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A Comparison of Methods for Finding SteadyState Solution in Power Electronic Circuits D.V. Malyna, J.L. Duarte, M.A.M. Hendrix, F.B.M. van Horck Electromechanics and Power Electronics Group Technical University of Eindhoven Den Dolech 2, 5600MB Eindhoven, The Netherlands
Abstract--This work concerns the fast steady-state determination of power electronic circuits. Switching converters operate in different modes. In each of these modes a converter is described as a linear time-invariant system (LTI). The steady state is found with computer simulations performed with Newton-based methods for accelerated steady-state determination. Two methods are investigated: exact and approximated Jacobians. The implementation of zero-crossing detection improved convergence.
Newton method. The first method uses the approximated Jacobians and the second method uses exact. The Jacobians are determined analytically in both cases. The comparison of various time-domain simulation approaches is covered in [6]. There it concluded that Newton s method with analytically determined Jacobians gives the fastest convergence. The development of fast power electronics simulator based on Jacobian calculation is reported in [7].
Index Terms--Simulation, steady-state, switched networks, state-space description, resonant converters.
II. THEORETICAL BACKGROUND
I. INTRODUCTION
T
he steady-state determination of a power electronic converter may seem a trivial problem. For the classical buck, boost, buck-boost, and uk converters the steady state can be easily found. However, for soft-switching resonant converters the procedure to find the solution is not as easy as it may look at first sight. The existing commercial tools suggest a brute-force approach i.e. simulation over hundreds or thousands of sequential periods, because transients usually decay very slowly [5]. This is very time-consuming and inefficient, especially when the number of simulations have to be large in the case of circuit optimization according to specific designer requirements. The steady-state can be determined in a few simulations over a single period and the algorithm for initial conditions determination for next period. This approach is called shooting . The first attempt to handle the problem in this way is described in [1]. The authors are using Newton s method for steady-state determination. The Jacobian is determined as the multiplication of the state transition matrices of topological sequences. The implementation of this method to power electronics problems is given in [2,9]; an approximated method for Jacobian determination is used. The exact Jacobian includes the sensitivity of the switching boundary conditions to the interval duration and initial state [4,3]; an implementation of the method is described in [11]. Both described approaches are based on
A. State-space description. Any linear electrical circuit can be described with the first order state-space differential equations.
dx(t) A x(t) B u(t) (2.1) dt x(t ),u(t ) : state-space and independent source vectors; A, B : state and independent sources matrices; The solution of the (2.1) is in the form: t
x(t )
x(t0 ) e A ( t
t0 )
e A (t
)
B u( )d
(2.2)
t0
x(t0 ) - state vector at the moment t0 . The structure of the circuit changes periodically in the case of power electronic converter. Some switches in the circuit opens and closes during the operation. Thus, matrices A and B have variant coefficients, depending on circuit configuration. We are interested in finding the steady-state of such circuit. When the steady state is achieved, the state vector value at the beginning of the period is equal to the state vector value at the end of the period: f (T , x 0 ) x 0 0 (2.3) Then from mathematical point of view to find a steadystate, means to find vector x 0 . The solution of (2.3) is the vector x 0 . Two possibilities will be discussed further.
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B. Method with approximated Jacobians Euler Backward Algorithm (EBA) is used for the integration of the (2.1). Implicit integration method avoids numerical instability [9]. According to the EBA the solution of (2.1):
x1
x0
h A x1 B u
(2.4)
x1 -state-space vector at the end of step h ; x0 -state-space vector at the beginning of the integration -1
x 0 h -1 B u I h A : state transition matrix
(2.5)
-1
xn-1
-1
h
(2.6)
B u
(2.6) in the terms of initial value x 0 for the constant u : xn
n
-1
x0
n
-1
h
-1
n 1
-1
B u (2.7)
According to (2.7) we can calculate xn directly from
x0 . Suppose f T , x0 is the value of the state vector at the end of the period, with initial value x 0 at the beginning of the period. x0j+1
x0j
1
J T , x0j
I
f T , x0j
x0j
(2.8)
f (T , x0j ) -state vector value at the end of the integration period of the j shoot. The Jacobian for (2.8) is determined: -1 n 1
-1 1
-1
n
(2.9)
Power electronics circuits are usually linear piecewise circuits. The intervals of continuous linear circuit structure could be recognized. Consider the circuit with two intervals. Then for each interval we can write (2.6) as:
x1n
-1 1
x 2n
2
x1n-1
-1
x 2n-1
h h
-1 1 -1 2
B1 u1 B2 u2
(2.10)
Iteration formula (2.8) for shooting algorithm remains the same. The only difference is in the Jacobian:
J
-1 1
k
-1 2
m
x0 j
(2.11)
k , m -the number of integration steps of size h during the phase 1) and 2) respectively; C. Method with exact Jacobians The second method treats the piece-wise system as a unity of topologies and provides the linearization. Consider the system of N switching topologies which is described by
J x (T , x 0 j )
1
R (T , x 0 j )
(2.14)
account boundary conditions for the switches: x(T , x 0 ) x0
x(T , x 0 ) T
c T
1
c x0
I (2.15)
Each interval is described by the set of matrices and values:
A1 , A 2 T1 , T2
An Tn
x(0), x(T1 )
x(TN )
(2.16)
Where N is the number of consecutive topologies during one switching period. The termination criteria for both methods is chosen when the energy flow in reactive components remains within specified precision: 1
Ej
x0j -initial value for the j shoot;
-1 n
1
The Jacobian J x is calculated with the taking to the
Ej
x0j 1 -initial value for the j 1 shoot;
J
The iterative formula for the next shoot is:
J x (T , x 0 )
General solution for the n-th integration step:
xn
Usually the steady state is approached in several consecutive iterations shoots . Then the difference between values of the state vector at the beginning and at the end of the period: (2.13) R (T , x 0 j ) f (T , x 0 j ) x 0 j
x0 j
step. Regrouping the terms in (2.4) we have:
x1
the nonlinear function and boundary conditions for the topology change: (2.12) f (T , x 0 ) , c(x0 , T ) 0
Ej 1
(2.17)
III. POWER ELECTRONICS DEVICE MODELING Typical power converter contains resistors, capacitors, inductors, voltage and current sources, transistors and diodes. Passive components and sources are modeled as ideal components. Major discussions deal about the switches. We have to distinguish two types of switches. The first one is a completely controlled switch. It turns on and off by the external signal (transistors). The second one is a noncontrolled switch. The state of the switch is determined by certain circuit parameters (i.e. voltages, currents). The diode is a non-controlled switch, because it turns on by the positive forward voltage and turns off by the current. The timing pattern for the completely controlled switch is known. But it is not the case for a non-controlled switch. Equation (2.2) has the integration constant as the function of the initial time t0 . The time t0 is known in the case of completely controlled switches and is unknown in the case of non-controlled switches.
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A fixed step method was chosen for integration. The minimum number of integration points is requested for speed, but for precision it is other way around. The biggest error is encountered for implicit switching conditions at the zero-crossings of the state vector. This problem can be solved with linear interpolation shown in Fig.2.1.
T3
T1
Lr D1 iLr
Ui
D2 Co
Cr uCr
T4
T2
uCo
R
D3
xNew
D4
Fig. 4.1. PRC circuit diagram. hNew
D1
D2
iLr
hOld
tn
iLr
Ui
uCr
Ui
uCo
uCr
uCo
tn
1
1)
D4
4)
D3
Fig 2.1. First-order linear interpolation. iLr
The new step size is determined as:
hNEW
iLr
Ui
x (tn ) hOLD x(tn ) x(tn 1 )
uCr
Ui
uCo
(3.1)
uCr
2)
5)
D2
As soon as the sign of the current through the diode changes, we calculate the new value of step h and new value of state-transition matrices. Then the new value of state vector can be calculated. The structural diagram of the routine is presented in Fig.2.2. iD changes sign?
hNew calculation
Gnew, Fnew, xNew calculation based on hNew
uCr
Ui
uCo
uCr
3)
D3
uCo
6)
D4
Fig. 4.2. PRC topology sequence.
For each of the topologies of Fig 4.2: A1
IV. SIMULATION RESULTS
A2
The abovementioned methods were implemented in MATLAB environment with application to the parallel load resonant converter, Fig 4.1. The environment has been chosen because of the focus on the methods comparison and mathematical background rather than specific computer platform optimization for performance. The selected converter has six possible piecewise-linear topologies shown in Fig.4.2. In state-space formulation:
iLr (t )
(4.1)
uCo (t )
Independent source vector is described by: ui (t ) ui We shall use state-space formulation: x A x B u
iLr
Ui
New set of Ai selection
uCr (t )
D1
iLr
Fig. 2.2. Zero-crossing detection flowchart
x(t )
uCo
(4.2)
1 Lr 1 R Cp
0 0
1 Cp
0
1 R Cp
0
1 Lr
1 Cr
0
0
0
0
1 R C0
0 1 Cp
A3
Cp
A4
Cr
0
1 Lr 1 R Cp
0 0
0
1 R Cp
1 Cp
1 Lr B1,2,3
0 0
C0
A1 A 4
A2 A6
A 3 B 4 ,5 , 6
B1,2 ,3
(4.4)
Steady-state determination for such topology is not obvious as it is for hard-switching converters. The flow diagram of the algorithm is shown in Fig.4.3. The rectangular shows the zero-crossing detection routine. The converter has been modeled with the parameters: Lr 31.7 H , Cr 37.6 nF , C0 1 F , R 27 ,
f
(4.3)
0 1 Cp
195kHz , U i
40V , T
0.005 ,
x 0 (iLr ; uCr ; uCo ) (0; 0;0.01)T . The simulation was performed with 50,100,150,200,250,300,350 fixed integration steps. Both algorithms with exact and simplified Jacobians were investigated.
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The algorithms were implemented in two stages: initialization and shooting. During initialization stage we performed brute force simulations until the sequence of topologies became definite. After that point the shooting approach takes over. A number of shoots of more than 25 has been considered as non-convergence of the fast steady-state determination algorithm, because brute-force simulation converges in 22-25 sequential periods for this particular example. The exact solution with exponential matrices was used for the method with exact Jacobians. Approximated Jacobians were used in EBA. EBA is quite convenient, because of the state transition matrices used for integration and for fast steady-state determination. The results are presented in Table 4.1. Both shooting methods without zero-crossing detection show an instable region depending on the step size (gray background). The method with EBA and simplified Jacobians is worse. Both methods are comparable in speed and convergence behavior when the zero-crossing detection is added.
Begin
Specification L,C,R,Ui,f Calculation A1..An and F,G
Initial conditions specification x0
Operating mode determination
Integration within one step x(j+1)=F*x(j)+G*u
C1...Cn satisfied ?
Yes
Registration of xi and transition moment Ti
Change of the state matrix Ai
No
No
End of the period ?
Yes
Precision reached?
Yes
X*=x(T,x0)
No Shooting with iterative formula x0=xShoot
Jacobian calculation
End
Fig. 4.3. Fast steady-state determination flowchart. TABLE 4.1. RESULTS FOR PRC SHOOTING FOR THE STEADY-STATE DETERMINATION Without zero-crossing detection
Steps
Periods
Time
Shoots
Time
Periods
Time
Shoots
Time
Shooting stage
Method
Initialization stage
Exact Jacobians
Shooting stage
50 100 150 200 250 300 350
4 4 4 4 4 4 4
0.016 0.016 0.031 0.031 0.047 0.047 0.062
5 25 4 3 3 25 4
0.031 0.156 0.032 0.031 0.031 0.360 0.063
2.958;23.080;23.080 2.997;25.068;25.068 2.922;24.953;24.953 2.953;25.442;25.442 2.972;25.677;25.677 2.953;25.349;25.349 2.953;25.718;25.718
4 4 4 4 4 4 4
0.015 0.015 0.031 0.031 0.047 0.063 0.063
4 4 4 4 4 4 4
0.016 0.032 0.047 0.047 0.062 0.062 0.078
3.003;27.172;27.172 2.973;26.514;26.514 2.961;26.300;26.300 2.969;26.213;26.213 2.962;26.144;26.144 2.957;26.095;26.095 2.954;26.060;26.060
Approximated Jacobians
Initialization stage
With zero-crossing detection
50 100 150 200 250 300 350
4 4 4 4 4 4 4
0.016 0.015 0.031 0.047 0.047 0.063 0.062
25 25 25 25 5 5 5
0.062 0.141 0.188 0.234 0.062 0.062 0.079
2.781;21.635;23.651 2.908;23.651;25.239 2.871;24.275;24.275 2.901;24.752;25.529 2.996;24.340;25.142 2.984;24.143;25.039 2.997;24.247;25.460
4 4 4 4 4 4 4
0.015 0.016 0.031 0.047 0.047 0.062 0.062
5 6 7 5 6 6 6
0.016 0.031 0.063 0.046 0.078 0.094 0.109
2.892;25.799;26.476 2.925;26.255;26.328 2.930;26.220;26.220 2.993;24.994;26.166 2.978;25.883;26.149 2.976;25.836;26.114 2.984;25.818;26.101
State-vector {Ilr;Ucr;Uc}
State-vector {Ilr;Ucr;Uc}
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V. FAST STEADY-STATE SIMULATOR The derivation of state-space matrices is straightforward for most practical circuit topologies, but it requires a lot of attention and time. The state determination itself and the formulation of the links between states requires good knowledge of the topology chosen. In the case of slight topology modification (i.e. snubber presence or absence, ESR of L and C, switches resistances) the complexity of state-space model increases significantly. This is the major source of human errors. It has been decided to give the job of state-space description formulation to a computer program. The program has to recognize standard network list files of popular schematic captures. The Schematics from Cadence has been chosen because of wide spread in the engineering community. A design process flowchart is shown in Fig.5.1.
Schematic entry
Netlist generation
Schematics
All possible states and links determination
Accelerated steady-state determination
M_SIM READ
M_SIM SHOOT
Fig. 5.1. Steady-State simulation process flow.
The simulator has been developed in MATLAB environment. Program (M_SIM READ) reads the netlistfile; determines the tree, position of switches, states; Nswitches
generates 2 sets for state space matrices and boundary conditions for the switches. Then the accelerated steady-state determination routine (M_SIM SHOOT) is performed. The algorithms of [8,9] were used. Library cells, used by the program are listed in the Table 5.1.
The method with approximated Jacobians has been chosen for implementation. It does not require analytical formulation of boundary conditions and their derivatives calculation. Buck, boost and PRC converters were used in the simulations. The parameters: a) buck converter U in 20V , L 330 H , C 100 F , Rload 10 , f
10kHz ,
0.5
b) boost converter U in 10V , L 75 H , C 100 F , Rload 10 , 0.5 f 10kHz , c) PRC U in 40V , Lr 21.7 H , Cr 37.6nF , C0 1 F Rload
27 , f
200kHz
Initial state for all state-space variables of all converters has been set to zero. The 250 integration points per period and zero-crossing detection for the EBA numerical integration routines were used. The simulations results are listed in the Table 5.2. TABLE 5.2. M_SIM RESULTS FOR DIFFERENT CONVERTERS Buck Boost PRC Shooting Br.force Shooting Br.force Shooting Br.force 1+3 66 2+6 40 3+7 30 Note: 1+3 for Buck converter means simulation over 1 initial period and 3 shoots to achieve steady state. 66 means the simulations over 66 sequential periods to achieve steady-state.
The result for M_SIM PRC state-trajectory progress is shown in Fig.5.2.
TABLE 5.1. M_SIM LIBRARY CELLS Passive
Switches T
Sources
Ron
R
Roff V
L
D
Ron Roff
J
C
Simulating
Shooting
Fig. 5.2. M_SIM state trajectory for the PRC over the number of periods.
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VI. CONCLUSIONS. Both steady-state shooting methods show good behavior and comparable speed. The method with exact Jacobians converges faster to the solution and introduces fewer errors because of the exact integration and exact Jacobians. The method with simplified Jacobians does not converge for a small number of integration steps. Inclusion of zero-crossing detection significantly improves both methods. As a rule the exact method converges in 4 shoots and the approximated method converges in 5 7 shoots . The simulation time for 50;100;150 integration steps is comparable for both methods and is enough for obtaining correct results. The method with simplified Jacobians gives correct results and is precise and fast, although slightly slower than the exact method. However it is more suitable for computer implementation because it does not require analytical formulation of the boundary conditions vectors. Fast steady-state simulator M_SIM has been developed and some converter topologies were successfully analyzed. VII. REFERENCES: [1]
T.J. Aprille, T.N. Trick, Steady-state analysis of nonlinear circuits with periodic inputs , Proceedings of the IEEE, vol. 60, no. 1, pp.108 114, Jan. 1972. [2] V. Rajagopalan, A. Jacob, A. Sevigny, K.S. Rao, Computation of Almost periodic steady-state response of power electronic converter systems , in ELECTRICAL MACHINES AND CONVERTERS Modelling and Simulation, Elsevier Science Publishers B.V. (NorthHolland)/ (C) IMACS, 1984. [3] G.C. Verghese, M.E. Elbuluk, J.G. Kassakian, A general approach to sampled-data modeling for power electronic circuits , IEEE Transactions on Power Electronics, vol. PE-1, no. 2, pp. 77 89, Apr. 1986 [4] J.L. Duarte, Small-signal modeling and analysis of switching converters using Matlab , International Journal Electronics, vol. 85, no. 2, pp. 231-269, 1998. [5] N. Mohan, T.M. Udeland, Simulation of power electronic and motion control systems an overview , Proceeedigs of the IEEE, vol. 82, no. 8, pp. 1287 1302, Aug. 1994. [6] D. Li, R. Tymerski, Comparison of simulation algorithms for accelerated determination of periodic steady state of switched networks , IEEE Transactions on Industrial Electronics, vol. 47, no 6, pp. 1278 1285, Dec. 2000. [7] D. Li, R. Tymerski, T. Ninomiya, PECS an efficient solution for simulating switched networks with nonlinear elements , IEEE Transactions on Industrial Electronics, vol. 48, no 2, pp. 367 376, Apr. 2001. [8] Leon O. Chua, Pen-Min Lin Computer aided analysis of electronic circuits , Prentice-Hall inc., 1975. [9] V. Rajagopalan, Computer-aided analysis of power electronic systems , Marcel Dekker inc., 1987. [10] J.L. Duarte, Sampled-data modeling and simulation of cyclically switched converters , EUT report 96-E-303, Eindhoven, Dec.1996. [11] J.L. Duarte, M.A.M. Hendrix Computer-aided optimization of switching converters for low-power applications , EUT report 00E-31, Eindhoven, Dec. 2000.
VIII. BIOGRAPHIES Dmytro V. Malyna was born in Kiev, Ukraine in 1977. He graduated from Kiev Polytechnic Institute (National Technical University), Kiev, Ukraine, with M.Sc. degree (honors) in Electronic Systems in 2000. Since 2000 he worked as design engineer for Applied Electronics Institute , private company in Kiev for power and industrial electronics development. In 20012003 he is with Philips Electronics, Eindhoven, The Netherlands as designer and researcher for consumer market SMPS (plasma screens, portable products). In 2003 D.V. Malyna started his Ph.D. study with Electromechanics and Power Electronics Group, Technical University of Eindhoven (TU/e). His research interests include power converter design methods, resonant topologies, magnetic design, modeling of electric circuits, analog and digital signal processing. Dr. Jorge Duarte (1955) is with the TU Eindhoven, Group Electromechanics and Power Electronics, as a member of the scientific staff since 1990. His teaching and research interests include modeling, simulation and design of power electronic systems. He received the M.Sc. degree in 1980 from the University of Rio de Janeiro, Brazil, and the Dr.Ing. degree in 1985 from the INPL-Nancy, France. In 1989 he was appointed a research engineer at Philips Lighting Central Development Laboratory, and since October 2000 he is consultant engineer (0.2 fte) at Philips Power Solutions in Eindhoven. Marcel Hendrix was born in 1956 in Weert, the Netherlands. He studied Electrotechnical Engineering at Eindhoven University of Technology from 1974-1981, specializing in electronic circuit design. In 1983 he joined Philips Lighting in Eindhoven, and started to work in the pre-development laboratory of Business Group Lighting Electronics and Gear (BGLE&G). Since that time he has been involved in the design and specification of switched power supplies for both low and high pressure gas-discharge lamps. In July 1998 he was appointed a part-time professor (UHD) in the MBS group, section Electromechanics and Power Electronics. His professional interests are with cost function based simulation and sampled-data, non-linear modeling, real-time programming and embedded control. Dr. Frank van Horck received the M.Sc. degree cum laude and the Ph.D. degree in electrical engineering both from Eindhoven University of Technology, the Netherlands. In October 1998, he joined Philips where his main research work involves power converters. Dr. Van Horck s research interests are focused on analytical and numerical solutions of electromagnetic theory, applied to power-electronic circuits and the electromagnetic environment.
