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Two Predictive Control Techniques for OutputVoltage Control and Improvement of the Source
Currents in an Indirect Matrix ConverterM. Rivera∗, J. Rodriguez†, J. Espinoza‡, A. Olloqui§, P. Wheeler¶, P. Zanchetta¶, C. Baier∗ and J. Munoz∗
∗ Universidad de Talca, Curico, CHILE, Email: [email protected], [email protected], [email protected]† Universidad Tecnica Federico Santa Marıa, Valparaıso, CHILE, Email: [email protected]‡ Universidad de Concepcion, Concepcion, CHILE, Email: [email protected]
§Tecnologico de Monterrey, Monterrey, Nuevo Leon, MEXICO, Email: [email protected]¶ University of Nottingham, Nottingham, UK, Email: [email protected], [email protected]
Abstract—This paper presents and compares two strategiesto generate sinusoidal output voltage waveforms and unitarydisplacement power factor on the input side using predictivecontrol with an indirect matrix converter. These objectives areaccomplished using two different predictive control schemes onthe input side: minimization of instantaneous reactive power andimposed input sinusoidal currents. Predictive control calculatesthe future values of the variables to be controlled in order tochoose the switching state that produces the minimum errorbetween the variables and their references. Both methods havebeen tested and compared in simulation, obtaining sinusoidaloutput voltage and achieving unitary input displacement factor,with a THD of less than 2% for both the input currents and theoutput voltages.
Index Terms—AC-AC power conversion, Voltage control, Ma-trix converter, Predictive control.
NOMENCLATURE
is Source current [𝑖𝑠𝐴 𝑖𝑠𝐵 𝑖𝑠𝐶 ]𝑇
vs Source voltage [𝑣𝑠𝐴 𝑣𝑠𝐵 𝑣𝑠𝐶 ]𝑇
ii Input current [𝑖𝐴 𝑖𝐵 𝑖𝐶 ]𝑇
vi Input voltage [𝑣𝐴 𝑣𝐵 𝑣𝐶 ]𝑇
𝑖𝑑𝑐 dc-link current𝑣𝑑𝑐 dc-link voltagei Output current [𝑖𝑎 𝑖𝑏 𝑖𝑐]
𝑇
v Output voltage [𝑣𝑎 𝑣𝑏 𝑣𝑐]𝑇
io Load current [𝑖𝑜𝑎 𝑖𝑜𝑏 𝑖𝑜𝑐]𝑇
vo Load voltage [𝑣𝑜𝑎 𝑣𝑜𝑏 𝑣𝑜𝑐]𝑇
i∗s Source current reference [𝑖∗𝑠𝐴 𝑖∗𝑠𝐵 𝑖∗𝑠𝐶 ]𝑇
v∗o Load voltage reference [𝑣∗𝑎 𝑣∗𝑏 𝑣∗𝑐 ]
𝑇
𝐶𝑓 Input filter capacitor𝐿𝑓 Input filter inductor𝑅𝑓 Input filter resistor𝐶𝑓𝑓 Output filter capacitor𝐿𝑓𝑓 Output filter inductor𝑅𝑓𝑓 Output filter resistor𝑅𝐿 Load resistance𝐿𝐿 Load inductance
I. INTRODUCTION
The direct matrix converter (DMC) is an ac-ac converter thathas been intensively studied over the last twenty years [1].
Its main advantages are: (1) the absence of dc-link storageelements, leading to a simple and compact power circuit,(2) the output voltage can have an arbitrary frequency andmagnitude (limited to a maximum of 86.66% of the inputvoltage), (3) the displacement factor at input can be adjusted toany value, including unity and, (4) four quadrant operation isobtained, which means that the converter can regenerate power[1]. The absence of dc-link storage elements means that theconverter can be used under a wide range of environmentalconditions, such as low pressure, high or low temperatures,etc. This makes the DMC extremely attractive for militaryand aerospace applications where a high degree of reliabilityis needed.
A very attractive variation of the DMC is the indirect matrixconverter (IMC), which has almost the same characteristics ofthe DMC and a similar topology to the traditional back-to-backconverter, but has bidirectional switches in the rectifier stageand no capacitor storage element in the dc-link. Moreover,an IMC can operate with lower commutation losses thana DMC, due to the implementation of a special switchingscheme known as zero dc-link current commutation, whichalso provides a safer option by minimizing the possibility ofopen circuiting the load’s current path while switching [2], [3].
With the advent of more powerful and faster microcon-trollers a new control method, predictive control, has be-come a feasible option. This is a nonlinear control methodwhich appears very appealing because of the simplicity andintuitiveness of its concept [4]. It is easy to implement andto include nonlinear constrains [5]. Predictive control alsohas some advantages over traditional linear controllers andPWM modulators, such as fast dynamic response and thesimple inclusion of additional constrains [6]. Model Predictivecontrol is a particular kind of predictive control which cantake into account the discrete nature of the converter and itsdigital implementation [7], [8]. Different implementations ofthis method have been shown in the literature [9]–[15], manyof them taking into account the instantaneous input reactivepower minimization, but it has been shown that this solutiondoes not guarantee sinusoidal input currents in presence of a
978-1-4799-2399-1/14/$31.00 ©2014 IEEE 1420
𝑅𝑓
𝐿𝑓
𝐶𝑓
𝐺
i𝑖 i𝐴
𝐵
𝐶
𝑆𝑟1 𝑆𝑟3 𝑆𝑟5
𝑆𝑟4 𝑆𝑟6 𝑆𝑟2
𝑆𝑖1 𝑆𝑖3 𝑆𝑖5
𝑆𝑖4 𝑆𝑖6 𝑆𝑖2
𝑎
𝑏
𝑐
𝑣𝑑𝑐 > 0
𝑖𝑑𝑐
𝑅𝑓𝑓
𝐿𝑓𝑓
𝐶𝑓𝑓
𝑅𝐿
𝐿𝐿
v𝑠i𝑠
v𝑖 v v𝑜i𝑜
𝑛
Fig. 1. Three-leg indirect matrix converter with output filter.
distorted input voltage [12], [16], [17]. In order to solve thisproblem a new method has been recently proposed in [18]which guarantees, in a very efficient way, sinusoidal inputcurrents with any desired input displacement angle irrespectiveof the input voltage.
This paper proposes a predictive output voltage control foran indirect matrix converter with an output filter. The maincontribution of this work is to present two different controlstrategies to achieve sinusoidal current and minimum THDfor the input side of the converter and sinusoidal load voltage.The first alternative corresponds to the minimization of theinstantaneous reactive power. The second alternative is a directcontrol of the source currents which are handled by imposing asinusoidal current reference on the input side. The feasibilityand characteristics of both methods are demonstrated usingsimulation results.
II. INDIRECT MATRIX CONVERTER MODEL
The topology under consideration is shown in Fig. 1. TheIMC can be divided in two stages: the rectifier and the inverter.This characteristic becomes an advantage when using thezero dc-link current switching scheme, which allows a safeoperation of the converter and a reduction in the switchinglosses. In particular, the mathematical model of the rectifierstage has the input phase voltages 𝑣𝐴, 𝑣𝐵 and 𝑣𝐶 and dc-linkcurrent 𝑖𝑑𝑐 as inputs and the dc-link voltage 𝑣𝑑𝑐 and inputcurrents 𝑖𝐴, 𝑖𝐵 and 𝑖𝐶 , as output:
𝑣𝑑𝑐 =[𝑆𝑟1 − 𝑆𝑟4 𝑆𝑟3 − 𝑆𝑟6 𝑆𝑟5 − 𝑆𝑟2
]vi (1)
ii =
⎡⎣ 𝑆𝑟1 − 𝑆𝑟4
𝑆𝑟3 − 𝑆𝑟6
𝑆𝑟5 − 𝑆𝑟2
⎤⎦ 𝑖𝑑𝑐 (2)
The inputs and outputs of each stage are related by theirswitching states. For the inverter this relations involves theoutput currents i and dc-link voltage (inputs) and the dc-linkcurrent 𝑖𝑑𝑐 and the output voltage v (outputs). This can beseen in equations (3) and (4):
𝑖𝑑𝑐 =[𝑆𝑖1 𝑆𝑖3 𝑆𝑖5
]i (3)
v =
⎡⎣ 𝑆𝑖1 − 𝑆𝑖4
𝑆𝑖3 − 𝑆𝑖6
𝑆𝑖5 − 𝑆𝑖2
⎤⎦ 𝑣𝑑𝑐 (4)
Not all the possible switching states are allowed. There aresome constraints which are mandatory for the safe operationof the converter:
∙ The input phases of the rectifier stage cannot be shortcircuited, thus only nine valid rectifier states can be used.
∙ The output phases of the inverter stage cannot be opencircuited thus, only eight inverter states are allowed.
Additionally, there must be a positive dc-link voltage inorder for the inverter switches to be able to commutate.Therefore, the valid rectifier states are reduced to only three atany instant, and the whole IMC has just twenty-four possiblevalid states.
III. CONTINUOUS TIME MODELS OF FILTERS AND LOAD
A. Input filter model
In order to prevent overvoltages and high frequency dis-tortion in the source current, an input filter is needed. Bymatching the filter poles to certain frequencies, a variety oftransfer functions can be obtained. Fig 1 shows a second-orderlow-pass input filter. Direct observation of the filter at the lineside of the converter shown in (1) establishes the followingrelations:
𝑑is𝑑𝑡
=1
𝐿𝑓(vs − vi)− 𝑅𝑓
𝐿𝑓is (5)
𝑑vi
𝑑𝑡=
1
𝐶𝑓(is − ii) (6)
B. Output filter and load models
In order to have a sinusoidal load voltage waveform, a low-pass output filter is needed. Fig. 1 shows the topology of thefilter used for each output phase of the IMC. The dynamicmodel of the filter is obtained by applying Kirchoff ’s currentand voltage laws on the output side of the circuit shown inFig. 1 (neglecting the 𝑅𝑓𝑓 value:
𝐶𝑓𝑓𝑑vo
𝑑𝑡= i− io (7)
1421
vs
is 𝑅𝑓 𝐿𝑓 ii
𝐶𝑓vi
𝑅𝑓𝑓 𝐿𝑓𝑓
𝐶𝑓𝑓
𝑅𝐿 𝐿𝐿
vo
vovo
vo∗
vo𝑘+1
io
io
vs is vi
i
i𝑞∗𝑠
𝑞𝑘+1𝑠
Output LoadInput Filter IMC
Source Voltage
SwitchingSignals
Voltage
Reactive PowerReference Reference
Load Voltage
ReactivePower
Prediction Prediction
OutputCost
FunctionOptimization
1
3
33
33 33
3
33
3
3 3
2424
12
Fig. 2. Proposed predictive voltage control scheme with instantaneousreactive power minimization.
𝐿𝑓𝑓𝑑i
𝑑𝑡= v − vo (8)
By assuming a passive 𝑅𝐿𝐿𝐿 load:
vo = (𝐿𝐿𝑑io𝑑𝑡
+𝑅𝐿) (9)
These equations can then be used to implement the proposedalgorithms.
IV. PREDICTIVE VOLTAGE CONTROL WITH
INSTANTANEOUS REACTIVE INPUT POWER MINIMIZATION
A. Control strategy
The first predictive control scheme used in this paper isshown in Fig. 2. It uses a discrete prediction model to predictthe values of the controlled variables in the next sample time(𝑡𝑘+1). In the case of the output filter voltage vo, this is donefor vo(𝑘 + 1) and in the input side the prediction is done for𝑞𝑠(𝑘+1). These values are then used to evaluate a cost functionfor each of the valid switching states of the converter, whereeach prediction is compared with the respective references.Finally, the state which gives the lowest value of the costfunction is selected and then applied to the converter.
B. Prediction models
Equations (5) and (6) give us the basis for establishing acontinuous time state space model for the input filter:[
vi
is
]= A
[vi
is
]+B
[vs
ii
](10)
where,
A =
[0 1/𝐶𝑓
−1/𝐿𝑓 −𝑅𝑓/𝐿𝑓
]
B =
[0 −1/𝐶𝑓
1/𝐿𝑓 0
] (11)
The discrete-time state space model is determined as,[vi(𝑘 + 1)is(𝑘 + 1)
]= Φ
[vi(𝑘)is(𝑘)
]+ Γ
[vs(𝑘)ii(𝑘)
](12)
with,Φ = 𝑒A𝑇𝑠 , Γ = A−1(Φ− I2𝑥2)B (13)
where 𝑇𝑠 correspond to the sampling time. For the output filtermodel, the Euler method is used in order to obtain a discrete-system:
𝑑𝑥
𝑑𝑡≃ 𝑥(𝑘 + 1)− 𝑥(𝑘)
𝑇𝑠(14)
thus, equation (7) becomes:
𝐶𝑓𝑓vo(𝑘 + 1)− vo(𝑘)
𝑇𝑠= i(𝑘)− io(𝑘) (15)
Hence, the prediction model for the load voltages is given as:
vo(𝑘 + 1) = vo(𝑘) +𝑇𝑠
𝐶𝑓𝑓(i(𝑘)− io(𝑘)) (16)
The prediction model for the instantaneous reactive inputpower is obtained as:
𝑞𝑠(𝑘+1) = 𝑣𝑠𝛼(𝑘+1)𝑖𝑠𝛽(𝑘+1)−𝑣𝑠𝛽(𝑘+1)𝑖𝑠𝛼(𝑘+1) (17)
with 𝑣𝑠𝛼, 𝑣𝑠𝛽 , 𝑖𝑠𝛼 and 𝑖𝑠𝛽 being the source voltages andcurrents in 𝛼𝛽 coordinates in (𝑘 + 1), respectively. Note thatif 𝑇𝑠 is very small, it can be assumed that 𝑣𝑠(𝑘) ≈ 𝑣𝑠(𝑘+1).
C. Cost function definition
The error between the predicted load voltage and its refer-ences can be expressed as:
△𝑣𝑜(𝑘 + 1) = ∣𝑣∗𝑜𝑎 − 𝑣𝑜𝑎∣+ ∣𝑣∗𝑜𝑏 − 𝑣𝑜𝑏∣+ ∣𝑣∗𝑜𝑐 − 𝑣𝑜𝑐∣ (18)
The term that minimizes the instantaneous reactive power isgiven by:
△𝑞𝑠(𝑘 + 1) = 𝑞∗𝑠 − (𝑣𝑠𝛼𝑖𝑠𝛽 − 𝑣𝑠𝛽𝑖𝑠𝛼) (19)
with 𝑞∗𝑠 , the instantaneous reactive power reference and 𝑣𝑠𝛼,𝑣𝑠𝛽 , 𝑖𝑠𝛼 and 𝑖𝑠𝛽 being the source voltages and currents in 𝛼𝛽coordinates, respectively.
The cost function used to validate this control scheme inthis paper is:
𝑔 = △𝑣𝑜(𝑘 + 1)2 + 𝜆𝑞△𝑞𝑠(𝑘 + 1)2 (20)
which allows for control of the load voltages and the mini-mization of the instantaneous reactive power on the input side.𝜆𝑞 is a weighting factor which is considered to give more orless priority to the control on the input side.
V. PREDICTIVE VOLTAGE CONTROL WITH IMPOSED
SINUSOIDAL SOURCE CURRENTS
A. Control strategy
The control scheme for voltage control with imposed sinu-soidal source current is shown in Fig. 3. In this case, the blockthat predicts the behaviour of the instantaneous reactive inputpower is replaced by a prediction model for the source currentswithout the need for new measurements. The prediction of thesource current is given by equation (12).
The error between the reference and predicted value of thesource current can be expressed as,
△𝑖𝑠(𝑘 + 1) = ∣𝑖∗𝑠𝛼 − 𝑖𝑠𝛼∣+ ∣𝑖∗𝑠𝛽 − 𝑖𝑠𝛽 ∣ (21)
1422
00 0
vs
is 𝑅𝑓 𝐿𝑓 ii
𝐶𝑓vi
𝑅𝑓𝑓 𝐿𝑓𝑓
𝐶𝑓𝑓
𝑅𝐿 𝐿𝐿
vo
vovo
io∗
vo𝑘+1
io
io
vs
vs
vs is vi
i
i
𝑖𝑘+1𝑠
Output LoadInput Filter IMC
Source Voltage
SwitchingSignals
VoltageCurrent
ReferenceLoad Voltage
Prediction Prediction
OutputCost
FunctionOptimization
3 3
3
3
33
33 33
3
33
3
3 3
24 24
12
Input
PLL eq.(22)
eq.(21)
𝜙 𝜃𝐼𝑠
𝐼𝑠
𝑖∗𝑠
Fig. 3. Proposed predictive voltage control scheme with imposed sourcecurrent control.
where, 𝑖∗𝑠𝛼 and 𝑖∗𝑠𝛽 correspond to the source current referencesand 𝑖𝑠𝛼 and 𝑖𝑠𝛽 are the source current predictions in sample𝑘 + 1.
Finally, the cost function in this case is defined as:
𝑔 = △𝑣𝑜(𝑘 + 1) + 𝛾𝑖△𝑖𝑠(𝑘 + 1) (22)
where 𝛾𝑖 is a weighting factor. Noting that 𝑔 = 0 (for anarbitrary value of 𝛾𝑖) gives perfect tracking of the load voltageand source currents, then by minimizing 𝑔, the optimumvalue for commutation state is guaranteed. In practice, withan appropriate selection of the weighting factor 𝛾𝑖, a giventotal harmonic distortion (THD) of the input and outputcurrents is obtained. The principal method for selecting theweighting factor and analysing the performance of the systemis presented in [19], where at first it is imposed equal to zeroin order to prioritize the control of the load voltage and later itis increased slowly aiming to obtain minimal THD of sourcecurrents and load voltages.
B. Determination of the input current’s reference amplitude
In order to set an appropriate reference for the input current,it is necessary at first to determine its amplitude. This canbe done by noting that the power at the input of the matrixconverter 𝑃𝑖, multiplied by the efficiency of the converter 𝜂,must be equal to the power at the output 𝑃𝑜. From figure 3,it is possible to show that the amplitude of the voltage in onephase of the input filter capacitor is:
𝑉𝑖 =(𝑉𝑠 −𝑅𝑓𝐼
∗𝑠 )
1− 4𝜋2𝑓2𝑖 𝐶𝑓𝐿𝑓
(23)
where 𝑓𝑖 is the fundamental frequency of the input voltageand 𝐼∗𝑠 is the amplitude of the input current reference. Then,the active power at the input of the converter is:
𝑃𝑖 = 𝑅𝑒 {𝑉𝑖𝐼𝑖} =3(𝑉𝑠 −𝑅𝑓𝐼𝑠)𝐼
∗𝑠
1− 4𝜋2𝑓2𝑖 𝐶𝑓𝐿𝑓
(24)
and the load’s active power is:
𝑃𝑜 = 3𝑅𝐿𝐼∗2𝑜 (25)
where 𝐼∗𝑜 is the amplitude reference for the output currentfundamental frequency. This current is obtained from therelationship relative to the reference output voltage amplitudegiven by:
𝐼∗𝑜 =𝑉 ∗𝑜√
(2𝜋𝑓2𝑜𝐿𝐿)2 +𝑅2
𝐿
(26)
Equations (24) and (25) can be put together by consideringthe converter efficiency 𝜂:
𝑃𝑖𝜂 = 𝑃𝑜 (27)
By doing
𝜎 =1
1− 4𝜋2𝑓2𝐶𝑓𝐿𝑓(28)
replacing it in equation (24) and then using this expression in(27), together with (25):
𝐼∗𝑠𝜎(𝑉𝑠 −𝑅𝑓𝐼∗𝑠 )𝜂 = 𝑅𝐿𝐼
∗2𝑜 (29)
from which it is possible to get 𝐼∗𝑠 :
𝐼∗𝑠 =−𝜎𝑉𝑠 ±
√(𝜎𝑉𝑠)2 − 4𝜎𝑅𝑓𝑅𝐿𝐼∗2𝑜 /𝜂
−2𝜎𝑅𝑓(30)
As it can be seen, the amplitude value for the input currentsreferences can be obtained as a function of the converterefficiency, the input filter’s parameters and the amplitude ofthe fundamental component of the input voltage and the outputcurrent reference. Finally, the input current references are:
𝑖∗𝑠𝐴 = 𝐼∗𝑠 sin(𝑤𝑠𝑡+ 𝜃)𝑖∗𝑠𝐵 = 𝐼∗𝑠 sin(𝑤𝑠𝑡− 2𝜋/3 + 𝜃)𝑖∗𝑠𝐶 = 𝐼∗𝑠 sin(𝑤𝑠𝑡+ 2𝜋/3 + 𝜃)
(31)
where 𝜃 is the phase angle of the source current.
VI. SIMULATION RESULTS
A Matlab-Simulink model of an indirect matrix converterhas been used to test the control methods mentioned above.Table I, shows the parameters used in the simulations.
TABLE ISIMULATION PARAMETERS
Variables Description Value
𝑇𝑠 Sampling time 10 [𝜇s]𝑉𝑠 Supply phase voltage 310 [V]𝑓𝑠 Supply frequency 50 [Hz]𝐿𝑓 Input filter inductance 400 [𝜇H]𝐶𝑓 Input filter capacitance 21 [𝜇F]𝑅𝑓 Input filter resistance 1 [Ω]𝐿𝑓𝑓 Output filter inductance 400 [𝜇H]𝐶𝑓𝑓 Output filter capacitance 21 [𝜇F]𝑅𝐿 Load resistance 5 [Ω]𝐿𝐿 Load inductance 15 [mH]𝑓𝑜 Output frequency 400 [Hz]𝜆𝑖 Weighting factor 7000𝜆𝑞 Weighting factor 0.08𝑉 ∗𝑜 Output Voltage reference 0.866𝑉𝑠 [V]
1423
a)
b)
c)
Time [s]0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
-2
0
2
-2
0
2
-2
0
2
Fig. 4. Simulation results of output voltage control with instantaneous inputreactive power minimization: a) source voltage 𝑣𝑠𝐴[V/150] and source current𝑖𝑠𝐴[A]; b) source voltage 𝑣𝑠𝐵[V/150] and source current 𝑖𝑠𝐵[A]; c) sourcevoltage 𝑣𝑠𝐶 [V/150] and source current 𝑖𝑠𝐶 [A].
a)
b)
c)
Time [s]0.244 0.246 0.248 0.25 0.252 0.254 0.256
0.244 0.246 0.248 0.25 0.252 0.254 0.256
0.244 0.246 0.248 0.25 0.252 0.254 0.256
-10
0
10
-2000
200
-500
0
500
Fig. 5. Simulation results of output voltage control with instantaneous inputreactive power minimization: a) output voltage 𝑣𝑎[V] of the indirect matrixconverter; b) load voltage references v∗
o[V] and measured value vo[V]; c)load currents io[A].
A. Predictive voltage control with instantaneous reactivepower minimization
By minimizing the reactive power, this method forces thesystem to operate with a unity input displacement factor, whichis extremely convenient, for most industrial applications. Thecost function considered in this case corresponds to (20) andthe weighting factor value is 𝜆𝑞=0.08.
As observed in Fig. 4, sinusoidal source currents can beobtained in phase with their respective source voltages, witha low harmonic distortion. The average THD value betweenthe three source currents is 𝑇𝐻𝐷𝑖=8.49%. This has beenaccomplished just including a term in the cost function thatreduces the instantaneous input reactive power of the system.
In Fig. 5, simulation results for the output side are presented.
a)
b)
c)
Harmonic [Hz]0 100 200 300 400 500 600 700 800 900 1000
0 100 200 300 400 500 600 700 800 900 1000
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
0
0.5
1
0
0.5
1
Fig. 6. Spectra analysis of main variables related to the output voltage controlwith instantaneous input reactive power minimization: a) source current 𝑖𝑠𝐴;b) output voltage 𝑣𝑎 of the indirect matrix converter; c) load voltage 𝑣𝑜𝑎.
a)
b)
c)
Time [s]0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
-2
0
2
-2
0
2
-2
0
2
Fig. 7. Simulation results of output voltage control with imposed sinusoidalsource currents: a) source voltage 𝑣𝑠𝐴[V/150] and source current 𝑖𝑠𝐴[A];b) source voltage 𝑣𝑠𝐵[V/150] and source current 𝑖𝑠𝐵[A]; c) source voltage𝑣𝑠𝐶 [V/150] and source current 𝑖𝑠𝐶 [A].
Fig. 5a, shows the output voltage of the indirect matrixconverter 𝑣𝑎 where it is possible see the PWM waveform dueto the commutation of the switches, with an average THDvalue of 𝑇𝐻𝐷𝑣=171.45%. Fig. 5b shows the controlled outputvoltages vo and their references v∗
o. A very good tracking ofthe load voltages to their respective references is obtained withan average THD value of 𝑇𝐻𝐷𝑣𝑜
=1.58%. Finally, sinusoidalload currents are obtained, as depicted in Fig. 5c, due to thenature of the applied load.
In Fig. 6 spectra analysis for the main variables involvesin the control strategy are showed, considering only thefrequencies up to 1kHz. As shown in this figure, a very lowharmonic distortion is observed in the source currents andload voltages. As expected a high THD value is observed forthe output voltage of the indirect matrix converter due to the
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a)
b)
c)
Time [s]0.244 0.246 0.248 0.25 0.252 0.254 0.256
0.244 0.246 0.248 0.25 0.252 0.254 0.256
0.244 0.246 0.248 0.25 0.252 0.254 0.256
-10
0
10
-2000
200
-500
0
500
Fig. 8. Simulation results of output voltage control with imposed sinusoidalsource currents: a) output voltage 𝑣𝑎[V] of the indirect matrix converter; b)load voltage references v∗
o[V] and measured value vo[V]; c) load currentsio[A].
a)
b)
c)
Harmonic [Hz]0 100 200 300 400 500 600 700 800 900 1000
0 100 200 300 400 500 600 700 800 900 1000
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
0
0.5
1
0
0.5
1
Fig. 9. Spectra analysis of main variables related to the output voltagecontrol with imposed sinusoidal source currents: a) source current 𝑖𝑠𝐴; b)output voltage 𝑣𝑎 of the indirect matrix converter; c) load voltage 𝑣𝑜𝑎.
commutation but they are high order harmonics, and for thisreason they are not observed in Fig. 6b.
B. Predictive voltage control with imposed sinusoidal inputcurrents
In this case the same parameters of the previous scheme areused, but instead of employing the cost function correspondingto equation (20), equation (22) is considered with a weightingfactor of 𝛾𝑖=7000.
One of the advantages of this control strategy is thepossibility to handle the phase of the source current (𝜃) asmost convenient. For this reason in order to demonstrate thefeasibility of the proposed strategy, we applied phase changesto the source currents while maintaining the control of theload voltages. This is shown in Fig. 7, where two step changes
are observed; the first is from phase zero to a displacementfactor with a phase of -30𝑜 and the second from -30𝑜 to 30𝑜.Zero phase is considered in order to obtain unity power factoroperation and, as shown in Fig. 7, the source current is inphase with respect to its respective source voltage. A veryfast dynamic response is observed in the three currents foreach of the two phase changes with an almost immediateresponse and without any overshoot. At the same time inall cases a very good tracking of the source currents to theirreferences is observed, which demonstrates the effectiveness ofthe proposed source current control. In order to compare thisbehavior with the instantaneous reactive power minimization,the operation with unity power factor is evaluated obtaining aTHD of 8.05% which is almost the same value than the oneobtained with the first strategy. The main advantage here isthe possibility to change the phase of the source current.
In Fig. 8, the simulation results for the output side aregiven. Fig. 8a, shows the output voltage of the indirect matrixconverter 𝑣𝑎 where it is possible see the PWM waveform dueto the commutation of the switches, with an average THDvalue of 𝑇𝐻𝐷𝑣=188.47%. Fig. 5b shows the controlled outputvoltages vo and their references v∗
o. A very good tracking ofthe load voltages to their respective references is obtained withan average THD value of 𝑇𝐻𝐷𝑣𝑜
=1.62%. This performancehas been obtained in despite the phase variations of the sourcecurrents, which demonstrates that predictive control can handleboth conditions without any problem, observing a very goodbehavior in both the source currents and load voltages. Finally,sinusoidal load currents are obtained, as depicted in Fig. 5c,due to the nature of the applied load.
VII. CONCLUSION
This paper has presented a model predictive control schemefor generating sinusoidal output voltages from an IndirectMatrix Converter. The main advantage of predictive control itsis simplicity, compared with other methods, and the simplicityof the inclusion of additional constraints. The algorithm usedcalculates each possible output voltage between samples, andthen chooses the state which matches the reference best. Thecalculations are made using a discrete model of the outputfilter, the output voltage has a delay of two samples. Also,two different methods to maintain input unity power factorare tested: minimization of reactive power, and impositionof sinusoidal input current sources. The latter method showsto be slightly better with the possibility to handle the phaseangle as needed. However, it has some complex details inimplementation, like the use of a PLL to obtain the inputvoltage phase.
ACKNOWLEDGMENTS
This publication was made possible by the Initia-tion FONDECYT Research Project 11121492 and CONI-CYT/BMBF PCCI 12048.
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