A Software Program for Modeling, Simulation, and ...

A Computer Simulation Program for Modeling, Analysis, and Space Vector Control of a New Family of Matrix Converters Osama A. Al-Naseem Electrical Engineering Department Kuwait University [email protected]

Abstract-A computer simulation program is developed to provide insight about the operation and control of a new family of matrix converters. Unlike the conventional matrix converter, the new matrix converter has 19683 possible switching combinations. Hence, computer simulation becomes a reasonable choice to analyze such a complex converter. A state-space model of the converter is incorporated within the simulator’s code. Space vector modulation is used to control the converter’s input and output voltages. Simulation results provide essential first-hand information for analyzing and understanding the operation and control of this family of matrix converters.

I. INTRODUCTION Ever since computers were in their early development stages, scientists from various fields showed interest in using computers as an aid for performing difficult and tedious tasks. Within the past three decades, interest in computer simulation grew extensively. Nowadays, most, if not all, sophisticated systems are verified using some method of computer simulation prior to final construction. This paper presents a computer simulation program that is developed to provide insight about the operation and control of a new family of matrix converters [1,2]. The program provides the user the freedom of entering different operating points for converter simulation. Additionally, the program outputs several plots and data files that illustrate the converter’s salient features. Based on these output plots and data files, it is possible to analyze the new converter and understand its operation and control limits. In this paper, a concise background section is presented describing the new family of matrix converters which has been simulated. The development of the simulation program is included in a another section. Future work and conclusions are included in the final two sections. II. BACKGROUND MATERIAL

Robert W. Erickson Colorado Power Electronics Center University of Colorado Boulder, CO 80309-0425, USA [email protected] the efficiency of a power electronic converter for variable-speed wind generators operating at light load. Efficient, high performance power electronic converters that can maximize operation efficiency at light load (i.e. at low wind speeds) are necessary in variable-speed wind generation applications [3]. Two notions influenced the advent of this converter: 1. The conventional matrix converter [4-7] 2. Multilevel converter techniques [8-13] The idea of a conventional matrix converter is based on direct ac-ac conversion. Hence, there is no need for energy-storage devices. Figure 1 shows a conventional matrix converter. There are 27 possible switching combinations with only three switch blocks are ON at any time. Output voltages are created from chopping input voltages, and input currents are created from chopping output currents [4-7]. N

IB

IA VA

IC VB

VC Va

S’Aa

S’Ab S’Ac

S’Ba

S’Bb S’Bc

S’Ca Vb S’Cb Vc S’Cc

Ia La Ib Lb Ic Lc

Nine four-quadrant switch blocks

Fig. 1. Conventional matrix converter.

The second notion, multilevel converter techniques, provides a way to increase the power rating of a converter by cascading semiconductor devices of smaller rating. An attractive approach is to incorporated multilevel converter techniques along with the conventional matrix converter theory [8-13]. Figure 2 shows a PWM waveform generated by a two-level converter and Fig.3 shows a PWM waveform generated by a five-level converter. L evel 1

v(t)

t

A. Motivation for a New Family of Matrix Converters The new family of matrix converters was proposed for use in variable-speed wind power applications. The major goal was to introduce a novel approach to improve

n

L evel 2

Fig. 2. Two-level PWM waveform.

This work was supported in part by the US DOE National Renewable Energy Laboratory under contract no. XCX-9-29204-01

UTILITY SIDE

v(t)

GENERATOR SIDE

Level 1 Level 2

Branch Aa

Phase A

Level 3

-240V+ Branch Aa

t

Level 4

(a)

Level 5

VCA = 0V Phase B

Phase b

VBC = 0V

Fig. 3. Five-level PWM waveform.

From Fig.2 and Fig.3, it can be seen that with higher number of levels, a waveform is closer to a sinusoid. In other words, multilevel converters produce waveforms with less harmonic content. Other advantages and disadvantages of multilevel converters are listed in Table I. A thorough investigation of two-level converter features compared to those of a three-level converters is available in [14]. TABLE I Advantages & Disadvantages of Multilevel Converters Advantages Disadvantages Increased Power Rating More Semiconductor Devices Reduced Switching Loss More Conduction Loss Reduced Magnetics Size Complex Control Reduced Harmonic Content Capacitor Voltage Unbalance Improved Efficiency for VariableSpeed Wind Generators Under Light Load

(b)

VCA = 0V

Phase C

Branch Cc -240V+ Branch Cc

Phase A

-240V+

Phase a

Phase B

Phase b

VBC = 0V

Vbc = 240V

-240V+

Phase A

Phase a Vab = 0V

VAB = 0V VCA = 0V Phase B VBC = 0V

Phase b

Vca = -480V

Vbc = 480V

Phase C

Phase c

+240V-

Fig. 5. Three different device switching combinations for one case of branch connections

C. Converter State-Space Model

Phase A

Figure 4 shows the basic configuration of the new family of matrix converters. There are nine switch cells. Each switch cell is of the bidirectional full-bridge configuration and can assume one out of four possible switch states. The new matrix converter has 19683 possible switching combinations [1,2]. Only five switch cells in the basic configuration may be used at any valid switching combination. Assuming each switch cell capacitor maintains 240Vdc, Fig. 5 shows three different switching combinations in which branches Aa, Ba, Ca, Cb, and Cc are connected while branches Ab, Ac, Bb, and Bc are disconnected. Note that branch Aa uses switch cell SAa, branch Ba uses switch cell SBa, etc. The converter has 81 possible cases of branch connections, with 243 possible device switching combination per branch connection. Proper operation of this converter requires converting three-phase variable-voltage variablefrequency waveforms into three-phase fixed-voltage, fixed-frequency waveforms while maintaining relatively the same dc voltage level across each switch cell capacitor. Further details are available in [1,2].

GENERATOR SIDE

iA

iAa

-iAa iAb

iAc

-iBa

iBa Phase B

iB

iBc

Phase C

Phase a

-ib

Phase b

-ic

Phase c

-iCa -iAb

iBb

iCa

-ia

-iBb -iCb

iCb -iBc

-iAc

iC iCc

-iCc

Fig. 6. Phase-branch current relationship.

From Kirchhoff’s current law and Fig. 6, it is apparent that each phase current of the multilevel matrix converter is the sum of three currents of the branches connected to that same phase. Hence, the following equations may be deduced: i A = i Aa + i Ab + i Ac i B = i Ba + i Bb + i Bc i C = i Ca + i Cb + i Cc

(1)

i a = i Aa + i Ba + i Ca

s w itch ce ll # 1

i b = i Ab + i Bb + i Cb Ia

O u tp u t S id e

i c = i Ac + i Bc + i Cc

IA VA

Va Ib

IB N VB

Vb Ic Vc

Vca = -240V

Phase c

0V

UTILITY SIDE

n

Phase c

Vab = 0V

Phase C

(c)

Vca = 0V

Vbc = 0V

VAB = 0V

B. Converter Structure & Operation

In p u t S id e

Phase a Vab = 0V

VAB = 0V

IC VC

Fig. 4. Basic configuration of the new family of matrix converters.

Equation (1) may be written in matrix form as follows: i Aa  i  Ab i A  1 1 1 0 0 0 0 0 0     i   0 0 0 1 1 1 0 0 0  i Ac   B  i Ba   (2) i C  0 0 0 0 0 0 1 1 1       i Bb =    ia  1 0 0 1 0 0 1 0 0   i   ib   0 1 0 0 1 0 0 1 0   Bc     i Ca    0 0 1 0 0 1 0 0 1     i c   i Cb  i   Cc 

This work was supported in part by the US DOE National Renewable Energy Laboratory under contract no. XCX-9-29204-01

Observing the system described in Eq. (2), one can infer that this is multi-input multi-output (MIMO) system. Equation (2) is written in the following form:    

 Branch  Connection   Main Matrix

 Phase   =  Current    Matrix 

 Branch  Current   Main Matrix

   

(3)

The branch connection main matrix is a 6 by 9 scalar matrix. On the other hand, the branch current main matrix is a 1 by 9 vector matrix. Recall that the multilevel matrix converter operates with only five branches conducting at any particular instant. Hence for each combination of branch connection, the branch connection main matrix reduces to a 6 by 5 scalar matrix. Call this new matrix A. Likewise, the branch current main matrix reduces to a 5 by 1 vector. Call this new matrix x. Matrix x is the converter state vector. The phase current matrix is a 6 by 1 matrix. Since only five phase currents are independent quantities, the phase current matrix reduces to a 5 by 1 matrix. Call this new matrix u. Matrix u is the converter input vector. Similarly, matrix A is reduced to a 5 by 5 matrix to include only five independent phases instead of six. In state space modeling, a control system can be described by associating the system’s input with the system’s state vector as in the following state space equation: .

(4)

x = Ax + Bu .

However, x =0 since the branch currents which form the state vector are considered constant during any single device switching combination. Hence, the state space equation reduces to the following: 1 0  A x + B u = 0 where B = -  0  0  0

0

0

0

1 0

0 1

0 0

0 0

0 0

1 0

0 0  0  0 1 

(5)

A second state space equation can be derived using the relationship between switch cell capacitor voltages and branch currents. To derive this second equation consider the switch cell drawn in Fig. 7. Q1

D1

iXY

X

Q3

D3

D2 + Vcap D4 icap

Q2 iXY

Q4

Y

Fig. 7.Multilevel converter switch cell showing capacitor voltage and current

The current icap is the switch cell capacitor current. Using the instantaneous capacitor current icap(t), the instantaneous capacitor voltage vcap(t)can be described as follows: t (6) 1 v cap ( t ) =

C

∫ icap (τ ) d τ

+ v cap ( t 0 )

t0

where C is the capacitance, t0 is the initial time of applying current through the capacitor, and vcap(t0) is the initial capacitor voltage prior to applying current. In

steady-state the value of vcap(t) is Vcap. Assume that vcap(t0) = Vcap since the capacitor voltage must be regulated to be always constant. Furthermore, because the line frequency is very low compared the frequency of switching between different device switching combinations, Eq. (6) can be approximated as follows: 1  (7) v (t ) = i (t ) T +V cap

 C

cap

switch

 

cap

where Tswitch is the period of using one device switching combination. The capacitor current icap(t) can be written in terms of the branch current iXY(t) as follows: (8) i cap (t ) = i XY (t ) SC state where SCstate is the state of a switch cell defined as follows: SCstate = 1 when only D1 & D4 or only Q1 & Q4 are ON (State1) SCstate = -1 when only D2 & D3 or only Q2 & Q3 are ON (State2) SCstate = 0 when only D1 & Q2 or only Q3 & D4 or only Q1 & D2 or only D3 & Q4 are ON (State3) or all devices are OFF (State 4)

Hence in state space form the following equation may be written:  vcap1 (t )   SC state1 v   0 ( ) t  cap 2   1 vcap (t )  = Tswitch  0 3   C  vcap 4 (t )  0 vcap (t )   0 5  

0

0

0

SC state2 0 0

0 SC state3 0

0 0 SC state4

0

0

0

  i1 (t )   i (t )  2   i3 (t )     i4 (t ) SC state5  i5 (t )  0

0 0 0

(9) which is of the form: y = C x + Du

(10)

where D = 0

State vector x is the reduced 5 by 1 matrix version of the state vector of Eq. (2). Because the conducting branches change from one branch connection to another, it is more appropriate to label the capacitor voltages and capacitor currents as is shown in Eq. (9).

III. SIMULATOR DEVELOPMENT A. Motivation for a Computer Program Because of the large number of possible switching combinations and due to the extremely tedious task of analyzing and sorting the countless possible converter operating points by hand, computer simulation seemed an attractive approach to unwind the complexity of this new family of matrix converters. Initially, the main objective was to write a program to determine the possible 19683 device switching combinations, sort them, store them into a look-up table, and transform the voltages synthesized by the device switching combinations into a d-q space vector diagram. Later, more operations were incorporated in the simulation program. B. Simulation Algorithm The entire 81 cases of branch connections were entered into the simulation program code manually. Each case of branch connection was an independent subroutine that was called by the main program. A loop was incorporated within each subroutine to account for the

This work was supported in part by the US DOE National Renewable Energy Laboratory under contract no. XCX-9-29204-01

243 different device switching combinations per branch connection. After having determined the possible 19683 device switching combinations, the next objective was to apply three-phase ac voltages to the generator and utility sides of the multilevel matrix converter. The converter must synthesize the same voltages at its generator and utility sides. Therefore, additional code was added to the simulation program to control the converter by space vector modulation. The simulator algorithm for this phase is as follows: (1) Read the instantaneous generator and utility side voltages. The rate at which these voltages are read constitutes the switching frequency. (2) Perform d-q transformation of these voltages into a space vector diagram. Therefore, two sets of d-q coordinates are deduced; one for the generator side voltages and another for the utility side voltages [1,2,7,15]. (3) Space vectors duty ratios are derived for space vector modulation, SVM, control process [1,2,7,15]. (4) Each switching period is divided into several subintervals. (5) Search for a device switching combination to synthesize the generator and utility side waveforms for the first switching period subinterval. (6) Increment the time and search for a device switching combination to synthesize the generator and utility side waveforms for the next switching period subinterval until the switching period is complete. The number of switching period subintervals defines the resolution of the PWM duty ratios. More subintervals means more resolution. (7) Read another set of instantaneous generator and utility side voltages. Through space vector modulation, the converter is able to synthesize pulse-width modulated waveforms with fundamental components that match the applied ac voltages. The simulator is able to perform this task successfully and only use one branch connection which included only 243 device switching combinations. The next main task has been to operate the multilevel matrix converter while maintaining capacitor voltages at approximately 240V. Therefore, generator and utility side phase currents must be employed in this phase in order to sense how much charge is added or removed from capacitors during any device switching combination. Using the converter’s state space model discussed Section II-C is necessary in order to determine currents and voltages of switch cell capacitors. Each device combination has its own reduced state matrix A, as is explained in Section II-C. After knowing the phase currents, the branch currents may be obtained by finding the inverse of matrix A. Therefore, additional code was added to the simulation program to find the inverse of matrix A for each different branch connection. The simulator then searches the device switching combinations that produce the desired input and output voltages. Each searched device switching combination is then tested to know if it still can maintain the capacitor voltages within ±5% variation from the desired 240V. Knowing the amount of charge that may be added or

removed from a switch cell capacitor as a result of selecting the searched device switching combination is necessary prior to selecting that device switching combination. To determine capacitor voltage variations during every device switching combination, matrix C of Eq. (9) is used as is explained in Section II-C. If a searched device switching combination cannot maintain the capacitor voltages within the ±5% set limit, a search for another device switching combination is performed. If no adequate device switching combinations is found, the ±5% limit is incremented by one to ±6%. This process may repeat until the limit reaches ±12%. If an adequate device switching combination is found, the limit is decremented by one. This process repeats until the limit falls back to ±5%. If in this process the limit reaches ±12%, and still there are no valid device switching combinations, the simulator searches for any random device switching combination that can return the capacitor voltages to within their original ±5% limit. Although this random device switching combination does not synthesize the appropriate input and output voltages, its effect on distorting the synthesized waveforms is negligible since it occurs for an infinitesimal period of time. This is confirmed through harmonic analysis of the synthesized waveforms up to the 26th harmonic order. Furthermore, device gating signal on and off times during a single switching period were allowed (if necessary) to be divided into randomly non consecutive time intervals in order to add more flexibility in finding a device switching combination that produces the appropriate voltages and still regulate capacitor voltages. The key point is that the average of the produced pulse-width modulated voltage must be the same. C. Simulation Results The simulator uses space vector modulation, SVM, to control the converter’s input and output voltages. Beside controlling the input and output voltages, the dc voltage of each switch cell capacitor is maintained within a small threshold from its nominal value, 240V in this case. Figure 9 shows simulator-generated voltage space vector diagram showing all possible space vectors of the basic configuration of Fig.4 assuming each switch cell capacitor maintains a dc voltage of 240V. Figures 10- 13 show simulator generated plots at 5kHz switching frequency.

Fig. 8. Simulator-generated voltage space vector diagram This work was supported in part by the US DOE National Renewable Energy Laboratory under contract no. XCX-9-29204-01

IV. FUTURE WORK Future additions include the use of neural network control to optimize converter operation. Another addition is to incorporate actual semiconductor models in the program. Hence, loss calculation (switching losses, conduction losses, etc) may be added. Finally, multilevel operation may also be applied to increase the converter power rating. This may be done by using series connection of two or more switch cells per branch.

Fig. 9. Simulator-generated output (utility) side voltage and current waveforms @ 240V, 11A, 60Hz, unity power factor.

V. SUMMARY & CONCLUSION The developed simulation program was used as a first-hand tool to analyze the properties and operation of this family of matrix converters, prior to actual design and construction. It provided extensive data files and plots showing the salient features of the new family matrix converter. The program may also be used by power electronics students to learn about this matrix converter. REFERENCES [1]

[2] [3]

Fig. 10. Simulator-generated harmonic spectrum of output (utility) side line-to-line voltage VAB

[4]

[5] [6] [7]

[8]

[9] Fig. 12. Simulator-generated input (generator) side voltage and current waveforms @ 50V, 20Hz, unity power factor.

[10] [11]

[12] [13] [14]

[15] Fig. 13. Capacitor voltage variations (within +/- 12% of Vcap)

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This work was supported in part by the US DOE National Renewable Energy Laboratory under contract no. XCX-9-29204-01