Predictive Control of an Indirect Matrix Converter Operating at Fixed Switching Frequency and Without Weighting Factors L. Tarisciotti, P. Wheeler, P. Zanchetta Dep. of Electrical and Electronic Engineering University of Nottingham, Nottingham, UK
[email protected]
M. Rivera Dep. of Industrial Technologies Universidad de Talca, Curic´o, Chile
[email protected]
Abstract—In this paper a predictive control technique for an indirect matrix converter is proposed. The control strategy allows the output current control and the instantaneous reactive input power minimization of the converter without any weighting factor in the controller and, at the same time, the operation at fixed switching frequency. Simulation results validate the proposal under both steady and transient states. Index Terms—current control, matrix converters, predictive control, finite control set model predictive control.
is vs ii vi 𝑖𝑑𝑐 𝑣𝑑𝑐 i v i∗ 𝐶𝑓 𝐿𝑓 𝑅𝑓 𝑅 𝐿
N OMENCLATURE Source current [𝑖𝑠𝐴 𝑖𝑠𝐵 𝑖𝑠𝐶 ]𝑇 Source voltage [𝑣𝑠𝐴 𝑣𝑠𝐵 𝑣𝑠𝐶 ]𝑇 Input current [𝑖𝐴 𝑖𝐵 𝑖𝐶 ]𝑇 Input voltage [𝑣𝐴 𝑣𝐵 𝑣𝐶 ]𝑇 𝑑𝑐-link current 𝑑𝑐-link voltage Load current [𝑖𝑎 𝑖𝑏 𝑖𝑐 ]𝑇 Load voltage [𝑣𝑎 𝑣𝑏 𝑣𝑐 ]𝑇 Load current reference [𝑖∗𝑎 𝑖∗𝑏 𝑖∗𝑐 ]𝑇 Input filter capacitor Input filter inductor Input filter resistor Load resistance Load inductance I. I NTRODUCTION
In comparison with the conventional back to back converter, the matrix converter (MC) features several advantages mainly due to the advantage of handling higher power densities and to operate in environments with harsh temperatures and pressures [1], [2]. This converter features sinusoidal input and output currents, bidirectional energy flow and controllable input displacement power factor [2], [3]. The indirect matrix converter (IMC) is another 𝑎𝑐-𝑎𝑐 converter which not only holds the same features of the conventional direct matrix converter (DMC) but also is easier to control and allows secure commutation [4]. Over the last few years, research on indirect matrix converters has benefited from advances in semiconductor technologies, which have mainly contributed towards enhancing efficiency. Compared with the standard matrix converter, this topology can use a simpler modulation scheme and does not need an additional 978-1-4673-7554-2/15/$31.00 ©2015 IEEE
over-voltage protection system. Conversely, the current path from input to output produces higher power losses. This can be partly mitigated by the use of semiconductors such as Reverse Blocking IGBTs (RB-IGBT) or Silicon Carbide (SiC), which have already been used in conventional MCs [5]–[8]. Additionally, some studies have focused on sparse matrix topologies, which can improve power density by reducing the number of semiconductors at the expense of functionally. This is the case of the sparse matrix [9], [10] the very sparse matrix [11], [12] and the ultra sparse matrix converter [13], which feature 15, 12 and 9 switches respectively. The IMC uses complex Pulse Width Modulation (PWM) schemes to achieve the goal of unity power factor and sinusoidal output current [14]. However, since power converters have a discrete nature, the application of predictive control constitutes a promising and better suited approach as compared to standard schemes that use mean values of the variables. Furthermore, predictive control utilizes a very intuitive control law that can easily deal with multi-variable cases, the treatment of constraints, and compensate for the dead time [15], [16]. Early practical applications of predictive control can be found in the 1980’s for the case of two-level inverters. Nowadays, it is possible to find applications in machine control, active rectifiers, matrix converters and even multilevel converters [17]–[19]. However, there are some issues which constitute a disadvantage of this control method. One of the main drawbacks of the finite-set model predictive control (FS-MPC) methods are that the control can choose only from a limited number of valid switching states because of the absence of a modulator. This generates noise as well as large voltage and current ripples. The variable switching frequency produces a spread spectrum, decreasing the performance of the system in terms of power quality [20]–[24]. In [25] the discrete space vector modulation is extended to be used with predictive control, where virtual vectors are considered in the control algorithm which are synthesized through an external modulator, obtaining constant switching frequency and an improved performance. Similar results are obtained in [26], where traditional PI controllers were substituted by a predictive controller but with a conventional modulation technique. In [21], [27] a deadbeat predictive controller is
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proposed in order to determine the duty cycle for the PWM pulses for a given reference current over the entire speed range of operation of a switched reluctance machine. In [28] a predictive controller is proposed with a cost function that includes the current error and additionally a penalization term which is used to control the switching frequency. In [20], [23], [29] the switching behaviour of PWM with triangular carriers is used to propose a predictive method with a fixed switching period which is divided into smaller evaluation steps to obtain improved performance for different power converter topologies. Different solutions have been proposed in the literature [22], [30], [31] which allow the operation at fixed switching frequency. However they result in complicated expressions for the switching time calculations and they are not intuitive as it is very complicated to introduce other objectives into the cost function. In order to solve these problems, this paper proposes a new solution which allows operation at a fixed switching frequency while maintaining the advantages of predictive control. The proposed method emulates the implementation of space vector modulation (SVM) with a linear PI controller. Using a suitable modulation scheme in the cost function minimization of the predictive algorithm for a selected number of switching states to generate the duty cycles for the vectors that are applied to the converter using a given switching pattern. The predictive control strategy is evaluated for each stage (rectifier and inverter) of the IMC which eliminates the necessity of weighting factors in the cost function, because each stage have different control objectives which are evaluated separately. II. I NDIRECT M ATRIX C ONVERTER M ODEL The topology of the IMC is shown in Fig. 1. The configuration of this converter is divided in two stages: the rectifier and the inverter. This characteristic becomes an advantage when using the zero 𝑑𝑐-link current switching scheme, which allows a safe operation of the converter and a reduction of the switching losses. In particular, the mathematical model of the rectifier stage has the input phase voltages vi and 𝑑𝑐-link current 𝑖𝑑𝑐 as inputs and the 𝑑𝑐-link voltage 𝑣𝑑𝑐 and input
𝑖𝑑𝑐 𝑆𝑟1
𝑆𝑟3
𝑆𝑟5
𝑆𝑖1
𝑆𝑖3
𝑆𝑖5
𝑖𝑎
𝑖𝐴 𝑖𝐵
𝑖𝑏
𝑣𝑑𝑐 > 0
𝑖𝐶
𝑆𝑟4
𝑖𝑐
𝑆𝑟6
𝑆𝑟2
𝑆𝑖4
𝑆𝑖6
𝑆𝑖2
Fig. 1. Power circuit of the indirect matrix converter.
currents ii as outputs. This is shown in equations (1) and (2): ] [ 𝑣𝑑𝑐 = 𝑆𝑟1 − 𝑆𝑟4 𝑆𝑟3 − 𝑆𝑟6 𝑆𝑟5 − 𝑆𝑟2 vi (1) ⎤ ⎡ 𝑆𝑟1 − 𝑆𝑟4 ii = ⎣ 𝑆𝑟3 − 𝑆𝑟6 ⎦ 𝑖𝑑𝑐 (2) 𝑆𝑟5 − 𝑆𝑟2 TABLE I VALID SWITCHING STATE ON THE RECTIFIER SIDE #
𝑆𝑟1 𝑆𝑟2 𝑆𝑟3 𝑆𝑟4 𝑆𝑟5 𝑆𝑟6
1 2 3 4 5 6
1 0 0 0 0 1
1 1 0 0 0 0
0 1 1 0 0 0
0 0 1 1 0 0
0 0 0 1 1 0
0 0 0 0 1 1
𝑖𝐴 𝑖𝐵 𝑖𝑑𝑐 0 -𝑖𝑑𝑐 -𝑖𝑑𝑐 0 𝑖𝑑𝑐
𝑖𝐶
0 -𝑖𝑑𝑐 𝑖𝑑𝑐 -𝑖𝑑𝑐 𝑖𝑑𝑐 0 0 𝑖𝑑𝑐 -𝑖𝑑𝑐 𝑖𝑑𝑐 -𝑖𝑑𝑐 0
𝑣𝑑𝑐 𝑣𝐴𝐶 𝑣𝐵𝐶 -𝑣𝐴𝐵 -𝑣𝐴𝐶 -𝑣𝐵𝐶 𝑣𝐴𝐵
TABLE II VALID SWITCHING STATE ON THE INVERTER SIDE # 1 2 3 4 5 6 7 8
𝑆𝑖1 𝑆𝑖2 𝑆𝑖3 𝑆𝑖4 𝑆𝑖5 𝑆𝑖6 1 1 0 0 0 1 1 0
1 1 1 0 0 0 0 1
0 1 1 1 0 0 1 0
0 0 1 1 1 0 0 1
0 0 0 1 1 1 1 0
1 0 0 0 1 1 0 1
𝑣𝑢𝑣 𝑣𝑣𝑤 𝑣𝑤𝑢 𝑣𝑑𝑐 0 -𝑣𝑑𝑐 -𝑣𝑑𝑐 0 𝑣𝑑𝑐 0 0
0 𝑣𝑑𝑐 𝑣𝑑𝑐 0 -𝑣𝑑𝑐 -𝑣𝑑𝑐 0 0
-𝑣𝑑𝑐 -𝑣𝑑𝑐 0 𝑣𝑑𝑐 𝑣𝑑𝑐 0 0 0
𝑖𝑑𝑐 𝑖𝑜𝑢 𝑖𝑜𝑢 +𝑖𝑜𝑣 𝑖𝑜𝑣 𝑖𝑜𝑣 +𝑖𝑜𝑤 𝑖𝑜𝑤 𝑖𝑜𝑢 +𝑖𝑜𝑤 0 0
Inputs and outputs of each stage are related by their switching states. For the inverter these relations involve the output currents i and 𝑑𝑐-link voltage 𝑣𝑑𝑐 as inputs, and the 𝑑𝑐-link current 𝑖𝑑𝑐 and the output voltage v as outputs. This can be seen in equations (3) and (4): ] [ (3) 𝑖𝑑𝑐 = 𝑆𝑖1 𝑆𝑖3 𝑆𝑖5 i ⎡ ⎤ 𝑆𝑖1 − 𝑆𝑖4 ⎣ 𝑆 v= (4) 𝑖3 − 𝑆𝑖6 ⎦ 𝑣𝑑𝑐 𝑆𝑖5 − 𝑆𝑖2 As expected not all the possible switching states are allowed. There are some constraints which are mandatory for the safe operation of the converter: ∙ Input phases of the rectifier stage cannot be short circuited thus, only nine valid rectifier states can be used. ∙ Output phases of the inverter stage cannot be disconnected thus, only eight states are allowed. Additionally, there must be a positive 𝑑𝑐-link voltage in order for the inverter switches to be able to commute. Thus, the valid rectifier states are reduced to only three at any instant, and the whole IMC has twenty-four possible valid states. In Table I and Table II are detailed each valid switching state for both the rectifier and inverter with their respective relations in terms of currents and voltages. In order to prevent over-voltages and harmonics distortions in the source current, an input filter is needed. By matching the filter poles to certain places, a variety of transfer functions can be obtained. A second-order low-pass filter at the input is
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io (𝑘)
vi (𝑘)
and input currents [32], [33]. Moreover the proposed technique evaluates separately for both the rectifier and inverter, the prediction of the two active vectors that conform each sector at every sampling time and evaluates the cost function separately for each prediction (for the inverter side is also evaluated the zero vector). For the rectifier two active vectors are considered, and thus two cost functions are evaluated (only positive 𝑑𝑐-link is possible in the IMC). For example, for sector I of the rectifier, the first prediction and cost function 𝑔𝑟1 is evaluated for vector v𝑟1 and the second prediction and cost function 𝑔𝑟2 is evaluated for vector v𝑟2 . With the evaluation of each cost function, it is possible to determine the duty cycles associated to each vector, which are calculated by solving:
3𝜙 AC Source is (𝑘)
vs (𝑘) Predictive Model
3𝜙 Filter
is (𝑘) 𝑄𝑝 (𝑘 + 1)
vi (𝑘) 𝑄∗ (𝑘)
vr 𝑜𝑝𝑡 , vi 𝑜𝑝𝑡 Cost Function Minimization
𝑑𝑟 , 𝑑𝑖
Switching Sequence
𝑆𝐴𝑎 (𝑘) .. . 𝑆𝐶𝑐 (𝑘)
IMC
i∗o (𝑘 + 1) ip o (𝑘 + 1)
Predictive Model
𝑑𝑟1 = 𝑔𝑟2 /(𝑔𝑟1 + 𝑔𝑟2 ) 𝑑𝑟2 = 𝑔𝑟1 /(𝑔𝑟1 + 𝑔𝑟2 ) 𝑑𝑟1 + 𝑑𝑟2 = 1
3𝜙 io (𝑘)
vo (𝑘)
where 𝑔𝑟1 is the cost function associated to switching state #1 which correspond to the error of the instantaneous reactive power when the reference is equal to zero (𝑄∗ = 0).
io (𝑘) Load
𝑅𝑜 𝐿𝑜
Fig. 2. Predictive current control of the indirect matrix converter with instantaneous input reactive power minimization.
considered in this work which allows to establish the following relations: 1 𝑑is 𝑅𝑓 = (vs − vi ) − is (5) 𝑑𝑡 𝐿𝑓 𝐿𝑓 1 𝑑vi = (is − ii ) 𝑑𝑡 𝐶𝑓
(8)
𝑔𝑟1 = [𝑣𝑠𝛼 (𝑘 + 1)𝑖𝑠𝛽 (𝑘 + 1) − 𝑣𝑠𝛽 (𝑘 + 1)𝑖𝑠𝛼 (𝑘 + 1)]2 (9) and 𝑔𝑟2 is the cost function associated to switching state #2. With these duty cycles, it is possible to evaluate the cost function 𝑔𝑟 which is evaluated at every sampling time, defined as: 𝑔𝑟 = 𝑑𝑟1 𝑔𝑟1 + 𝑑𝑟2 𝑔𝑟2 (10) This is the cost function that is evaluated and minimized. Once the optimal cost function is selected, the time that each vector is applied is obtained as:
(6)
𝑡𝑟1 = 𝑇𝑠 𝑑𝑟1 /𝑇𝑚 𝑡𝑟2 = 𝑇𝑠 𝑑𝑟2 /𝑇𝑚 𝑡𝑟1 + 𝑡𝑟2 = 𝑇𝑠
From Fig. 1, the load mathematical model is given as: 𝑑i + 𝑅i (7) 𝑑𝑡 III. P ROPOSED P REDICTIVE M ETHOD WITH F IXED S WITCHING F REQUENCY v=𝐿
Such as indicated in [4], it is possible to define each available vector for the rectifier and inverter side of the IMC (Table I and Table II) in the 𝛼-𝛽 plane. With these representations six sectors are defined given by two adjacent vectors in each side of the converter. Let’s define for the rectifier the first sector as the one between vector v𝑟1 and vector v𝑟2 , which correspond to the current generated by rectifier switching state #1 and rectifier switching state #2, respectively, based on eq. (1) and eq. (2) and Table I. Similarly, let’s define for the inverter the first sector the one between vector v𝑖1 and vector v𝑖2 , which correspond to the voltage generated by inverter switching state #1 and inverter switching state #2, respectively, based on eq. (3) and eq. (4) and Table II. The proposed method for the IMC is shown in Fig. 2. This strategy is almost the same idea as the classical predictive control method because it uses the same prediction of the load
(11)
𝑜𝑝𝑡 where, 𝑡𝑟1 , 𝑡𝑟2 correspond to the time that optimal vectors 𝑣𝑟1 𝑜𝑝𝑡 and 𝑣𝑟2 are applied, respectively. For the discrete implementation a number of steps 𝑇𝑚 is defined during each sampling time 𝑇𝑠 . Similar is the implementation of the control strategy for the inverter side, but in this case two active vectors are considered in each sector plus a zero vector. For example, for sector I of the inverter, the first prediction and cost function 𝑔𝑖0 is evaluated only once for vector v𝑖0 , followed by the prediction and cost function 𝑔𝑖1 evaluated for vector v𝑖1 and the cost function 𝑔𝑖2 evaluated for vector v𝑖2 . With the evaluation of each cost function, it is possible to determine the duty cycles associated to each vector, which are calculated by solving:
𝑑𝑖0 = 𝑇𝑠 𝑔𝑖1 𝑔𝑖2 /(𝑔𝑖0 𝑔𝑖1 + 𝑔𝑖1 𝑔𝑖2 + 𝑔𝑖0 𝑔𝑖2 ) 𝑑𝑖1 = 𝑇𝑠 𝑔𝑖0 𝑔𝑖2 /(𝑔𝑖0 𝑔𝑖1 + 𝑔𝑖1 𝑔𝑖2 + 𝑔𝑖0 𝑔𝑖2 ) 𝑑𝑖2 = 𝑇𝑠 𝑔𝑖0 𝑔𝑖1 /(𝑔𝑖0 𝑔𝑖1 + 𝑔𝑖1 𝑔𝑖2 + 𝑔𝑖0 𝑔𝑖2 ) 𝑑𝑖0 + 𝑑𝑖1 + 𝑑𝑖2 = 1
(12)
where 𝑔𝑖0 is the cost function associated to switching state #7 or #8, which correspond to the error of the load current
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from eq. (1)- eq. (4): ] [ ] [ ] [ v˙i vi vs =A +B is ii i˙s
v𝑟1 1 0 v𝑟2
1
[
where,
0 𝑇𝑟1
0 1/𝐶𝑓 A= −1/𝐿 −𝑅 𝑓 𝑓 /𝐿 [ ]𝑓 0 −1/𝐶𝑓 B= 1/𝐿𝑓 0
𝑇𝑟2 a)
v𝑖1
1 0 1
(16)
] (17)
Assuming that
v𝑖2 0 1 v𝑖0 0
vs = vs (𝑘𝑇𝑠 ) = vs (𝑘) ii = ii (𝑘𝑇𝑠 ) = ii (𝑘) 𝑇𝑖0 4
𝑇𝑖1 2
𝑇𝑖2 2
𝑇𝑖0 2
𝑇𝑠
𝑇𝑖2 2
𝑇𝑖1 2
for 𝑘𝑇𝑠 ≤ 𝑡 ≤ (𝑘 + 1)𝑇𝑠 , with 𝑇𝑠 being the sampling time, the discrete-time state space model is determined as: ] [ ] [ ] [ vi (𝑘) vs (𝑘) vi (𝑘 + 1) =Φ +Γ (19) is (𝑘 + 1) is (𝑘) ii (𝑘)
𝑇𝑖0 4
b)
Fig. 3. Switching pattern: a) for the rectifier side; b) for the inverter side.
where,
predictions respect to their respective references: 𝑔𝑖0 = [i∗ − i]2
(18)
(13)
and 𝑔𝑖1 , 𝑔𝑖2 are the cost functions associated to switching state #1 and #2 of the inverter, respectively. With these duty cycles, it is possible to evaluate the cost function 𝑔𝑖 which is evaluated at every sampling time, defined as:
Φ = 𝑒A𝑇𝑠 ,
Γ = A−1 (Φ − I2𝑥2 )B.
(20)
On the inverter side, the output current prediction can be obtained using a forward Euler approximation in eq. (7), such as: (21) i(𝑘 + 1) = 𝑐1 v(𝑘) + 𝑐2 i(𝑘) where, 𝑐1 = 𝑇𝑠 /𝐿 and 𝑐2 = 1 − 𝑅𝑇𝑠 /𝐿, are constants dependent on load parameters and the sampling time 𝑇𝑠 . IV. S IMULATION R ESULTS
𝑔𝑖 = 𝑑𝑖1 𝑔𝑖1 + 𝑑𝑖2 𝑔𝑖2
(14)
This is the cost function that is evaluated and minimized for the inverter stage. Once the optimal cost function is selected, the time that each vector is applied is obtained as: 𝑡𝑖0 = 𝑇𝑠 𝑑𝑖0 /𝑇𝑚 𝑡𝑖1 = 𝑇𝑠 𝑑𝑖1 /𝑇𝑚 𝑡𝑖2 = 𝑇𝑠 𝑑𝑖2 /𝑇𝑚 𝑡𝑖0 + 𝑡𝑖1 + 𝑡𝑖2 = 𝑇𝑠
In order to validate the effectiveness of the proposal method, simulation results were carried out in both steady and transient conditions. The simulation parameters are shown in Table III. TABLE III PARAMETERS OF THE IMPLEMENTATION
(15)
where 𝑡𝑖0 , 𝑡𝑖1 , 𝑡𝑖2 correspond to the time that optimal vectors 𝑜𝑝𝑡 𝑜𝑝𝑡 𝑜𝑝𝑡 , 𝑣𝑖1 and 𝑣𝑖2 are applied, respectively. After obtaining 𝑣𝑖0 the duty cycles and selecting the optimal two vectors to be applied, a switching pattern procedure, such as the one shown in Fig. 3, is adopted with the goal of applying the optimal vectors in both the rectifier and inverter stages [34]. A similar idea has been first proposed for a three-phase active rectifier and a seven-level converter in [35], [36] respectively and now it is extended to the IMC. The complexity in this paper is given by two controllers working in parallel and a more complex switching pattern procedure. At difference of the classical predictive current control strategies proposed for the IMC, with this proposal no weighting factors are needed. Since the predictive controller is formulated in discrete time, it is necessary to derive a discrete time model for the loadconverter system. The rectifier stage can be represented by a state space model with the state variables is and vi obtained
Variables
Description
Value
𝑉𝑠 𝐶𝑓 𝐿𝑓 𝑅𝑓 𝑅 𝐿 𝑇𝑠 𝑓𝑠
Amplitude 𝑎𝑐-voltage Input filter capacitor Input filter inductor Input filter resistor Load resistance Load inductor Sampling time Switching frequency∗ Simulation step
311[V] 21[𝜇F] 400[𝜇H] 0.5[Ω] 10[Ω] 10[𝜇H] 50 [𝜇s] 20[kHz] 1 [𝜇s]
A. Results in Steady State Fig. 4 and Fig. 5 show simulation results in steady state for the proposed predictive controller. In Fig. 4a is shown the source voltage and current. The source current 𝑖𝑠𝐴 is almost sinusoidal and in phase with the source voltage 𝑣𝑠𝐴 , demonstrating the good performance of the control strategy when zero instantaneous input reactive power is imposed. Additionally, it is verified the performance of the input filter which eliminates the high frequency harmonics of the input currents present in the converter due to the commutation of the switches as depicted in Fig. 4b. Positive 𝑑𝑐-link voltage
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a)
40
40
20
20 a)
0 -20 -40 0.02
0.03
0.04
0.05
0
-40 0.02
0.06
40
-20 0.03
0.04
0.05
-40 0.02
0.06
c)
0
0 0.03
0.04 Time [s]
0.05
0.06
0.02
15
15 a) 10
5
5 0.03
0.04
0.05
0 0.02
0.06
500 0
0.03
0.04
0.05
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06 Time [s]
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06 Time [s]
0.07
0.08
0.09
0.1
0
-500 0.02
0.06
20 c)
0 -20 0.02
0.1
500 b)
20 c)
0.09
Fig. 6. Simulation results in transient state: a) source voltage 𝑣𝑠𝐴 [V/10] and source current 𝑖𝑠𝐴 [A]; b) capacitor voltage 𝑣𝐴 [V/10] and input current 𝑖𝐴 [A]; c) 𝑑𝑐-link voltage [V].
a) 10
-500 0.02
0.08
400 200
Fig. 4. Simulation results in steady state: a) source voltage 𝑣𝑠𝐴 [V/10] and source current 𝑖𝑠𝐴 [A]; b) capacitor voltage 𝑣𝐴 [V/10] and input current 𝑖𝐴 [A]; c) 𝑑𝑐-link voltage [V].
b)
0.07
600
400
0 0.02
0.06
0
200
0.02
0.05
-20
600 c)
0.04
20 b)
0 -40 0.02
0.03
40
20 b)
0 -20
0.03
0.04 Time [s]
0.05
0 -20 0.02
0.06
Fig. 5. Simulation results in steady state: a) 𝑑𝑐-link current 𝑖𝑑𝑐 [A]; b) load voltage 𝑣𝑎 [V]; c) load currents io [A].
Fig. 7. Simulation results in steady state: a) 𝑑𝑐-link current 𝑖𝑑𝑐 [A]; b) load voltage 𝑣𝑎 [V]; c) load currents io [A].
is ensured all the time ensuring the safe operation of the converter (Fig. 4c). In Fig. 5a is observed a very good tracking of the load current i to its respective references i∗ which are established as 16[A] and with a reference frequency of 50[Hz]. It is possible to observe that the load current ripple for the proposed predictive scheme is very low. This can also be seen in Fig. 5b which has an almost sinusoidal load voltage 𝑣𝑎𝑛 and presents a fixed switching frequency.
a source current in phase with respect to its respective source voltage is obtained and also a good tracking of the load currents to their respective references.
B. Results in Transient Condition In order to demonstrate the performance of the proposed strategy in terms of dynamic response, transient state analysis is done as depicted in Fig. 6 and Fig. 7. A step change from 16[A] to 12[A] is applied at instant 𝑡 = 0.06[s] and as expected, very fast dynamic response is observed. Again,
V. C ONCLUSION In this paper a new predictive control scheme has been proposed which allows the operation of the converter with a fixed switching frequency while maintaining the advantages of the classical finite-state model predictive control techniques such as fast dynamic response and easy inclusion of nonlinearities. Simulations results demonstrate that this is a viable alternative to avoid linear controllers and performs well in both steady and transient conditions with a good tracking to its references and a reduced ripple. By considering the proposed strategy,
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a new alternative has emerged for the control of an indirect matrix converter. ACKNOWLEDGMENTS The authors would like to thank the financial support of FONDECYT Initiation into Research 11121492 Project and Universidad de Talca. R EFERENCES [1] P. Wheeler, J. Rodriguez, J. Clare, L. Empringham, and A. Weinstein, “Matrix converters: a technology review,” Industrial Electronics, IEEE Transactions on, vol. 49, no. 2, pp. 276–288, Apr 2002. [2] L. Empringham, J. Kolar, J. Rodriguez, P. Wheeler, and J. Clare, “Technological issues and industrial application of matrix converters: A review,” Industrial Electronics, IEEE Transactions on, vol. 60, no. 10, pp. 4260–4271, Oct 2013. [3] J. Rodriguez, M. Rivera, J. Kolar, and P. Wheeler, “A review of control and modulation methods for matrix converters,” Industrial Electronics, IEEE Transactions on, vol. 59, no. 1, pp. 58–70, Jan 2012. [4] J. Kolar, T. Friedli, J. Rodriguez, and P. Wheeler, “Review of threephase pwm ac-ac converter topologies,” Industrial Electronics, IEEE Transactions on, vol. 58, no. 11, pp. 4988–5006, Nov 2011. [5] A. Musing, M. Heldwein, T. Friedli, and J. Kolar, “Steps towards prediction of conducted emission levels of an rb-igbt indirect matrix converter,” in Power Conversion Conference - Nagoya, 2007. PCC ’07, April 2007, pp. 1181–1188. [6] T. Friedli, M. Heldwein, F. Giezendanner, and J. Kolar, “A high efficiency indirect matrix converter utilizing rb-igbts,” in Power Electronics Specialists Conference, 2006. PESC ’06. 37th IEEE, June 2006, pp. 1–7. [7] K. Sung, R. Iijima, S. Nishizawa, I. Norigoe, and H. Ohashi, “Experimental investigation of normally-on type bidirectional switch for indirect matrix converters,” in Power Electronics Conference (IPEC-Hiroshima 2014 - ECCE-ASIA), 2014 International, May 2014, pp. 117–122. [8] A. Escobar-Mejia, C. Stewart, J. Hayes, S. Ang, J. Balda, and S. Talakokkula, “Realization of a modular indirect matrix converter system using normally off sic jfets,” Power Electronics, IEEE Transactions on, vol. 29, no. 5, pp. 2574–2583, May 2014. [9] E. Lee, K.-B. Lee, J.-S. Lim, Y. Lee, and J.-H. Song, “Predictive current control for a sparse matrix converter,” in Power Electronics and Motion Control Conference (IPEMC), 2012 7th International, vol. 1, June 2012, pp. 36–40. [10] E. Lee and K.-B. Lee, “Fault diagnosis for a sparse matrix converter using current patters,” in Applied Power Electronics Conference and Exposition (APEC), 2012 Twenty-Seventh Annual IEEE, Feb 2012, pp. 1549–1554. [11] T. Nguyen and H.-H. Lee, “Carrier-based pwm method for four-leg very sparse matrix converter,” in Power Electronics and ECCE Asia (ICPE ECCE), 2011 IEEE 8th International Conference on, May 2011, pp. 1703–1710. [12] M. Aner, E. Nowicki, and D. Wood, “Employing a very sparse matrix converter for improved dynamics of grid-connected variable speed small wind turbines,” in Power and Energy Conference at Illinois (PECI), 2012 IEEE, Feb 2012, pp. 1–7. [13] J. Schonberger, T. Friedli, S. Round, and J. Kolar, “An ultra sparse matrix converter with a novel active clamp circuit,” in Power Conversion Conference - Nagoya, 2007. PCC ’07, April 2007, pp. 784–791. [14] F. Schafmeister and J. Kolar, “Novel hybrid modulation schemes significantly extending the reactive power control range of all matrix converter topologies with low computational effort,” Industrial Electronics, IEEE Transactions on, vol. 59, no. 1, pp. 194–210, Jan 2012. [15] R. Aguilera and D. Quevedo, “Predictive control of power converters: Designs with guaranteed performance,” Industrial Informatics, IEEE Transactions on, vol. 11, no. 1, pp. 53–63, Feb 2015. [16] H. Young, M. Perez, J. Rodriguez, and H. Abu-Rub, “Assessing finitecontrol-set model predictive control: A comparison with a linear current controller in two-level voltage source inverters,” Industrial Electronics Magazine, IEEE, vol. 8, no. 1, pp. 44–52, March 2014. [17] M. NARIMANI, B. Wu, V. Yaramasu, Z. Cheng, and N. Zargari, “Finite control-set model predictive control (fcs-mpc) of nested neutral point clamped (nnpc) converter,” Power Electronics, IEEE Transactions on, vol. PP, no. 99, pp. 1–1, 2015.
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