Steady-state algorithm for switching power electronic devices

J.C.Contreras-Sampayo, J. Usaola-Garcia and A.R.Wood

Abstract: Obtaining the steady-state operation of a power electronic device by means of brute force computer simulation is not feasible in many practical cases. Fast steady-state algorithms that formulate the steady-state problem as a boundary problem and solve it using Newton’s method have been proposed to overcome this difficulty. These algorithms are known as shooting algorithms. An extension of the shooting algorithm for piecewise linear circuits is provided. The complete Jacobian matrix that takes into account the switching instants variation is analytically derived for a state variable formulation of the steady-state problem of a piecewise linear circuit. A computer program PWiseSS based on this algorithm is used to solve a previously proposed test circuit of difficult convergence as well as to solve a realistic six-pulse converter of interest to the power electronics engineer.

1 Introduction

Determining the steady-state of a nonlinear circuit is a problem still not yet fully satisfactorily solved, which con- tinues to attract the interest of researchers [l-31. The obvi- ous approach involves simulating the circuit in the time domain until all transients have died out. This approach, known as brute-force, is very ineficient, and expensive in terms of computing effort in many practical cases when the presence of widely differing time constants makes the prob- lem difficult. Power electronic converters typically present time constants that are much longer than the period of the steady-state solution. To overcome this, more efficient solu- tion methods known as fast steady-state methods have been proposed. These methods fall into two categories: time domain and frequency domain.

Frequency-domain methods [ 1, 4-61 first assume a limited spectrum in the response of the circuit, then use optimisation techniques to determine the frequency compo- nents of the response. Limiting the frequency spectrum may introduce error in the calculated switching instants, which may in turn put the accuracy of the considered frequencies in doubt. Usually, a much wider spectrum than required for the solution must be used in the computation.

Time-domain methods formulate the steady-state prob- lem as a two-point boundary value (TPBV) problem. The boundary condition is the steady-state condition, i.e. the initial state must be equal to the final state. The shooting solution of the TBPV problem involves simulating the cir- cuit across a single period and correcting the initial state iteratively towards the steady-state. Aprille and Trick [7]

0 IEE, 2001 IEE Proceedings online no. 20010188 DOL lO.l049/ipepa:200 10 1 88 Paper received 20th October 2oM) J.C. Contreras-Sampayo and J.Usaola-Gmia are with the Univedad Carlos 111 de Madrid, AV. de la Universidad 30, 2891 1 LeganCs, Madrid, Spain A.R. Wood is with the University of Canterbury, Department of Electrical Engineering, Private Bag 4800, Christchurch, New Zealand

were among the first to propose a shooting approach, using Newton’s method to correct the initial state. This approach involves computing the sensitivity of the final state to the intial state, which is not trivial; these sensitivities are numerically computed as the simulation advances through the period.

An envelope-following approach [8] substantially improves efficiency over previous time-domain algorithms when the low-frequency envelope of the response is rela- tively smooth and thus a number of transient cycles can be skipped by extrapolation.

Several previous works [9-1 I] have shown that power electronic devices can be accurately modelled as piecewise linear circuits. Armanazi [12] suggested an algorithm for piecewise linear circuits based on state-space formulation and a Newton solution. Armanazi’s approach computes the required sensitivity matrix assuming the switchmg instants to be independent of the initial state. More recently, the importance of taking into account the switch- ing instant variations in the computation of the sensitivities has been reported in the literature [13, 141. Failure to con- sider these variations leads to reduced efficiency or lack of convergence in some cases. El-Bidweihy and Al-Badwaihy [ 151 and Wong [ 1 13 identified and partially overcame this deficiency, but with some limitations on the nature of the circuit being solved [13]

This work presents an extension of the state-variable- based shooting algorithm to determine the steady-state of a periodically excited piecewise linear circuit that contributes the complete sensitivity matrix of the final state to the ini- tial state. This matrix is analytically derived. It is used to attain faster and more reliable convergence to the steady state. The algorithm yields a solution that consists of the complete set of switching instants, state vectors at the switching instants, and a model of each of the linear stages. This solution can be quickly evaluated off-line for any time instant required.

The algorithm has been implemented in a computer pro- gram, PWiseSS, which accepts PSPICE-type circuit descrip- tion files and has been written in MATLAB. Simulation

IEE Proc.-Electr. Power Appl., Vol. 148, No. 2, March 2001 245

times for realistic devices are relatively fast in a desktop computer.

2 Formulation of the problem

The response of a power electronic device can be accurately obtained by modelling its semiconductors as piecewise lin- ear elements [lo], i.e. as resistors of either high or low resist- ance, depending on the state of the switch. It is assumed that the circuit has n energy storage elements and that it has a periodic steady-state of period T. The period of the response is assumed to be known. In these conditions, all state variables have a piecewise linear evolution. Each stage is bounded by two consecutive switching instants. The order in which the switchings occur is, in general, not known beforehand.

Each linear stage i is characterised by a particular combi- nation of the status of the switches, called the switch com- bination, which remains constant throughout the stage. The final state at the end of the period X,, is a nonlinear function F of the initial state X,. Neither the times at which the semiconductors switch nor the order in which they switch are known a priori; they both depend on the initial state A',. The number of stages ns, the order in which the semiconductors switch, and the associated stage sequence are a function of X, and all unknowns. In general, the ini- tial instant will bisect a stage thus making the apparent number of stages ns one greater than the actual number of stages.

The steady-state problem can be formulated as a two- point boundary value problem. An initial (and final) state X,* must be found such that it satisfies the following boundary constraint:

F(X,*) = x,* (1) A mismatch function Y, defined as "(3 = F ( 3 - X,, allows eqn. 1 to be formulated as a homogeneous equation:

9 ( X , * ) = F(X,*) - x; = 0 (2) To solve eqn. 2, two subproblems must be solved:

The first is to evaluate Y(X,) for any given initial state A',. The evaluation algorithm described below evaluates W(X,), i.e. yields the final state after one period given the initial state. Ths algorithm is quite general and does not require the switching sequence to be known. It computes all switching instants, the switching sequence and all transi- tion matrices for each linear stage within the period. This algorithm is based on state-transition-matrix computation and evaluation of the system differential equations, rather than on numerical simulation.

Fhe second step involved in the Newton solution of eqn. 2 is obtaining the set of sensitivities of the final state to the initial state. In Section 4 the expression of the sensi- tivity (Jacobian) matrix of the final state to the initial state is derived for a generic piecewise linear circuit using a state variable formulation.

3 Evaluation algorithm

The response of a piecewise linear circuit, and thus the state vector at the end of the period, can be obtained by comput- ing the state transition matrices and evaluating the solution to the system differential equations The algebraic expres- sion of the solution to the differential equatious of the system is known for any particular linear stage i, although not for the complete response because the stage sequence and switching instants are, in general, unknown. Direct evaluation of the algebraic solution is not very efficient

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because it requires a matrix exponential to be computed at each time instant. A substantial increase in efficiency over direct evaluation can be achieved using the eigensystem of the state matrix if the evaluation is performed at regularly spaced points. Using this evaluation approach, the switch- ing instants can be accurately solved for by solving a simple equation [16] but only if it is known which semiconductor switches in the first place. Determining which semiconduc- tor switching must be solved for is not trivial in a general case, when no previous knowledge of the operation of the device is available.

3. I Evaluation of the algebraic solution The evolution of the state variables for any particular linear stage i is defined by the following set of differential equa- tions:

X ( t ) = Ai . X ( t ) + Bi . S ( t ) + C, . S ( t ) (3) 5' is the vector of independent sources and X is the state vector. Matrix Ci is omitted in previously proposed state- space-based steady-state algorithms [ 1 I , 151 but is non-zero when all-capacitor cutsets or all-inductor loops exist. Matri- ces A, B and C can be automatically computed for any cir- cuit that consists of any combination of resistors, inductors, capacitors and independent sources using the systematic formulation provided in [17] or [18]. The solution to eqn. 3 has two terms; a transient term ( X ) and a steady-state term (J3:

X ( t ) = X" t ) + x-(t) (4)

3.7.1 Transient term evaluation: The transient term is an exponential term of the form:

x'((t) = eA*.t . KX ( 5 ) A; is the state matrix and Kix is a constant that depends upon the initial condition for the stage. In previously pro- posed algorithms [Ill direct evaluation of eqn. 5 has been suggested. This involves computing the exponential of an n x n matrix (Ai) for each time step, which is computation- ally very demanding. It is possible to evaluate this term more efficiently using the diagonal eigensystem of Ai. The following equation relates Ai to its eigensystem:

A i . V, = V, . Ai ( 6 ) A; is the diagonal matrix of eigenvalues of A; i2nd V, is a full matrix whose columns are the corresponding eigenvec- tors. Ai can be made the system matrix by means of a vari- able change from the original vector X( t ) to a new set of variables Y(t) = V, . X(t). The evolution of the diagonal- ised variables Y is given by

y"t) = &(t ) . KY- 2 1 = v, . hT,x (7)

(t> ( 8 )

The values of Y at two consecutive grid points t and t + h are related through the exponential of matrix A;:

Yt ( t + h) = ,At (t+ h) . KY = &it . . K y = (:Ai h . Yt

Ai is diagonal and thus e";' is computed as the exponential of n scalars. Once the transient term has been evaluated at the first time point of the stage, all subsequent values at the remaining time points within the same conduction period can be evaluated as a product of M scalars using eqn. 8 6ecause grid points are equally spaced.

The eigensystem of A ; (matrices A,, V, and V;') is computed only once throughout the entire steady-state algorithm. This computation is performed the first time a particular switch combination occurs. After computation these matrices are saved for reuse.

IEE Proc.-Electr. Power Appl . . Vol. 148. No. 2, March 2001

3.1.2 Steady-state term evaluation: The problem of evaluating the steady-state term for a given stage is the problem of finding the steady-state response of a periodi- cally excited linear network. This is a well known problem, which can be efficiently solved in the frequency domain by solving the frequency-domain version of eqn. 3 for all frequencies present in the spectrum of the exciting sources [19]:

(9) I represents the identity matrix of size n x n. Using the diagonal cigensystem of A to solve for the diagonalised variables Y rather than for the original state variables X notably increases efficiency because the new state matrix AY is diagonal and thus [iMil- Aiy]-' can be computed via the inverse of n scalars. This can be done at almost no extra cost because the eigensystem of A has already been computed for the evaluation of the transient term. The required time domain values F ( t ) at grid points are finally obtained by evaluation in the time domain and addition of all contributions at different frequencies.

X ( j w ) = [ j w I - A,]-' * [B, + jwCi] . S ( ~ W )

3.2 Advancing evaluation across switchings The change of status in a switch, and thus the end of stage i, is defined by a linear condition over a combination of voltages andor currents of the circuit that ceases to be met. This condition is the stage inequality that is met for all time instants within the same stage. The stage inequality has, in general, the following expression:

D, . X ( t ) + E, . S ( t ) + G, > 0 (10) Rajagopalan [I71 describes a systematic method to compute the output matrices. These output matrices allow any cur- rent or voltage in the circuit to be obtained as a linear com- bination of the state (X) vector and the sources (S) vector. This method is used to compute matrices D,E and G for any given switch combination.

The diagonal variables Y, rather than the original state variables X have been computed, so eqn. 10 must be modi- fied to refer to the diagonal variables Y:

D,.Y-'.Y(t) + E , . S ( t ) + G , > 0 (11)

If eqn. 11 is met at one point of the time grid but not at the next one, both points cannot belong to the same stage. The switch combination and the value of the state variables at the next point of the grid need to be determined. In addi- tion, all stages in between need to be accurately defined. Once it has been determined which semiconductor switch- ing determines the end of the current stage, the problem of finding the switching instant is trivial. It is not trivial, how- ever, to determine which semiconductor defines the end of the stage. This is specially difficult if the time step h is large relative to switching intervals and more than one switching exists between two consecutive grid points.

There can be any number of switchings between two consecutive points of the gnd, and it is not known a priori which switch changes state first. The time instants at which the stage inequation is evaluated are adjusted such that the earliest switchng is identified in most cases, and new valid switch combinations are determined before and after the switch change. The problem of advancing the evaluation across switchings is thus narrowed to that of finding an individual zero crossing in a given interval.

4 The sensitivity matrix

In this Section, the expression of the sensitivity matrix of the TPBV problem defined in eqn. 2 will be analytically

derived using a different formulation of the steady-state problem. First, this formulation will be defined, then the Jacobian matrix of this formulation will be derived and finally both Jacobians will be related.

Consider a formulation of the steady-state problem whose unknowns are the state vectors at all switching instants ( X I , X,, ..., X,J, the switching instants themselves (TI , T2, ..., Tns-,) and the initial and final states (Xi = X,,). T h s formulation consists of two subsets of equations.

The first subset corresponds to the relationship that ties the state variables at both ends of each stage. This relation- ship is the solution to the differential equations of the system for each stage. This solution, comprehensive of transient and steady-state terms, is denoted here f ; to preserve clarity:

= f i ( X % - l , Z , ~ , - l ) (12)

At the solution, the final state is equal to the initial state and thus they are both denoted Xo. The first equation cor- responds to the mismatch between the initial and final states; the boundary constrain mismatch Y(Xo). This first subset is constituted by ns vector equations or n x ns scalar equations:

-xo + fns(Xns-1,Tns-1) = o -x1 + f l (X0 ,T l ) = o -x2 + f2(Xl,T2,Tl) = o -x3 + f 3 ( x 2 , T 3 , T 2 ) = o

-x, + f , ( X , - l , ~ t , T , - l ) = 0

-Xns--l + fns-1(Xns-2, Tns, Tns-1) = 0 (13)

. . .

The second subset corresponds to the conditions for a par- ticular time instant to be a stage boundary: the stage ine- quality for the switch that changes its state at a boundary must be null. This subset is constituted by y1s - 1 scalar equations :

= o = o

d l X 1 + elS(T1) - g1 daXz + e2S(T2) - 9 2

. . . diXi + eiS(T,) - g, = o . . . dns--lXn,-l + ens-lS(Tns-l) - gns--l = 0

(14) where dj, e; and gi are the rows of matrices D, E and G in eqn. 10 that correspond to the switch whose change of state determines the end of stage i and the beginning of stage i + 1.

Denoting the vector of unknowns U and the mismatch function 4> the complete set of eqns. 13 and 14 can be . formulated as:

Vector U is composed of 11s state vectors 4. of size n x 1 and ns - 1 time instants T, (scalar unknowns):

(16) The length of vector U is nu = ns + ns - 1. The TPBV

@ ( U ) = 0 (15)

U = [Xo, Xi, . . . , Xns-i, Ti, Ti, . . . , Z-i]

IEE Proc.-Electr. Power Appl. , Vol. 148. No. 2, Murch 2001 241

formulation, considering only the initial and final condi- tions, is equivalent and can be derived from the extended formulation by the elimination of all intermediate variables.

The number of equations, and thus the size of the Jaco- bian matrix depends on the number of stages. As the TPBV formulation is equivalent to the extended one, the Newton update for the extended formulation is also the Newton update for the TPBV formulation.

The structure of the Jacobian JQ of function @ is repre- sented below. It is very sparse because each of the nus- match equations depends only on a few neighbouring variables:

Ja =

- I 7Y-l 71’ -I

r; -1

-I r,ns-2

(17) In eqn. 17 only non-zero entries have been represented and I stands for an identity matrix. JQ is divided into four square submatrices corresponding to the four combinations of the two types of variables and two types of mismatch equations that form its origins. Each of these submatrices has a band structure with five types of relevant (non-zero, non-one) entries. These relevant entries have been denoted zl ... z,. The algebraic expression of these non-zero entries is obtained by differentiating eqns. 13 and 14

All matrix exponentials in eqn. 18 can be eficiently computed using the diagonal matrix of eigenvalues. The eigensystem of A has already been computed for the time evaluation and thus the sensitivity matrix is obtained at virtually no extra cost. The derivative of the steady state of the Ith stage, Xim(Ti), is computed using the frequency- domain solution that has been saved during the grid evalu- ation. This solution is differentiated in the frequency domain and then evaluated at the required switching instant. The same approach is used to evaluate S(T,J. After applying the grid evaluation algorithm across the period T, the Jacobian J@( U), and the mismatches a( U) are computed for the given initial condition.

Consider now the Newton iterative solution of the extended formulation:

248

u p + l = u p - J ; y u p ) q U p ) (19) The switching instants have been solved for within the desired tolerance and thus all mismatches, except those that correspond to the equation for Xo in eqn. 13, are negligible. The first n components of mismatch @ correspond to the boundary constraint mismatch. i.e. the difference between the initial and final states, i.e. the TPBV formulation mis- match Y The first n equations of eqn. 19 can be separated from the rest, splitting the inverse of the Jacobian matrix in four submatrices:

The first n components of vector U (U,), correspond to the initial state Xo thus, extracting the first n equations from eqn. 20:

X:” = X,P - JG, i , a (Up) . Q ( U P ) (21) Finally, identifying terms with the standard TPBV formula- tion (eqn. 2) allows both sensitivity matrices to be related:

The update, not the Jacobian, is required, thus instead of computing the matrix inverse, only the solution of an n x n linear system by Gaussian elimination needs to be com- puted. Using sparse matrix techniques to solve this linear system notably reduces computer load. The analytically derived Jacobian matrix J@ has been checked against a numerical approximation to validate it.

4. I Solving the steady-state problem Once the Newton update for the initial state has been com- puted two different approaches can be followed:

Update the initial state only (X,) discarding the other val- ues that are obtained (the rest of state variable vectors at all switchmg instants and the switching instants).

Update all the variables involved in the extended formula- tion. The optimal approach is to combine both. Until the steady-state stage sequence is known the former approach should be used. Once it is known the latter (faster) approach can be used. In the first iterations the initial state is, in general, far from the steady-state, so it will be changed substantially and thus the stage sequence may change. If the succession of stages is not the steady-state one, then the problem @(U) = 0 is not properly formulated and thus its solution, if it exists, is meaningless.

The problem of determining when the correct sequence has been reached is not trivial, unless previous information on the response of the circuit is available. The following approach is used: 1. The first two iterations are computed using the TPBV formulation. Ths has heuristically proved to be more effi- cient for common power electronic circuits. 2. If the maximum relative steady-state mismatch is less than a predetennined value, the extended solution is solved alone saving the first state vector. 3. If the problem diverges, which is likely when the switch- ing sequence is incorrect, the saved state vector is used to proceed with another TPBV update.

IEE Proc -Electr Power A p p l , Vol 148, No 2, March 2001

4. Once the extended formulation converges, the grid evalu- ation algorithm is run again at least once. This is a check against mistaking a meaningless solution for the correct one. For a six-pulse rectifier typically two grid evaluation itera- tions are required to reach the correct steady-state stage sequence.

5 Simulation results

5. I Voltage controlled switch The circuit represented in Fig. 1 was proposed [13] as an example of a circuit that cannot be solved by the standard algorithms in [l l , IS]. The switch is in position A until the capacitor charges to the constant voltage Eref = 3V at which time the switch moves to position B where it remains until the start of the next period. This is modelled in PWiseSS by using two switches. Despite the large (1OI2) on/ off resistance ratio of the switches, an accurate solution was attained withn two iterations.

0

'IO pF T

" Fig. 1 Voltage controlled switch

Table 1: Voltage controlled switch simulation results

Iteration Error, p.u. Type Flops

- N = 5 1 1 .o 3526

2 2.02 10-14 red. 597

3 0 red. . 413 N=50 1 1 .o - 10,897

2 2.9347 x red. 1001

3 1.6304 x red. 873 4 1.6304 x ext. 230

5 1.6304 x ext. 230

'red.' indicates only initial state updated and 'ext.' complete vector U updated

3.5 r

L 0

U 0

U

c .- a

0.5 " O I / o.oK I I I

0.0 0.2 0.4 0.6 0.8 1.0 time,s XIO-3

Fig. 2 Cupucitor voltage for N = 5

Assuming no prior knowledge of the response is availa- ble, the initial state vector (capacitor voltage) is set to zero. Simulation results for N = S grid points and N = 50 grid points are presented in Table 1 and the capacitor voltage evolution for N = S is plotted in Fig. 2. The computer load

IEE Proc.-Electr. Power Appl. , Vol. 148, No. 2, March 2001

depends upon the number of points per period, although the solution attained is almost the same in both cases and can be evaluated oMine to obtain the required time values or frequency-domain representation. The minimum attain- able relative error depends on the number of grid points but it is negligible in both cases. When using 50 grid points the accuracy of the solution could not be improved beyond 1.6 x 10-l6. The error column in the simulation results table (Table 1) represents the relative error to the exact solution. The computer load is smaller (230kflops) for the extended formulation iterations than for the TPBV formulation iter- ations (873kflops). This example shows how the hybrid TPBV-extended approach provides an advantage over using the TPBV formulation alone.

5.2 Six-pulse rectifier Simulation results for the six-pulse rectifier represented in Fig. 3 are presented in Table 2 for N = 40 grid points. Using only TPBV updates resulted in a 7% increase in the total computer load, as shown in the three rightmost col- umns of Table 2. The diodes are modelled as binary resis- tors with Ron = 10mQ, Roff = 1 MQ and a forward voltage drop of 0.7V. Null initial conditions are used and the rela- tive steady-state error is assessed using the average value of the state vector at the switching instants. The waveforms for the AC current and DC voltage across the load resistor are plotted in Fig. 4.

10 mH

Fig. 3 Six-puke r e c t f b with snubbers and LCfilter

U 6

0.000 0.OOL 0.008 0.012 0.016 0.020 time,s

Fig. 4 AC (current) and DC (voltage) in the rectfk

6 Conclusions

An extended shooting fast steady-state algorithm for piece- wise linear circuits is presented. It contributes the analytical derivation of the sensitivity (Jacobian) matrix of the final state after one period to the initial state. This matrix is used to attain robust and fast convergence to the steady-state. The algorithm computes state transition matrices for the linear stages as needed using a systematic, computer- oriented method [ 171. Using eigenanalysis, the solution to

249

Table 2: Six-pulse rectifier simulation results

Iteration Error, p.u. Type kflops Error, p.u. Type kflops

- - N=40 1 2.93 930.3 2.93 930.3 2 7.09 x IO-I red. 925.7 7.09 x IO-’ red. 925.7 3 8.41 x IO-’ red. 470.7 8.41 x IO-’ red. 470.7 4 1.57 x red. 467.4 1.57 x red. 467.4 5 2.53 x IO-’ ext. 355.2 3.97 x red. 467.2 6 3.64 x IO-” ext. 355.0 8.73 x red. 467.2 7 . 1.44~ red. 112.0 1.24~ red. 111.5

‘red.‘ indicates only initial state updated and ‘ext.’ complete vector U updated

the system differential equations is evaluated at regularly spaced time instants. The use of eigenanalysis and the uniformity of the grid points at which the solution is evaluated produces a significant increase in eficiency over standard, direct evaluation. Switchng instants are solved for explicitly, and no previous knowledge of the operation sequence of the circuit is required.

The algorithm initially uses a Newton-based shooting two-point boundary value approach to get near to the correct solution. The final approach to the steady-state solution is achieved using an extended formulation, that updates not just initial and final state variables, but the intermediate switching instants, and the state variables at those instants. Providing the correct switching sequence has been obtained, the second stage provides fast and reliable convergence.

The use of numerical methods is narrowed to solving three well framed mathematical problems; finding the eigenvalues and eigenvectors of a positive definite matrix, solving a linear system by Gaussian elimination and invert- ing a matrix.

The solution attained by the algorithm comprises the ini- tial conditions, the switching instants and the state variables at those instants, and a set of equations that can be used to specify the circuit variables at any point in between the switching instants. The equation set must be evaluated to compute the value of any circuit variable at any desired time instant. However, ths is extremely fast, and the time spacing between data points has little impact on the overall computer load.

7 Acknowledgments

The support and advice received from Bruce C. Smith, Prof. Jos Arrillaga arid Simon Todd from the University of Canterbury is greatly appreciated. The first author spent 1997 at the University of Canterbury supported by a schol- arship from the Ministerio de Educaci6n y Cultura Espa- no1 (Spanish Ministry of Education and Science). This research has been possible thanks to project PB95-0281 of the same Ministry.

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250 IEE Proc.-Eleclr. Power Appl., Vol. 148, No. 2, Mnrch 2001