v, M )] ,
(16)
where: det Z 6= 0 T
U = (u1 , u2 , ..., um ) z1,1 ... z1,m ... ... Z = ... zm,1 ... zm,m a zi,j |i6=j = i,m+1 ij C zi,j |i=j =
(10) (11)
(13)
ai,m+1 ij C
·
+
(v+v0 ) Li
m P
ai,k
k=1
¸T E1 Em T E= , ..., A3 = [a31 , ..., a3m ] L1 Lm µ ¶ ν+v0 M (t) a3i =ai,m+1 − + −ai,m+2 η · v Rext C C i, j = 1, m. (17)
(12)
From a mathematical standpoint, synergetic control design [10] is based on a new method for generating control laws ψi = ψi (i1 , i2 , ..., im , v) or feedbacks that direct the system from arbitrary initial conditions into a vicinity of manifolds ψi (i1 , i2 , ..., im , v) = 0 and then ensure asymptotically stable motion along these manifolds toward the end attractors. On these attractors the desired properties of the controlled system are guaranteed. In short, the synergetic control design procedure is as follows. For system (9) the designer’s specifications are formulated as a set of macro-variables (10) based on the system state variables. The number of macro-variables in the set equals the number of control channels. Controls are found as a joint solution of the system of functional equations (12) and system model (9). ψi =
(15)
n=1
Control design uses synergetic control theory [10] and implements active current sharing [8] under which power distribution among converters is based on adjusting of the reference voltage [7]. ψi = ψi (i1 , i2 , ..., im , v) ; i = 1, m
m X
(14)
As a result of synergetic control design procedure, we get a general nonlinear PI control law. As it follows from the synergetic control theory, for any Ti > 0 and as long as vectors λi satisfy conditions for linear independency (15) for any i = 1, m, synergetic control law (16) moves the system representing point, which describes the system motion in its state space, onto the intersection of the manifold (11), where the system becomes of reduced order. C. Closed-loop Behavior There are two important components in the behavior of a closed-loop system: the stability of its behavior and the way parameters of the control law are to be tuned to satisfy designers specifications.
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1) Stability: The closed-loop system goes first into the vicinity of the manifolds and along the manifolds towards the equilibrium point. The stability analysis for these two parts of the system motion is done separately. a) Motion Towards Manifolds: We construct Lyapunov function (18) using (6). Equation (19) shows the derivative of the Lyapunov function (18) and summarizes the stability conditions. X ψi V = 0.5ψi2 > 0 and Ti ψ˙i = − (18) Ti V˙ = −
m X ψ2 i
i=1
Ti
< 0 if Ti > 0
i=1: ψ1 = i1 + a1,m+1 v + a1,m+2 M (t) (v + v0 ) ai,m+1 z1,1 = − + i1 Li C ψi = i1 − ii Ei = E1 ; Li = L1 ai,m+1 z1,i = ii C (v + v0 ) (v + v0 ) zi,1 = − ; zi,i = L1 Li zi,j = 0 ; ∀j = 2, m ; j 6= i
(20)
Moreover, since all ψi = 0 and inductances Li are equal as well as feeding voltages Ei , it can be shown that on manifolds controls and currents are equal as shown in (21). uiψ = u1ψ , iiψ = i1ψ ; i = 2, m
(21)
Using all the above (20,21) simplification we get the following expression for the control that directs the closedloop system on the manifold intersection towards the equilibrium:
u1ψ = ³
a1,m+1 C
Mψ vψ +v0 C − Rext C
where: f (vψ , Mψ ) =
d2 vψ dvψ + b1 + b2 vψ = 0, dt2 dt where: mE1 a21,m+1 mL1 a31,m+1 + Rext C v0 C Rext C 2 2 ηmL1 a1,m+1 a1,m+2 ηmE1 a1,m+1 a1,m+2 η b2 = − − . Rext C 2 v0 C C (24)
(a1,m+1 vψ +a1,m+2 Mψ ) m+
(vψ +v0 ) L1
´.
(22)
1
b1 =
−
In accordance with Hurwitz stability criterion [13], the system (24) is asymptotically stable if coefficients b1 and b2 are greater than zero as shown in (25). mE1 a21,m+1 mL1 a31,m+1 + >0 Rext C v0 C Rext C 2 2 ηmL1 a1,m+1 a1,m+3 ηmE1 a1,m+1 a1,m+2 η b2 = − − >0 Rext C 2 v0 C C (25)
1
−
Conditions shown in (25) represent limiting constrains for selecting control law coefficients to ensure a stable motion of the closed-loop system along the intersection of manifolds. 2) Current Sharing: Above, to simplify analysis, we considered the case with equal current sharing and masterslave operation, and selected macro-variable coefficients accordingly. However, in this section, on an example of a two-converter system, we examine how the selection of coefficients in macro-variables influences the current sharing in the system. We also show that synergetic control laws provide power systems with a dynamic management of responsibilities, including dynamic reconfiguration and allocation of current sharing. For a two-converter system the general macro-variable structure is as follows: ψ1 = a1,1 i1 + a1,2 i2 + a1,3 v + a1,4 M ψ2 = a2,1 i1 + a2,2 i2 + a2,3 v + a2,4 M
´ + a1,m+2 ηvψ
(vψ +v0 ) u1ψ L1
Linearization of (23) in the vicinity of the equilibrium point where v and M approximately equal to zero gives us the following second order system:
b1 =
i = 2, m :
E1 L1 +a1,m+1
E 1 +a1,m+2 η · vψ −f (vψ , Mψ )+ dvψ L1 µ ¶ = Mψ (t) vψ +v0 dt (1+ai,m+1 ) − C Rext C (23) dMψ (t) = η · vψ dt
(19)
It can be seen from (19) that the motion of the closedloop system towards manifolds (10) is stable as long as the time constants Ti are greater than zero. b) Motion Along the Manifolds: During the second part of the motion, the system is moving along the intersection of manifolds (10). The resulting motion can be described by a reduced order dynamic model as shown in (23). To simplify the derivation of the reduced order model for an m-paralleled boost converter system, we consider the most common case of the system operation, when paralleled converters are fed from the same source, inductors have the same values of their inductances, and an equal current sharing among converters is enforced. All these assumptions lead us to the following simplifications:
³
The resulting closed-loop behavior is defined by the equation (23) where control u1ψ is as shown in (22).
(26)
After the transients are finished, the representing point of the two-converter system inevitably reaches the intersection of the manifolds (27). The equations (27) define
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the steady state part of the closed-loop system motion along the manifolds towards the equilibrium point. ψ1 = a1,1 i1ψ + a1,2 i2ψ + a1,3 vψ + a1,4 Mψ = 0 ψ2 = a2,1 i1ψ + a2,2 i2ψ + a2,3 vψ + a2,4 Mψ = 0
(27)
Thus, the system currents that we can calculate from (27) are functions of the output voltage error v and an integral of the voltage error M as shown in the following: a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = −a2,3 vψ − a2,4 Mψ or A · I = B · X.
(a) varying inductance
(28)
Moreover, the coefficients of matrices A and B in (28) define the properties of current sharing between the two converters on the manifolds. In systems with two parallel-connected converters, the only possible basic current sharing regimes are masterslave and democratic. When the system is on the manifolds, either of these can be achieved by a particular choice of the coefficients in (28). In particular, master-slave current sharing occurs when the master converter maintains the output voltage and the slave converter supplies current according to the rating assigned to it [6], which is usually defined for the slave converter by the master. On the other hand, democratic current sharing occurs when each of the two converters provides current in proportion to its current rating (21) and each converter independently accounts for the output voltage error [19]. a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = 0
(29)
a1,1 i1ψ + a1,2 i2ψ = −a1,3 vψ − a1,4 Mψ a2,1 i1ψ + a2,2 i2ψ = −a2,3 vψ
(30)
Equations (29) and (30) show the manifolds’ structure for master-slave and democratic current sharing respectively, and give important insights not only for the current sharing properties but also for the responsibility of each converter. For example, coefficients a11 , a12 , a21 , and a22 define what fraction of the current is allocated to each of the converters, and if a2,1 = −a2,2 equal current sharing is assigned between the converters. Moreover, coefficients a13 and a23 define which converter is responsible for managing errors of the output voltage. Equations (29) and (30) demonstrate that the difference between mater-slave and democratic current sharing corresponds to coefficient a23 . In the case of democratic sharing, this coefficient is not zero; hence, during the transition to the equilibrium state, coefficient a23 influences the rate of change of converters’ currents. Since the output voltage error equals zero at equilibrium, it does not affect static current sharing in the system.
(b) varying capacitance
Fig. 2. Output voltage error transients under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
(a) varying inductance
(b) varying capacitance
Fig. 3. State portraits (i1 ,v plain) of the system under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
IV. S IMULATION R ESULTS AND D ISCUSSION For simulation of a two-converter system, parameters of which are presented in (31), we performed control design using the algorithm proposed in (16). The resulting control laws can be easily reconstructed using (16) and are not presented here to save space. The control law parameters are presented in (32). L1 = L2 = Ln = 0.001H C = Cn = 0.001F E1 = E2 = En = 12V v0 = 15V Rext = 10Ω
(31)
a1,1 = 1 a1,2 = 1 a1,3 = 5 a1,4 = 10 a2,1 = 1 a2,2 = −1 a2,3 = 0 a2,4 = 0 T1 = T2 = Tn = 0.001s
(32)
A. Simulation Results Using Maple, we performed a number of simulations, in which we studied the ability of the control to withstand system parameters’ variation and flexibility of current sharing. 1) Robustness Against Parameter Variation: In this simulations we varied independently by two times each of the following parameters: inductance L1 = (0.5, 1, 2)Ln and capacitance C = (0.5, 1, 2)Cn , while the rest of the parameters were kept constant. Simulation results are presented in Figs. 2-5 which accordingly are output voltage transients, state portraits in state plain (i1 ,v), inductor current transients, and state portraits in state plain (i1 ,i2 ).
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(a) varying inductance
(b) varying capacitance
Fig. 4. Inductor current transients under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
(a) varying inductance
(b) varying capacitance
Fig. 5. State portraits (i1 ,i2 plain) of the system under varying inductance (L1 = (0.5, 1, 2)Ln ) and capacitance (C = (0.5, 1, 2)Cn )
a) Voltage and Current Transients: It can be seen from figures (2, 3 and 4) that on the one hand, the variation of the converter inductance does not impact the closedloop system voltage and current transients. On the other hand, the variation of the load capacitance directly and proportionally influences the settling time of the voltage and current transients. b) Current Sharing: The state portraits in state plane (i1 ,i2 ) presented in Fig.6 illustrate the current sharing between two converters. As it was predicted by the theory, the figure shows that the system’s representing point first goes into vicinity of the manifold i1 = i2 and along this manifold goes to the equilibrium point. The variation of the inductance results in insignificant impact on current sharing, and the capacitance variation does not affect the current sharing specified during the control design. 2) Current Sharing: State portraits in state plain (i1 ,i2 ) presented in Fig.6 illustrate operation of different current sharing strategies. Fig.6(a) shows master-slave current sharing with different current distribution among converter: case (1) illustrates equal current sharing (dash line) and in case (2) the current of the first converter i1 is twice the current of the second converter i2 (solid line). Fig.6(b) presents democratic current sharing. As it was predicted in section III-C2, in democratic current sharing the current sharing ratings during the transition depend on the output voltage error, and in steady state operation it becomes equal as specified in control design in our case. Fig.6(c) shows the state portrait for the system
(a) state portrait of the system (b) state portrait of the system under master-slave current sharing under democratic current sharing with different current ratings (MS1 (i1 = i2 , a23 = 3) i1 = i2 and MS2 i1 = 2i2 )
(c) state portrait of the system un- (d) transient behavior of the output der dynamic current sharing (a21 = voltage under different current shar0, a22 = 1, a23 = 3) ing Fig. 6. Transient behavior of the closed-loop system under different current sharing strategies
with dynamic current sharing. It can be seen that current withdrawn from the second converter in steady state equals zero. However, during transients in the system the second converter supplies an additional amount of current to the load. Thus, the second converter behaves as a capacitor by supplying the energy to the system load dynamically. Fig.6(d) presents voltage transients of the closed-loop system under the above mentioned four different current sharing strategies. As it can be seen from the figure, the deviation of the voltage transient responses from each other is insignificant. Thus, as it was predicted in section III-C2, the structure of macro-variables defines the properties of current sharing in the system. For example, structure of macro-variables presented in (29) results in master-slave current sharing and structure (30) leads to democratic current sharing. In particular, coefficients a11 , a12 , a21 , and a22 define what fraction of the current is allocated to each of the converters, and coefficients a13 and a23 define which converter is responsible for managing the error of the output voltage. During transients, coefficient a23 influences the rate of change of the converters’ currents. Thus, synergetic control proposed in this paper provides flexible means for selecting and adjusting energy flow within the system. B. Discussion As it was predicted by the theory, the simulations show that the system’s representing point first goes into vicinity
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of the manifolds intersection and along this intersection goes to the equilibrium point. Moreover, the manifold created in the closed-loop system state space shows very robust properties under 400% variation of the system parameters. Thus, synergetic control designed in the paper provides a good insensitivity for the closed-loop system to the converters parameters’ variations. Moreover, control law coefficients provide extraordinary flexibility not only for the allocation of current sharing and tuning the system transients but also for selecting the type of current sharing properties and responsibilities of each converter. The resulting flexibility in current sharing provided by synergetic control laws can benefit reconfigurable power systems by allocating current sharing and managing responsibilities dynamically. V. C ONCLUSION This paper presented synergetic control design for an m-paralleled boost converter system under active current sharing. The presented design overcame such problems of the system as multi-connectivity, nonlinearity, and high dimensionality. Multi-connectivity and nonlinearity of the system were taken into account by the presented set of manifolds, which define laws of interaction among converters. Moreover, the introduced set of macro-variables accommodates different current sharing regimes such as master-slave and democratic current sharing simply by changing coefficients in the macro-variables. Contraction of the system state space under synergetic control made it possible to draw a stability analysis for an arbitrary number of paralleled boost converters. The stability analysis and current sharing analysis presented in this paper explained the choice of control law coefficients. The simulation results for a two-converter system validated the theory and showed that the closed loop systems are characterized by stable operation; fast response; good voltage regulation; ability to nullify steady state error not only of output voltage but also of current sharing; capability to control the current withdrawn from each converter; and ability to change current sharing during operation, and more importantly, capability to withstand the variation of the system parameters by 400%.
[5] A. Kolesnikov, G. Veselov, A. Monti, F. Ponci, E. Santi, and R. Dougal, “Synergetic synthesis of dc-dc boost converter controllers: theory and experimental analysis,” in Applied Power Electronics Conference and Exposition, 2002. APEC 2002. Seventeenth Annual IEEE, vol. 1, 2002, pp. 409–415 vol.1. [6] J. Rajagopalan, K. Xing, Y. Guo, F. Lee, and B. Manners, “Modeling and dynamic analysis of paralleled dc/dc converters with master-slave current sharing control,” in Applied Power Electronics Conference and Exposition, 1996. APEC ’96. Conference Proceedings 1996., Eleventh Annual, vol. 2, 1996, pp. 678–684 vol.2. [7] V. Thottuvelil and G. Verghese, “Stability analysis of paralleled dc/dc converters with active current sharing,” in Power Electronics Specialists Conference, 1996. PESC ’96 Record., 27th Annual IEEE, vol. 2, 1996, pp. 1080–1086 vol.2. [8] I. Batarseh, K. Siri, and H. Lee, “Investigation of the output droop characteristics of parallel-connnected dc-dc converters,” in Power Electronics Specialists Conference, PESC ’94 Record., 25th Annual IEEE, 1994, pp. 1342–1351 vol.2. [9] M. di Bernardo and F. Vasca, “Discrete-time maps for the analysis of bifurcations and chaos in dc/dc converters,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on [see also Circuits and Systems I: Regular Papers, IEEE Transactions on], vol. 47, pp. 130–143, 2000. [10] ed. by A.A. Kolesnikov, Sovremennaya Prikladnaya Teoria Uprablenia: Chasty I Sinergeticheskii podhod v teorii upravlenia. Moscow - Taganrog: TRTU, 2000, vol. 2. [11] A. Kolesnikov, Synergetic Control Theory, 1st ed. Moscow– Taganrog: TSURE, 1994, vol. 1. [12] G. Kondratiev, Geometrical Synthesis Theory Of Optimal Stationary Smooth Control Systems. Moscow: Fizmatlit, 2003. [13] R. Clark, “The Routh-Hurwitz stability criterion, revisited,” Control Systems Magazine, IEEE, vol. 12, pp. 119–120, 1992.
ACKNOWLEDGEMENT The authors acknowledge the support of ONR grants N00014-02-1-0623, N00014-03-1-0434. R EFERENCES [1] J. G. Ciezki and R. W. Ashton, “Selection and stability issues associated with a navy shipboard dc zonal distribution system,” IEEE Tr. on Power Delivery, vol. 15, pp. 665–669, April 2000. [2] C. Tse, Complex Behavior of Switching Power Converters, 1st ed. CRC Press, 2003, vol. 1. [3] J. G. Ciezki and R. W. Ashton, “The design of stabilizing controls for shipboard dc-to-dc buck choppers using feedback linearization techniques,” in Proc. PESC’98, pp. 335–341, 1998. [4] S. K. Mazunder, A. H. Nayfeh, and D. Borojevic, “Robust control of parallel dc-dc buck converters by combining integral-variablestructure and multiple-sliding-surfaces control schemes,” IEEE Tr. on Power Electron., vol. 17, pp. 428–437, May 2002.
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