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H-5 L inverter based on satisfactory optimization algorithm
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithmR Guifeng Wang∗, Jianguo Jiang, Wei Wu School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200030, China
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Satisfactory optimization NPC/H Nonlinear FCS-MPC Common-mode voltage suppression
a b s t r a c t In the design of the cost function in the nonlinear finite control set model predictive control (FCS-MPC) system, the traditional method based on weighting factors demonstrates some limitations, such as the weighting factors adjusting and heavy predictive calculation due to the increased number of voltage vectors applied in controlling multilevel converters. This paper proposes a simplified FCS-MPC method based on common mode voltage satisfactory optimization, which could considerably reduce the predictive calculation by the optimized switch combination and simplify the cost function design. Moreover, satisfactory optimization is adopted to achieve the accuracy control of common-mode voltage amplitude without adjusting process of weighting factors. The simulation and experimental results verify the feasibility of this control strategy. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Model predictive control (MPC) has now more than three decades of sustained development and is considered as one of the most important advances in process control. It is widely used in the multivariable, nonlinear, strong coupling industrial control fields with multi constraint conditions, including chemical industry, metallurgy, aerospace, mechanical manufacturing, electric power and so on [1–4]. It was first proposed in 2000 that MPC technology can be applied to power electronics and power transmission [5]. In 2003, the model predictive method with ergodic optimization for the combination of all the switching states of matrix converter was first applied to control the matrix converter [6]. In [7], the finite control R This work was supported by the National High Technology Research and Development Program of China (863 Program) under Grant no. 2011AA050403 and Doctoral Program of Higher Specialized Research Fund (new teachers class) Project (20110073120034). ∗ Corresponding author. Tel.: +86 13813459613; fax: +86 51683882453. E-mail address: [email protected], [email protected] (G. Wang).
set model predictive control (FCS-MPC) of converter was described in detail which has many advantages, such as intuitive modeling, easy to understand, direct control, easy to handle the multi-constraint conditions and multi-objective. Moreover, compared with the traditional control algorithm, it has no PWM modulator and there is no need to adjust PI parameters. Therefore, FCS-MPC has become a popular research direction in the field of predictive control of power converters. There are many researches about converter topology with FCS-MPC system, such as two-level voltage source inverter (2L-VSI) [8,9], three-level neutral point clamped inverter (3L-NPC) [10,11], flying-capacitor inverter (FCI) [14,15], active-front-end rectifiers (AFE) [18], cascaded Hbridge multilevel converters (CHB) [12,13], matrix converters (MC) [6], five-level active neutral-point-Clamped Inverter (5L-ANPC) [16,17]. This paper studies the threelevel neutral-point-clamped H-bridge (NPC/H) cascaded five-level inverter. H-bridge cascaded multi-level topology is one of the sophisticated topological structures. Compared with the two-level H-bridge topology, NPC/H cascaded multi-level topology requires less number of independent DC powers, which is conducive to simplifying the
http://dx.doi.org/10.1016/j.chaos.2015.12.021 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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Twelve pulse rectification system
NPC/H five-level inverter system R
6 pulse rectifier
A B C
Three phase load
6 pulse rectifier
Fig. 1. NPC/H five-level inverter main circuit topology.
Table 1 Multilevel converter FCS-MPC prediction and optimization times. Name
Two-level inverter (such as: 2L-VSI) Three-level inverter(such as: 3L-NPC) Five-level converter (such as: NPC/H-5 L) Seven-level converter (such as 3H -CHB) (2C+1) Level converter
Output phase voltage level
2 3 5 7 2C+1
structure of converter. ACS5000 inverter of ABB Company adopts the NPC/H five-level topology with the main circuit structure, as shown in Fig. 1. The basic control strategy of FCS-MPC system is to conduct predictive computation for the optimized switch combination with the traversing method and compare the objective functions to select the best one. The difficulty of FCS-MPC controller design lies in the weighting factor adjusting in the cost function. However, the root cause that obstructs the application of FCS-MPC system is its heavy computation, especially in the field of multilevel converter control [12]. Repeated computation 7 times should be conducted in one sampling cycle even using the two-level voltage source inverter (2L-VSI) which has the simplest structure and has been most widely employed in industry, and ignoring the problems of redundant voltage vectors and multistep prediction. Compared with the traditional PI control system, the computation is obviously heavy. As the power electronic technology develops towards the direction of high voltage and large capacity, the converter also develops towards the multi-level and multiplex direction, and the increase of the levels causes the exponential growth of switch combination and rise of computation. Table 1 shows the multilevel converter FCSMPC prediction and optimization times with the traversing method. Some complex applications such as motor drive control require more complex control system model, for the controller should complete the flux observation, coordinate transformation, speed closed loop, closed-loop PI control
Voltage vector number All voltage vector
Basic voltage vector
8 27 125 343 (2C + 1)3
7 19 61 127 12C2 + 6C + 1
algorithm in addition to the FCS-MPC predictive control operations, which means greater computational burden. According to FCS-MPC predictive control principle, it is necessary to improve the sampling frequency of the system based on the hardware requirement in order to get better control performance. However, the inherent problem of FCS-MPC – heavy computation, limits the room for improvement. Therefore, to promote the application of FCS-MPC in the field of multilevel converter control, heavy computation should be reduced first. At present, algorithms of reducing the computation of FCS-MPC mainly focus on specific applications, thus lacking generality. The cost function design and the weight adjusting problem in the multi objective control are ignored as a result [10,12,20,21]. As it is difficult to conduct offline adjusting for the cost function established by the traditional weighting factor method, in [19], the fuzzy control concept was introduced in the cost function design, and then the membership function replaced the weight function to achieve the online adjusting of cost function. In [20] the satisfactory optimization theory was combined with the fuzzy control concept to simplify the membership function design. On the basis of the satisfactory optimization control theory, this paper aims at the simplification of cost function design and explores the optimal switch combination to reduce the computation. Then, the satisfactory optimization concept is introduced to replace the adjusting process of weighting factors by the common mode voltage satisfactory optimization to solve this problem.
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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3
o
Vdc C1 2
Vdc C2 2
Sa2
Sa1
1
1
Sa22
Sa12
Sa23
Sa13
Sa24
Sa14
Vdc C3 2
Vdc C4 2
Sb2 1
Sb11
Sb22
Sb12
Sb23
Sb13
Sb24
Sb14
Vdc C5 2
Vdc C6 2
Sc2
Sc1
1
1
Sc22
Sc12
Sc23
Sc13
Sc24
Sc14
L
L
L
R
R
R
n Fig. 2. The topology of NPC/H five-level inverter with resistive inductive load.
function in theαβ two-phase static coordinate system after the 3/2 transformation is shown as follows
2. FCS-MPC controller design of NPC/H five-level inverter
uα uβ
2.1. FCS-MPC predictive model of NPC/H five-level inverter The main circuit topology of NPC/H five-level inverter with resistive inductive load is as shown in Fig. 2. Assuming the three-phase load is balanced, the mathematical model based on the three-phase static coordinate system is:
2 V = (vao + avbo + a2 vco ) 3
(1)
2 I = (ia + aib + a2 ic ) 3
(2)
2E 1 = 3 0
−1/2 √ 3/2
(3)
a = e j2π /3 is the complex operator, vao , vbo and vco represent the voltage of the inverter output terminal to the DC side midpoint o. R and L respectively represents the resistance and inductance of one-phase load. Adopt first-order forward Euler formula to discretize load current differential term dI/dt and obtain the predictive value Ip (k + 1 )of the current at the moment of k + 1.
Ip ( k + 1 ) = 1 −
RTs Ts I (k ) + V (k ) L L
(4)
Ts is the cycle of the control system. Define the switching function of NPC/H five-level inverterSx , x ∈ {a, b, c},and Sx ∈ {−2, −1, 0, 1, 2}. The five possible values of Sx correspond five possible output level states of each phase of the inverter, and then there is:
uxo = Sx E
(5)
E is half of the DC side voltage. According to the equivalent principle of amplitude, the relationship between the voltageuα , uβ and the switching
Sa Sb Sc
(6)
In the same way, the Eq. (4) is converted to the αβ two-phase static coordinate system and obtain the discrete domain prediction model of the resistance–inductance load with[iα (k), iβ (k)]T as the state variable can be obtained and the switching function combination [Sa , Sb , Sc ]T ∈ {−2, −1, 0, 1, 2}3 based on the state space.
Sa ( k ) iα (k + 1 ) i (k ) = (ATs + I ) α + Ts B Sb (k ) iβ (k + 1 ) iβ (k ) Sc ( k ) In the equation
dI V = RI + L dt
−1/2 √ − 3/2
A=
−L/R 0
2E 1 0 ,B = −L/R 3L 0
−1/2 √ 3/2
−1/2 √ − 3/2
(7)
2.2. FCS-MPC cost function design of NPC/H five-level inverter The optimal performance indicators of NPC/H five-level variable frequency drive system include: current following control, dv/dtjump limit, reduction of switching loss, common mode voltage suppression and capacitor voltage balance control. At this point, NPC/H-FCS-MPC of NPC/H fivelevel inverter is transformed into the online solution of multi objective optimization problem. Generally, the traditional FCS-MPC system uses the weight method to construct the cost function. This paper adopts the main circuit topology, as shown in Fig. 1. The neutral point of the capacitor directly connects to the neutral point of series 12 pulse rectifier, which could ensure the balance of the capacitor voltage. However, NPC/H five-level inverter single-phase outputs two levels ± E with two different working modes which have different influence on the capacitor voltage. Therefore, it is necessary to carry out switching mapping rationally. In this paper, the simplest method is adopted, that is, two kinds of
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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working mode are alternatively used to ensure the voltage balance control, so as to reduce an optimal performance indicator that needs to be considered for the NPC/H fivelevel inverter drive system. Reducing switching loss means to reduce the switching frequency in each control cycle. The switching frequency for each phase of the inverter meets the following conditions:
f sx = 2|sx (k + 1 ) − sx (k )|, x ∈ {a, b, c}
(8)
For the NPC/H five-level inverter topology, the phase voltage and line voltage for each switching cannot exceed two-level jump in principle. The former is harmful to the inverter circuit and causes the damage to the switch device; the latter is harmful to the motor, since the over highdv/dt can easily damage the insulation of the motor and produce serious electromagnetic interference (EMI). Therefore, the limit of output phase voltage dv/dtof the inverter should meet the demand|sx (k + 1 ) − sx (k )| < 2, x ∈ {a, b, c}; the limit of output line voltagedv/dt should meet the demand|sa (k + 1 ) − sb (k + 1 )| < 2, |sb (k + 1 ) − sc (k + 1 )| < 2, |sb (k + 1 ) − sc (k + 1 )| < 2. Then the two optimal performance indicators, thedv/dt jump limit and the reduction of switching loss, can be unified as:
⎧ ∞ |sx (k + 1 ) − sx (k )| ≥ 2, x ∈ {a, b, c} ⎪ ⎪ ⎪ ⎪ ⎨∞ |sa (k + 1 ) − sb (k + 1 )| ≥ 2, or |sb (k + 1 ) − sc (k + 1 )| ≥ 2, or fswitch = |sb (k + 1 ) − sc (k + 1 )| ≥ 2 ⎪ ⎪ ⎪ ⎪ ⎩2(|sa (k + 1 ) − sa (k )| + |sb (k + 1 ) − sb (k )| +|sc (k + 1 ) − sc (k )| )
(9)
The NPC/H-FCS-MPC cost function established by the weighting method is shown as follows:
g=
[m3Gdc;January 23, 2016;16:49]
∗
iα (k + 1 ) − iαp (k + 1 ) + i∗β (k + 1 ) − i p (k + 1 ) β
+ksw fswitch + kcmv fcmv
(10)
In Eq. (10), ksw and kcmv are the weighting factor; i∗α (k + 1 )and i∗β (k + 1 ) represent the given current in
theαβ coordinate system at the moment oftk+1 and they are unknown value. In the actual implementation process of the algorithm, as the sampling time of the system is short enough, it can be approximated to be the given current at the moment of tk , that is, i∗α (k + 1 ) ≈ i∗α (k ), i∗β (k +
1 ) ≈ i∗β (k ). iα (k + 1 )and iβ (k + 1 ) represent the predictive
current in theαβ coordinate system corresponding to the different switch combination of the inverter at the moment of tk+1 , which can be obtained by Eq. (7). fcmv is the common mode voltage and can be obtained by the following equation.
fcmv =
1 E (uao + ubo + uco ) = (sa + sb + sc ) 3 3
(11)
Different values of weighting factorksw andkcmv in Eq. (10) enable different control effect of NPC/H five-level inverter to be obtained. The weighting method is a multiobjective optimization control method which regards the optimal solution of the cost function as the optimization
objective, thus ensuring the uniqueness of the optimal solution. However, this method has obvious shortcomings, and it is difficult to configure a single weighting factor for the multi-objective control task and achieve different control effect expected by designers. Besides, the control effect of each objective is closely related to the selection of weighting factor, which leads to the difficulty of weighting factor adjusting. In addition, due to the conflict between the optimal performance indicators, the over optimization of a certain performance indicator of NPC/H five-level inverter will inevitably lead to the control quality deterioration of other indicators. 3. Simplified FCS-MPC strategy based on satisfactory optimization of common mode voltage 3.1. Optimal switch combination design The cost function of the NPC/H five-level inverter is as shown in Eq. (10) where two weighting factors are required to be adjusted. The current following is regarded as the core control objective, while the switching loss and the common mode voltage suppression are regarded as the target of performance optimization. By the predictive model, the redundant switch combination has the same effect on current following performance, but its effect on switching loss and common mode voltage is different. Therefore, the cost function of NPC/H five-level inverter established by the weighting method can be analyzed separately from two aspects, aiming to explore the control strategy which is conducive to the design of cost function and the reduction of the computation. Remove the redundant vectors with great common mode voltage to reduce the common mode voltage of the motor. Then 125 vectors can be simplified to 61 independent vectors, as shown in Fig. 3. However, for the FCS-MPC control system, the calculation burden is still too heavy. As the control period of the FCS-MPC system is short enough and the given instruction in the adjacent control cycle has slight variation when the system is operating at steadystate, voltage vectors should be selected from adjacent vectors. The adjacent vector method [21] could be used to optimize the switch combination design in order to reduce the computation. To adopt this method, there are no rules to follow by the analysis in Fig. 3. Thus, the table lookup method is used to respectively list adjacent vectors of 61 vectors. The selected adjacent vectors have different influence on the common voltage and switching loss, so the weighting factor adjusting process cannot be simplified. Therefore, this method is not universal. The switching loss is expected to be lower possible, especially the high-power transmission device. Based on the idea of [22], only one phase or one level transformation in each control period is allowed, that is, there are no more than two switches operating in one control cycle. This optimized switch combination design method can realize the minimum switching loss in a single control cycle, and the maximum number of optional vector is 2 ∗ N + 1 whereNis the number of bridge arms of the inverter. If the optimal switch combination in the last cycle is s(k ) = (sa (k ), sb (k ), sc (k )), the possible switch
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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-11-1
-120
-221
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-110
-210
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Fig. 3. Vector map based on minimum common mode voltage of NPC/H inverter.
combination is:
⎧ s(k + 1 )1 = (sa (k ) sb (k ) sc (k )) ⎪ ⎪ ⎪ s(k + 1 )2 = (sa (k ) + 1 sb (k ) sc (k )) ⎪ ⎪ ⎪ ⎨s(k + 1 )3 = (sa (k ) − 1 sb (k ) sc (k )) s(k + 1 )4 = (sa (k ) sb (k ) + 1 sc (k )) ⎪ ⎪ s(k + 1 )5 = (sa (k ) sb (k ) − 1 sc (k )) ⎪ ⎪ ⎪ ⎪ ⎩s(k + 1 )6 = (sa (k ) sb (k ) sc (k ) + 1 ) s(k + 1 )7 = (sa (k ) sb (k ) sc (k ) − 1 )
(12)
This method can be applied to any topological structure and the following conditions should be satisfied:
⎧ Two − leveltopology : sx (k ) ± 1 ∈ [0 1] ⎪ ⎪ ⎪ ⎪ ⎨T here − leveltopology : sx (k ) ± 1 ∈ [−1 1] F ive − leveltopology : sx (k ) ± 1 ∈ [−2 2] ⎪ ⎪ Se v en − le v eltopology : s ( k ) ± 1 ∈ [−3 3] ⎪ x ⎪ ⎩ (2C + 1 ) − leveltopology : sx (k ) ± 1 ∈ [−C C]
is 7, for example, the possible switch combinations of (1 0 −1)include (1 0 −1), (0 0 −1), (2 0 −1), (1 1 −1), (1 −1 −1), (1 0 0), (1 0 −2). For the above optimized switch combination design algorithm, there is only one switching in one bridge arm in each control cycle, which can avoid the voltage jump and guarantee the lowest switching loss. Therefore, FCS-MPC cost function of NPC/H five-level inverter can be simplified as:
+kcmv fcmv x ∈ {a, b, c}
(13) Therefore, two-level topology needs only 4 prediction and comparison operations and the multilevel (above three-level) topology needs no more than 7 prediction and comparison operations, and the selected vectors are all adjacent vectors. Take the NPC/H five-level inverter in Fig. 4 as an example. According to the above optimized switch combination design algorithm, the number of optimized switch combinations corresponding to 6 vertices of the outermost hexagon is 4, for example, the possible switch combinations of (−2 2 −2) include (−2 2 −2), (−1 2 −2), (−2 2 −1), (−2 1 −2). The number of optimized switch combinations of other vectors of the outermost hexagon except 6 vertices is 5, for example, the possible switch combinations of (−2 0 2) include (−2 0 2), (−2 1 2), (−2 −1 2), (−2 0 1), (−1 0 −2). The number of optimized switch combinations other vectors except the outermost hexagon
min(g) = i∗α (k + 1 ) − iαp (k + 1 ) + i∗β (k + 1 ) − iβp (k + 1 )
(14)
This optimized switch combination algorithm not only greatly reduces the computation, but also simplifies the control algorithm. Only the weighting factor of common mode voltage needs to be adjusted. 3.2. Satisfactory optimization of common mode voltage The satisfactory optimization theory was first proposed by H.A. Simon, the Nobel Prize laureate in economics, in 1978 in the economic organization’s research on the practical decision-making. The satisfactory optimization is more in line with the way to solve actual problem. The results that people pursue are satisfactory not the optimal. The satisfactory optimization idea is ubiquitous in the real life, and has been widely used in industrial production scheduling, power network planning, mechanical optimization design and other fields [23,24]. Eq. (11) indicates that the common mode voltage amplitude of NPC/H five-level inverter has 7 levels: 2E, 5E/3, 4E/3, E, 2E/3, E/3, 0. Establish cost function by the above optimized switch combination selection algorithm and the traditional weight factor method. If the weight factor of the common mode voltage suppression is too small,
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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0-12 -1-21
-1-22
1-12 0-21
0-22
2-12 1-21
1-22
2-21 2-22
Fig. 4. Optimized switch combination selection schematic diagram.
it is difficult to guarantee that the switch combination (2 2 2) and (−2 −2 −2)are not involved in the control, which may lead to large current fluctuations, and the common mode voltage amplitude range cannot be precisely controlled. Therefore, the satisfactory optimal control method could be used to replace the tradition weight factor adjusting by the satisfactory optimization of common mode voltage amplitude. Assume that the satisfactory function for the common mode voltage design suppression is: ⎧
μcmv =
⎨0
⎩∞
E |sa + sb + sc | ≤ E 3 E |sa + sb + sc | > E 3
(15)
voltage suppression, and the given current value changes from 15 A to 40 A at 0.062 s. Fig. 5 (b) and (c) show the simulation waveform of the traditional weight factor method with the weight factor of 0.01 and 0.1. This indicates that the traditional weight factor method fails to control the amplitude range of the common mode voltage. The common mode voltage suppression cannot be achieved if the weight factor is too small, while the current control effect will be affected if the weight factor is too large. As shown in Fig. 5 (d) and (e), under the same load and given conditions, THD with the satisfactory optimization strategy of common mode voltage is 2.16%, while when the traditional weight factor method with the weight factor of 0.1 is adopted, THD increases to 3.37%.
The cost function based on the satisfactory optimization of common mode voltage is:
g=
⎧
⎨ i∗α (k + 1 ) − iαp (k + 1 ) + i∗ (k + 1 ) − i p (k + 1 ) β
⎩∞
β
E |sa + sb + sc | ≤ E 3 E |sa + sb + sc | > E 3
4. Simulation and experiment 4.1. Simulation study Establish the simplified FCS-MPC simulation model of NPC/H five-level inverter based on common mode voltage in Matlab/Simulink to verify the effectiveness of the algorithm. The simulation parameters are shown as follows: DC side capacitor voltage E = 300 V, three-phase load R = 15 , L = 9 mH, control period Ts = 0.0001 s, filter capacitor C1 = C2 = 3300 uF. (1)Simulation analysis of common mode voltage control The satisfactory optimization strategy of common mode voltage proposed in this paper and the traditional weight factor method are used to simulate the common mode
(16)
The simulation also verifies that FCS-MPC has good dynamic performance and the current response time of the two strategies is about 2 ms. Therefore, by the simplified FCS-MPC method based on the satisfactory optimization strategy of common mode voltage, it is easy to realize the common mode voltage amplitude control of NPC/H5 L inverter without the weight factor adjusting. This could greatly simplify the design process of the controller. (2)The simplified FCS-MPC strategy analysis based on the satisfactory optimization strategy of common mode voltage (a) The relationship between THD, switching frequency and current modulation depth The traditional PWM modulation strategy THD decreases with the increase of the voltage modulation depth.
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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Fig. 5. Simulation waveform about common mode voltage control.
The switching frequency also varies with the change of the voltage modulation depth, while the FCS-MPC is the direct current control strategy and there is no voltage modulation depth. To this end, the current modulation depth is defined as:
mi = Ipeak_re f /Ipeak_ max
(16 )
In this equation, I peak_re f is the given current peak andI peak_ max is the maximum current peak determined by the circuit. Fig. 6 shows the current control effect and simulation waveform about THD, switching frequency of FCS-MPC strategy in full current modulation based on the satisfactory optimization strategy of common mode voltage. The
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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Switching Frequency[Hz]
8
[m3Gdc;January 23, 2016;16:49]
THD/%
60 40 20 0
0
0.5 Mi
500 400 300 200 100 0
1
0
0.5 Mi
1
Fig. 6. Full current modulation simulation waveform about THD, switching frequency (b) The relationship between THD, switching frequency and L/R.
R=15 R=1.5 R=150
THD/%
60 40 20 0
0
0.2
0.4
0.6
0.8
1
Mi
Switching Frequency[Hz]
80
500 R=1.5
400 300 200
R=15
100 0
R=150 0
0.2
0.4
0.6
0.8
1
Mi
(a) THD
(b) Switching frequency
Fig. 7. Full current modulation simulation waveform under different L/R. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
current modulation depth increases by 1% and its maximum value is 1.1. Thus, it can be seen that the harmonic distortion rate decreases with the increase of current modulation depth and it then rises gradually after the over modulation. The switching frequency is low in the full modulation depth, less than 420 Hz. In the low modulation depth (mi < 0.2), the harmonic distortion is relatively great which is because FCS-MPC belongs to the direct current control and the effect of the current component is far less than that of voltage vector in the predictive model. Fig. 7 shows full current modulation simulation waveform under different L/R. The blue waveform is the simulation waveform when L/R remains unchanged and the capacitor voltage decreases from E = 300 V to E = 30 V, equivalent to the fact that L and R increase by ten times at the same time; the red waveform is simulation waveform when L/R rises by ten times; the black waveform is the simulation waveform when L/R decreases by ten times. Compared with Fig. 6, the distribution law of THD and switching frequency under certain L/R is unique; THD decreases with the increase of L/R; the switching frequency increases with the increase of L/R. (c) Relationship between THD, switching frequency, fundamental frequency and sampling frequency Fig. 8 shows the simulation waveform under different fundamental frequency and sampling frequency. The blue waveform is the simulation waveform with the fundamental frequency of 50 Hz and the sampling frequency of 10 kHz. The red waveform is the simulation waveform with the fundamental frequency of 25 Hz and the sampling frequency of 10 kHz. The black waveform is the simulation waveform with the fundamental frequency of 50 Hz and the sampling frequency of 20 kHz. Therefore, THD and the
Table 2 The main components and parameters. Name
Specifications and parameters
Rectifier module DC-side capacitor IGBT module Resistance-inductance load Voltage sensor Current sensor DSP FPGA
1600 V, 75 A C1 = C2 = 3300 uF 1200 V, 50 A R = 15 , L = 9 mH LV28-P, Rated voltage 270 V LA25-NP, Rated current 12 A TMS320F28335, 50 MHz XC3S500E, 50 MHz
switching frequency have nothing to do with the fundamental frequency. THD decreases with the increase of sampling frequency; the switching frequency rises with the increase of sampling frequency. 4.2. Experimental verification TPS2012B Tektronix oscilloscope is used to observe the given value i∗a and the actual value ia of a-phase current. Fluke43B power quality analyzer is used to detect the load common mode voltage and current harmonic distortion rate. Ac-side of the uncontrolled rectifier is 27 V and the main components and parameters are as shown in Table 2. Fig. 9 (a) shows current following experimental waveform when the reference current has sudden change. The result that the actual current strictly follows the given change is basically identical with simulation analysis in Fig. 5 (a), and verifies that FCS-MPC has good dynamic performance. Fig. 9 (b) is the corresponding common mode voltage waveform, the common mode voltage amplitude is always controlled within the ±E range, which indicate the
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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50Hz 50Hz 25Hz 25Hz20kHz 20Hz
THD/%
50 40 30 20 10 0
1000 Switching Frequency[Hz]
60
0.2
0.4
0.6
0.8
1
20k
800 600 400
50Hz
200 0
0
9
25Hz 0
0.2
0.4
0.6
0.8
1
Mi
(a) THD
(b) Switching frequency
Fig. 8. Simulation waveform under different fundamental frequency, sampling frequency. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Experimental waveforms.
proposed control strategy based on satisfactory optimization algorithm could realize accuracy control of commonmode voltage amplitude. Fig. 9 (c) and (d) show the measured value of THD under different modulation depth, THD is relatively large with the low current modulation depth and it decreases with the increase of current modulation depth. The experimental results are consistent with the simulation analysis, which verifies the characteristics and feasibility of the control strategy proposed in this paper. 5. Conclusion In the design of the cost function in the FCS-MPC system, the traditional method based on weighting factors demonstrates some limitations, such as the weighting
factors adjusting and heavy predictive calculation due to the increased number of voltage vectors applied in modulating multilevel converters. This paper proposes a simplified FCS-MPC method based on common mode voltage satisfactory optimization, which could considerably reduce the predictive calculation by the optimized switch combination and simply the cost function design. Moreover, satisfactory optimization is adopted to achieve the accuracy control of common-mode voltage amplitude without adjusting process of weighting factors and simplify the controller design process. Then it carries out detailed analysis of the relationship between harmonic distortion rate, switching frequency and current modulation depth, coupled with the relationship between harmonic distortion rate, fundamental frequency and sampling frequency. The
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simulation and experimental results verify the characteristics and feasibility of this control strategy. References [1] Culter CR, Ramaker BL. Dynamic matrix control – A computer control algorithm. In: Proceedings of the joint automatic control conference; 1980. [2] Qin SJoe, Badgwell ThomasA. A survey of model predictive control technology. Control Ind Eng Pract 2003;11(7):733–64. [3] Maciejowski JM. Predictive control with constraints. Englewood Cliffs, NJ: Prentice-Hall; 2002. [4] Yugeng Xi. Predictive control. Beijing: National Defence Industry Press; 2013. p. 1–8. [5] Kennel R, Linder A. Predictive control of inverter supplied electrical drives, In: Proceedings of the thirty first IEEE power electronic specialists conference, vol. 2, p. 2000 . [6] Muller S, Ammann U, Rees S. New modulation strategy for a matrix converter with a very small mainsfilter. In: Proceedings of the IEEE power electronic specialists conference; 2003. p. 1275–80. [7] Rodriguez J, Pontt J, Silva C, Correa P, Lezana P, Cortes P, Ammann U. Predictive current control of a voltage source inverter. IEEE Trans Ind Electron 2007;54(1):495–503. [8] Wolbank TM, Stumberger R, Lechner A, Machl J. Novel approach of constant switching-frequency inverter control with optimum current transient response. In: Proceedings of the international symposium on power electronics, electrical drives, automation and motion (SPEEDAM); 2010. p. 803–8. [9] Zhang Y, Zhu J, Hu J. Model predictive direct torque control for grid synchronization of doubly fed induction generator. In: Proceedings of the 2011IEEE international conference on electric machines drives (IEMDC); 2011. p. 765–70. [10] Vargas R, Cortes P, Ammann U, Rodriguez J, Pontt J. Predictive control of a three-phase neutral-point-clamped inverter. IEEE Trans Ind Electron 2007;54(5):2697–705. [11] Miranda H, Teodorescu R, Rodriguez P, Helle L. Model predictive current control for high-power grid-connected converters with output LCL filter. In: Proceedings of the 2009 thirty fifth annual conference of IEEE industrial electronics; 2009. p. 633–8. [12] Cortes P, Wilson A, Kouro S, Rodriguez J, Abu-Rub H. Model predictive control of multilevel cascaded H-bridge inverters. IEEE Trans Ind Electron 2010;57(8):2691–9.
[13] Townsend C, Summers T, Betz R. Multi-goal heuristic model predictive control technique applied to a cascaded H-bridge statcom. IEEE Trans Power Electron 2012;27(3):1191–200. [14] Defay F, Llor AM, Fadel M. Direct control strategy for a fourlevel three-phase flying-capacitor inverter. IEEE Trans Ind Electron 2010;57(7):2240–8. [15] Thielemans S, Vyncke TJ, Melkebeek JAA. Voltage quality analysis of a three-level flying capacitor inverter with model based predictive control. In: Proceedings of the IEEE eighth international conference on power electronics ECCE Asia (ICPE and ECCE); 2011. p. 124–31. [16] Kieferndorf F, Karamanakos P, Bader P, Oikonomou N, Geyer T. Model predictive control of the internal voltages of a five-level active neutral point clamped converter. In: Proceedings of the 2012 IEEE energy conversion congress and exposition (ECCE); 2012. p. 1676–83. [17] Oikonomou N, Gutscher C, Karamanakos P, Kieferndorf FD, Geyer T. Model predictive pulse pattern control for the fivelevel active neutral-point-clamped inverter,. IEEE Trans Ind Appl 2013;49(6):2583–92. [18] Cortes P, Rodriguez J, Antoniewicz P, Kazmierkowski M. Direct power control of an AFE using predictive control. IEEE Trans Power Electron 2008;23(5):2516–23. [19] Villarroel F, Espinoza JR, Rojas CA, Rodriguez J, Rivera M, Sbarbaro D. Multi objective switching state selector for finite-states model predictive control based on fuzzy decision making in a matrix converter. IEEE Trans Ind Electron 2013;60(2):589–99. [20] Xiaodong Cao, Guojun Tan, Conggang Wang, Hao Li. A low switching frequency PWM rectifier with satisfactory predictive control strategy. Chin J Electr Eng 2013;27:69–77. [21] Zhang Yalin, Lin Hua. Simplified model predictive current control method of voltage-source inverter. In: Proceedings of the 2011 IEEE eighth international conference on power electronics and ECCE Asia (ICPE & ECCE); 2011. p. 1726–33. [22] Iqbal A, Abu-Rub H, Ahmed SKM, Cortes P, Rodriguez J. Model predictive current control of a three-level five-phase NPC VSI using simplified computational approach. In: Proceedings of the 2014 twentyninth annual IEEE applied power electronics conference and exposition (APEC); 2014. p. 2323–30. [23] Ping Ren. Satisfactory criterion in optimization theory. Fuzzy Math 1983;4:111–12. [24] Shaoyuan Li, Yugeng Xi. Satisfaction control based on fuzzy goals and fuzzy constraint. Control Decis 2000;06:674–7.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithmR Guifeng Wang∗, Jianguo Jiang, Wei Wu School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200030, China
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Satisfactory optimization NPC/H Nonlinear FCS-MPC Common-mode voltage suppression
a b s t r a c t In the design of the cost function in the nonlinear finite control set model predictive control (FCS-MPC) system, the traditional method based on weighting factors demonstrates some limitations, such as the weighting factors adjusting and heavy predictive calculation due to the increased number of voltage vectors applied in controlling multilevel converters. This paper proposes a simplified FCS-MPC method based on common mode voltage satisfactory optimization, which could considerably reduce the predictive calculation by the optimized switch combination and simplify the cost function design. Moreover, satisfactory optimization is adopted to achieve the accuracy control of common-mode voltage amplitude without adjusting process of weighting factors. The simulation and experimental results verify the feasibility of this control strategy. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Model predictive control (MPC) has now more than three decades of sustained development and is considered as one of the most important advances in process control. It is widely used in the multivariable, nonlinear, strong coupling industrial control fields with multi constraint conditions, including chemical industry, metallurgy, aerospace, mechanical manufacturing, electric power and so on [1–4]. It was first proposed in 2000 that MPC technology can be applied to power electronics and power transmission [5]. In 2003, the model predictive method with ergodic optimization for the combination of all the switching states of matrix converter was first applied to control the matrix converter [6]. In [7], the finite control R This work was supported by the National High Technology Research and Development Program of China (863 Program) under Grant no. 2011AA050403 and Doctoral Program of Higher Specialized Research Fund (new teachers class) Project (20110073120034). ∗ Corresponding author. Tel.: +86 13813459613; fax: +86 51683882453. E-mail address: [email protected], [email protected] (G. Wang).
set model predictive control (FCS-MPC) of converter was described in detail which has many advantages, such as intuitive modeling, easy to understand, direct control, easy to handle the multi-constraint conditions and multi-objective. Moreover, compared with the traditional control algorithm, it has no PWM modulator and there is no need to adjust PI parameters. Therefore, FCS-MPC has become a popular research direction in the field of predictive control of power converters. There are many researches about converter topology with FCS-MPC system, such as two-level voltage source inverter (2L-VSI) [8,9], three-level neutral point clamped inverter (3L-NPC) [10,11], flying-capacitor inverter (FCI) [14,15], active-front-end rectifiers (AFE) [18], cascaded Hbridge multilevel converters (CHB) [12,13], matrix converters (MC) [6], five-level active neutral-point-Clamped Inverter (5L-ANPC) [16,17]. This paper studies the threelevel neutral-point-clamped H-bridge (NPC/H) cascaded five-level inverter. H-bridge cascaded multi-level topology is one of the sophisticated topological structures. Compared with the two-level H-bridge topology, NPC/H cascaded multi-level topology requires less number of independent DC powers, which is conducive to simplifying the
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Twelve pulse rectification system
NPC/H five-level inverter system R
6 pulse rectifier
A B C
Three phase load
6 pulse rectifier
Fig. 1. NPC/H five-level inverter main circuit topology.
Table 1 Multilevel converter FCS-MPC prediction and optimization times. Name
Two-level inverter (such as: 2L-VSI) Three-level inverter(such as: 3L-NPC) Five-level converter (such as: NPC/H-5 L) Seven-level converter (such as 3H -CHB) (2C+1) Level converter
Output phase voltage level
2 3 5 7 2C+1
structure of converter. ACS5000 inverter of ABB Company adopts the NPC/H five-level topology with the main circuit structure, as shown in Fig. 1. The basic control strategy of FCS-MPC system is to conduct predictive computation for the optimized switch combination with the traversing method and compare the objective functions to select the best one. The difficulty of FCS-MPC controller design lies in the weighting factor adjusting in the cost function. However, the root cause that obstructs the application of FCS-MPC system is its heavy computation, especially in the field of multilevel converter control [12]. Repeated computation 7 times should be conducted in one sampling cycle even using the two-level voltage source inverter (2L-VSI) which has the simplest structure and has been most widely employed in industry, and ignoring the problems of redundant voltage vectors and multistep prediction. Compared with the traditional PI control system, the computation is obviously heavy. As the power electronic technology develops towards the direction of high voltage and large capacity, the converter also develops towards the multi-level and multiplex direction, and the increase of the levels causes the exponential growth of switch combination and rise of computation. Table 1 shows the multilevel converter FCSMPC prediction and optimization times with the traversing method. Some complex applications such as motor drive control require more complex control system model, for the controller should complete the flux observation, coordinate transformation, speed closed loop, closed-loop PI control
Voltage vector number All voltage vector
Basic voltage vector
8 27 125 343 (2C + 1)3
7 19 61 127 12C2 + 6C + 1
algorithm in addition to the FCS-MPC predictive control operations, which means greater computational burden. According to FCS-MPC predictive control principle, it is necessary to improve the sampling frequency of the system based on the hardware requirement in order to get better control performance. However, the inherent problem of FCS-MPC – heavy computation, limits the room for improvement. Therefore, to promote the application of FCS-MPC in the field of multilevel converter control, heavy computation should be reduced first. At present, algorithms of reducing the computation of FCS-MPC mainly focus on specific applications, thus lacking generality. The cost function design and the weight adjusting problem in the multi objective control are ignored as a result [10,12,20,21]. As it is difficult to conduct offline adjusting for the cost function established by the traditional weighting factor method, in [19], the fuzzy control concept was introduced in the cost function design, and then the membership function replaced the weight function to achieve the online adjusting of cost function. In [20] the satisfactory optimization theory was combined with the fuzzy control concept to simplify the membership function design. On the basis of the satisfactory optimization control theory, this paper aims at the simplification of cost function design and explores the optimal switch combination to reduce the computation. Then, the satisfactory optimization concept is introduced to replace the adjusting process of weighting factors by the common mode voltage satisfactory optimization to solve this problem.
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3
o
Vdc C1 2
Vdc C2 2
Sa2
Sa1
1
1
Sa22
Sa12
Sa23
Sa13
Sa24
Sa14
Vdc C3 2
Vdc C4 2
Sb2 1
Sb11
Sb22
Sb12
Sb23
Sb13
Sb24
Sb14
Vdc C5 2
Vdc C6 2
Sc2
Sc1
1
1
Sc22
Sc12
Sc23
Sc13
Sc24
Sc14
L
L
L
R
R
R
n Fig. 2. The topology of NPC/H five-level inverter with resistive inductive load.
function in theαβ two-phase static coordinate system after the 3/2 transformation is shown as follows
2. FCS-MPC controller design of NPC/H five-level inverter
uα uβ
2.1. FCS-MPC predictive model of NPC/H five-level inverter The main circuit topology of NPC/H five-level inverter with resistive inductive load is as shown in Fig. 2. Assuming the three-phase load is balanced, the mathematical model based on the three-phase static coordinate system is:
2 V = (vao + avbo + a2 vco ) 3
(1)
2 I = (ia + aib + a2 ic ) 3
(2)
2E 1 = 3 0
−1/2 √ 3/2
(3)
a = e j2π /3 is the complex operator, vao , vbo and vco represent the voltage of the inverter output terminal to the DC side midpoint o. R and L respectively represents the resistance and inductance of one-phase load. Adopt first-order forward Euler formula to discretize load current differential term dI/dt and obtain the predictive value Ip (k + 1 )of the current at the moment of k + 1.
Ip ( k + 1 ) = 1 −
RTs Ts I (k ) + V (k ) L L
(4)
Ts is the cycle of the control system. Define the switching function of NPC/H five-level inverterSx , x ∈ {a, b, c},and Sx ∈ {−2, −1, 0, 1, 2}. The five possible values of Sx correspond five possible output level states of each phase of the inverter, and then there is:
uxo = Sx E
(5)
E is half of the DC side voltage. According to the equivalent principle of amplitude, the relationship between the voltageuα , uβ and the switching
Sa Sb Sc
(6)
In the same way, the Eq. (4) is converted to the αβ two-phase static coordinate system and obtain the discrete domain prediction model of the resistance–inductance load with[iα (k), iβ (k)]T as the state variable can be obtained and the switching function combination [Sa , Sb , Sc ]T ∈ {−2, −1, 0, 1, 2}3 based on the state space.
Sa ( k ) iα (k + 1 ) i (k ) = (ATs + I ) α + Ts B Sb (k ) iβ (k + 1 ) iβ (k ) Sc ( k ) In the equation
dI V = RI + L dt
−1/2 √ − 3/2
A=
−L/R 0
2E 1 0 ,B = −L/R 3L 0
−1/2 √ 3/2
−1/2 √ − 3/2
(7)
2.2. FCS-MPC cost function design of NPC/H five-level inverter The optimal performance indicators of NPC/H five-level variable frequency drive system include: current following control, dv/dtjump limit, reduction of switching loss, common mode voltage suppression and capacitor voltage balance control. At this point, NPC/H-FCS-MPC of NPC/H fivelevel inverter is transformed into the online solution of multi objective optimization problem. Generally, the traditional FCS-MPC system uses the weight method to construct the cost function. This paper adopts the main circuit topology, as shown in Fig. 1. The neutral point of the capacitor directly connects to the neutral point of series 12 pulse rectifier, which could ensure the balance of the capacitor voltage. However, NPC/H five-level inverter single-phase outputs two levels ± E with two different working modes which have different influence on the capacitor voltage. Therefore, it is necessary to carry out switching mapping rationally. In this paper, the simplest method is adopted, that is, two kinds of
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working mode are alternatively used to ensure the voltage balance control, so as to reduce an optimal performance indicator that needs to be considered for the NPC/H fivelevel inverter drive system. Reducing switching loss means to reduce the switching frequency in each control cycle. The switching frequency for each phase of the inverter meets the following conditions:
f sx = 2|sx (k + 1 ) − sx (k )|, x ∈ {a, b, c}
(8)
For the NPC/H five-level inverter topology, the phase voltage and line voltage for each switching cannot exceed two-level jump in principle. The former is harmful to the inverter circuit and causes the damage to the switch device; the latter is harmful to the motor, since the over highdv/dt can easily damage the insulation of the motor and produce serious electromagnetic interference (EMI). Therefore, the limit of output phase voltage dv/dtof the inverter should meet the demand|sx (k + 1 ) − sx (k )| < 2, x ∈ {a, b, c}; the limit of output line voltagedv/dt should meet the demand|sa (k + 1 ) − sb (k + 1 )| < 2, |sb (k + 1 ) − sc (k + 1 )| < 2, |sb (k + 1 ) − sc (k + 1 )| < 2. Then the two optimal performance indicators, thedv/dt jump limit and the reduction of switching loss, can be unified as:
⎧ ∞ |sx (k + 1 ) − sx (k )| ≥ 2, x ∈ {a, b, c} ⎪ ⎪ ⎪ ⎪ ⎨∞ |sa (k + 1 ) − sb (k + 1 )| ≥ 2, or |sb (k + 1 ) − sc (k + 1 )| ≥ 2, or fswitch = |sb (k + 1 ) − sc (k + 1 )| ≥ 2 ⎪ ⎪ ⎪ ⎪ ⎩2(|sa (k + 1 ) − sa (k )| + |sb (k + 1 ) − sb (k )| +|sc (k + 1 ) − sc (k )| )
(9)
The NPC/H-FCS-MPC cost function established by the weighting method is shown as follows:
g=
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∗
iα (k + 1 ) − iαp (k + 1 ) + i∗β (k + 1 ) − i p (k + 1 ) β
+ksw fswitch + kcmv fcmv
(10)
In Eq. (10), ksw and kcmv are the weighting factor; i∗α (k + 1 )and i∗β (k + 1 ) represent the given current in
theαβ coordinate system at the moment oftk+1 and they are unknown value. In the actual implementation process of the algorithm, as the sampling time of the system is short enough, it can be approximated to be the given current at the moment of tk , that is, i∗α (k + 1 ) ≈ i∗α (k ), i∗β (k +
1 ) ≈ i∗β (k ). iα (k + 1 )and iβ (k + 1 ) represent the predictive
current in theαβ coordinate system corresponding to the different switch combination of the inverter at the moment of tk+1 , which can be obtained by Eq. (7). fcmv is the common mode voltage and can be obtained by the following equation.
fcmv =
1 E (uao + ubo + uco ) = (sa + sb + sc ) 3 3
(11)
Different values of weighting factorksw andkcmv in Eq. (10) enable different control effect of NPC/H five-level inverter to be obtained. The weighting method is a multiobjective optimization control method which regards the optimal solution of the cost function as the optimization
objective, thus ensuring the uniqueness of the optimal solution. However, this method has obvious shortcomings, and it is difficult to configure a single weighting factor for the multi-objective control task and achieve different control effect expected by designers. Besides, the control effect of each objective is closely related to the selection of weighting factor, which leads to the difficulty of weighting factor adjusting. In addition, due to the conflict between the optimal performance indicators, the over optimization of a certain performance indicator of NPC/H five-level inverter will inevitably lead to the control quality deterioration of other indicators. 3. Simplified FCS-MPC strategy based on satisfactory optimization of common mode voltage 3.1. Optimal switch combination design The cost function of the NPC/H five-level inverter is as shown in Eq. (10) where two weighting factors are required to be adjusted. The current following is regarded as the core control objective, while the switching loss and the common mode voltage suppression are regarded as the target of performance optimization. By the predictive model, the redundant switch combination has the same effect on current following performance, but its effect on switching loss and common mode voltage is different. Therefore, the cost function of NPC/H five-level inverter established by the weighting method can be analyzed separately from two aspects, aiming to explore the control strategy which is conducive to the design of cost function and the reduction of the computation. Remove the redundant vectors with great common mode voltage to reduce the common mode voltage of the motor. Then 125 vectors can be simplified to 61 independent vectors, as shown in Fig. 3. However, for the FCS-MPC control system, the calculation burden is still too heavy. As the control period of the FCS-MPC system is short enough and the given instruction in the adjacent control cycle has slight variation when the system is operating at steadystate, voltage vectors should be selected from adjacent vectors. The adjacent vector method [21] could be used to optimize the switch combination design in order to reduce the computation. To adopt this method, there are no rules to follow by the analysis in Fig. 3. Thus, the table lookup method is used to respectively list adjacent vectors of 61 vectors. The selected adjacent vectors have different influence on the common voltage and switching loss, so the weighting factor adjusting process cannot be simplified. Therefore, this method is not universal. The switching loss is expected to be lower possible, especially the high-power transmission device. Based on the idea of [22], only one phase or one level transformation in each control period is allowed, that is, there are no more than two switches operating in one control cycle. This optimized switch combination design method can realize the minimum switching loss in a single control cycle, and the maximum number of optional vector is 2 ∗ N + 1 whereNis the number of bridge arms of the inverter. If the optimal switch combination in the last cycle is s(k ) = (sa (k ), sb (k ), sc (k )), the possible switch
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-12-2
-22-2 -22-1 -220
-11-1
-120
-221
02-1
-12-1
-110
-210
12-2
02-2
22-2
010
21-2
11-2
01-2
01-1
5
11-1
10-1
00-1
20-2
10-2
2-1-2
20-1
3 -222
-111
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-202 -2-12
-2-22
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0-12
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1-1-1
100
001
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000
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0-10 0-11
1-21
2-2-2
2-2-1
2-10
1-20
1-11
0-21
0-22
1-10
2-1-1
2-20 2-21
2-22
1-22
Fig. 3. Vector map based on minimum common mode voltage of NPC/H inverter.
combination is:
⎧ s(k + 1 )1 = (sa (k ) sb (k ) sc (k )) ⎪ ⎪ ⎪ s(k + 1 )2 = (sa (k ) + 1 sb (k ) sc (k )) ⎪ ⎪ ⎪ ⎨s(k + 1 )3 = (sa (k ) − 1 sb (k ) sc (k )) s(k + 1 )4 = (sa (k ) sb (k ) + 1 sc (k )) ⎪ ⎪ s(k + 1 )5 = (sa (k ) sb (k ) − 1 sc (k )) ⎪ ⎪ ⎪ ⎪ ⎩s(k + 1 )6 = (sa (k ) sb (k ) sc (k ) + 1 ) s(k + 1 )7 = (sa (k ) sb (k ) sc (k ) − 1 )
(12)
This method can be applied to any topological structure and the following conditions should be satisfied:
⎧ Two − leveltopology : sx (k ) ± 1 ∈ [0 1] ⎪ ⎪ ⎪ ⎪ ⎨T here − leveltopology : sx (k ) ± 1 ∈ [−1 1] F ive − leveltopology : sx (k ) ± 1 ∈ [−2 2] ⎪ ⎪ Se v en − le v eltopology : s ( k ) ± 1 ∈ [−3 3] ⎪ x ⎪ ⎩ (2C + 1 ) − leveltopology : sx (k ) ± 1 ∈ [−C C]
is 7, for example, the possible switch combinations of (1 0 −1)include (1 0 −1), (0 0 −1), (2 0 −1), (1 1 −1), (1 −1 −1), (1 0 0), (1 0 −2). For the above optimized switch combination design algorithm, there is only one switching in one bridge arm in each control cycle, which can avoid the voltage jump and guarantee the lowest switching loss. Therefore, FCS-MPC cost function of NPC/H five-level inverter can be simplified as:
+kcmv fcmv x ∈ {a, b, c}
(13) Therefore, two-level topology needs only 4 prediction and comparison operations and the multilevel (above three-level) topology needs no more than 7 prediction and comparison operations, and the selected vectors are all adjacent vectors. Take the NPC/H five-level inverter in Fig. 4 as an example. According to the above optimized switch combination design algorithm, the number of optimized switch combinations corresponding to 6 vertices of the outermost hexagon is 4, for example, the possible switch combinations of (−2 2 −2) include (−2 2 −2), (−1 2 −2), (−2 2 −1), (−2 1 −2). The number of optimized switch combinations of other vectors of the outermost hexagon except 6 vertices is 5, for example, the possible switch combinations of (−2 0 2) include (−2 0 2), (−2 1 2), (−2 −1 2), (−2 0 1), (−1 0 −2). The number of optimized switch combinations other vectors except the outermost hexagon
min(g) = i∗α (k + 1 ) − iαp (k + 1 ) + i∗β (k + 1 ) − iβp (k + 1 )
(14)
This optimized switch combination algorithm not only greatly reduces the computation, but also simplifies the control algorithm. Only the weighting factor of common mode voltage needs to be adjusted. 3.2. Satisfactory optimization of common mode voltage The satisfactory optimization theory was first proposed by H.A. Simon, the Nobel Prize laureate in economics, in 1978 in the economic organization’s research on the practical decision-making. The satisfactory optimization is more in line with the way to solve actual problem. The results that people pursue are satisfactory not the optimal. The satisfactory optimization idea is ubiquitous in the real life, and has been widely used in industrial production scheduling, power network planning, mechanical optimization design and other fields [23,24]. Eq. (11) indicates that the common mode voltage amplitude of NPC/H five-level inverter has 7 levels: 2E, 5E/3, 4E/3, E, 2E/3, E/3, 0. Establish cost function by the above optimized switch combination selection algorithm and the traditional weight factor method. If the weight factor of the common mode voltage suppression is too small,
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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121 221 210 021 20-1 010 110 2-1-2 10-1 -110 -10-1 00-1 0-1-2 1-1-2 -20-1 -2-1-2 222-1-1-2 122 211 200 022 111 2-2-2 011 -122 -222 100 1-1-1 2-1-1 000 -111 -100 -211 0-1-1 -1-1-1 0-2-2 1-2-2 -200 -2-1-1 -1-2-2 -2-2-2 112 212 201 012 -112 001 101 2-10 2-2-1 1-10 -101 -201 -1-10 0-10 1-2-1 -212 0-2-1 -2-10 -2-2-1 -1-2-1 002 102 202 -102 2-11 -202 -1-11 0-11 1-11 -2-11 1-20 2-20 -2-20 -1-20 0-20 -121 -210
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0-12 -1-21
-1-22
1-12 0-21
0-22
2-12 1-21
1-22
2-21 2-22
Fig. 4. Optimized switch combination selection schematic diagram.
it is difficult to guarantee that the switch combination (2 2 2) and (−2 −2 −2)are not involved in the control, which may lead to large current fluctuations, and the common mode voltage amplitude range cannot be precisely controlled. Therefore, the satisfactory optimal control method could be used to replace the tradition weight factor adjusting by the satisfactory optimization of common mode voltage amplitude. Assume that the satisfactory function for the common mode voltage design suppression is: ⎧
μcmv =
⎨0
⎩∞
E |sa + sb + sc | ≤ E 3 E |sa + sb + sc | > E 3
(15)
voltage suppression, and the given current value changes from 15 A to 40 A at 0.062 s. Fig. 5 (b) and (c) show the simulation waveform of the traditional weight factor method with the weight factor of 0.01 and 0.1. This indicates that the traditional weight factor method fails to control the amplitude range of the common mode voltage. The common mode voltage suppression cannot be achieved if the weight factor is too small, while the current control effect will be affected if the weight factor is too large. As shown in Fig. 5 (d) and (e), under the same load and given conditions, THD with the satisfactory optimization strategy of common mode voltage is 2.16%, while when the traditional weight factor method with the weight factor of 0.1 is adopted, THD increases to 3.37%.
The cost function based on the satisfactory optimization of common mode voltage is:
g=
⎧
⎨ i∗α (k + 1 ) − iαp (k + 1 ) + i∗ (k + 1 ) − i p (k + 1 ) β
⎩∞
β
E |sa + sb + sc | ≤ E 3 E |sa + sb + sc | > E 3
4. Simulation and experiment 4.1. Simulation study Establish the simplified FCS-MPC simulation model of NPC/H five-level inverter based on common mode voltage in Matlab/Simulink to verify the effectiveness of the algorithm. The simulation parameters are shown as follows: DC side capacitor voltage E = 300 V, three-phase load R = 15 , L = 9 mH, control period Ts = 0.0001 s, filter capacitor C1 = C2 = 3300 uF. (1)Simulation analysis of common mode voltage control The satisfactory optimization strategy of common mode voltage proposed in this paper and the traditional weight factor method are used to simulate the common mode
(16)
The simulation also verifies that FCS-MPC has good dynamic performance and the current response time of the two strategies is about 2 ms. Therefore, by the simplified FCS-MPC method based on the satisfactory optimization strategy of common mode voltage, it is easy to realize the common mode voltage amplitude control of NPC/H5 L inverter without the weight factor adjusting. This could greatly simplify the design process of the controller. (2)The simplified FCS-MPC strategy analysis based on the satisfactory optimization strategy of common mode voltage (a) The relationship between THD, switching frequency and current modulation depth The traditional PWM modulation strategy THD decreases with the increase of the voltage modulation depth.
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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Fig. 5. Simulation waveform about common mode voltage control.
The switching frequency also varies with the change of the voltage modulation depth, while the FCS-MPC is the direct current control strategy and there is no voltage modulation depth. To this end, the current modulation depth is defined as:
mi = Ipeak_re f /Ipeak_ max
(16 )
In this equation, I peak_re f is the given current peak andI peak_ max is the maximum current peak determined by the circuit. Fig. 6 shows the current control effect and simulation waveform about THD, switching frequency of FCS-MPC strategy in full current modulation based on the satisfactory optimization strategy of common mode voltage. The
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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Switching Frequency[Hz]
8
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THD/%
60 40 20 0
0
0.5 Mi
500 400 300 200 100 0
1
0
0.5 Mi
1
Fig. 6. Full current modulation simulation waveform about THD, switching frequency (b) The relationship between THD, switching frequency and L/R.
R=15 R=1.5 R=150
THD/%
60 40 20 0
0
0.2
0.4
0.6
0.8
1
Mi
Switching Frequency[Hz]
80
500 R=1.5
400 300 200
R=15
100 0
R=150 0
0.2
0.4
0.6
0.8
1
Mi
(a) THD
(b) Switching frequency
Fig. 7. Full current modulation simulation waveform under different L/R. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
current modulation depth increases by 1% and its maximum value is 1.1. Thus, it can be seen that the harmonic distortion rate decreases with the increase of current modulation depth and it then rises gradually after the over modulation. The switching frequency is low in the full modulation depth, less than 420 Hz. In the low modulation depth (mi < 0.2), the harmonic distortion is relatively great which is because FCS-MPC belongs to the direct current control and the effect of the current component is far less than that of voltage vector in the predictive model. Fig. 7 shows full current modulation simulation waveform under different L/R. The blue waveform is the simulation waveform when L/R remains unchanged and the capacitor voltage decreases from E = 300 V to E = 30 V, equivalent to the fact that L and R increase by ten times at the same time; the red waveform is simulation waveform when L/R rises by ten times; the black waveform is the simulation waveform when L/R decreases by ten times. Compared with Fig. 6, the distribution law of THD and switching frequency under certain L/R is unique; THD decreases with the increase of L/R; the switching frequency increases with the increase of L/R. (c) Relationship between THD, switching frequency, fundamental frequency and sampling frequency Fig. 8 shows the simulation waveform under different fundamental frequency and sampling frequency. The blue waveform is the simulation waveform with the fundamental frequency of 50 Hz and the sampling frequency of 10 kHz. The red waveform is the simulation waveform with the fundamental frequency of 25 Hz and the sampling frequency of 10 kHz. The black waveform is the simulation waveform with the fundamental frequency of 50 Hz and the sampling frequency of 20 kHz. Therefore, THD and the
Table 2 The main components and parameters. Name
Specifications and parameters
Rectifier module DC-side capacitor IGBT module Resistance-inductance load Voltage sensor Current sensor DSP FPGA
1600 V, 75 A C1 = C2 = 3300 uF 1200 V, 50 A R = 15 , L = 9 mH LV28-P, Rated voltage 270 V LA25-NP, Rated current 12 A TMS320F28335, 50 MHz XC3S500E, 50 MHz
switching frequency have nothing to do with the fundamental frequency. THD decreases with the increase of sampling frequency; the switching frequency rises with the increase of sampling frequency. 4.2. Experimental verification TPS2012B Tektronix oscilloscope is used to observe the given value i∗a and the actual value ia of a-phase current. Fluke43B power quality analyzer is used to detect the load common mode voltage and current harmonic distortion rate. Ac-side of the uncontrolled rectifier is 27 V and the main components and parameters are as shown in Table 2. Fig. 9 (a) shows current following experimental waveform when the reference current has sudden change. The result that the actual current strictly follows the given change is basically identical with simulation analysis in Fig. 5 (a), and verifies that FCS-MPC has good dynamic performance. Fig. 9 (b) is the corresponding common mode voltage waveform, the common mode voltage amplitude is always controlled within the ±E range, which indicate the
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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50Hz 50Hz 25Hz 25Hz20kHz 20Hz
THD/%
50 40 30 20 10 0
1000 Switching Frequency[Hz]
60
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20k
800 600 400
50Hz
200 0
0
9
25Hz 0
0.2
0.4
0.6
0.8
1
Mi
(a) THD
(b) Switching frequency
Fig. 8. Simulation waveform under different fundamental frequency, sampling frequency. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Experimental waveforms.
proposed control strategy based on satisfactory optimization algorithm could realize accuracy control of commonmode voltage amplitude. Fig. 9 (c) and (d) show the measured value of THD under different modulation depth, THD is relatively large with the low current modulation depth and it decreases with the increase of current modulation depth. The experimental results are consistent with the simulation analysis, which verifies the characteristics and feasibility of the control strategy proposed in this paper. 5. Conclusion In the design of the cost function in the FCS-MPC system, the traditional method based on weighting factors demonstrates some limitations, such as the weighting
factors adjusting and heavy predictive calculation due to the increased number of voltage vectors applied in modulating multilevel converters. This paper proposes a simplified FCS-MPC method based on common mode voltage satisfactory optimization, which could considerably reduce the predictive calculation by the optimized switch combination and simply the cost function design. Moreover, satisfactory optimization is adopted to achieve the accuracy control of common-mode voltage amplitude without adjusting process of weighting factors and simplify the controller design process. Then it carries out detailed analysis of the relationship between harmonic distortion rate, switching frequency and current modulation depth, coupled with the relationship between harmonic distortion rate, fundamental frequency and sampling frequency. The
Please cite this article as: G. Wang et al., Nonlinear FCS-MPC strategy of NPC/H-5 L inverter based on satisfactory optimization algorithm, Chaos, Solitons and Fractals (2016), http://dx.doi.org/10.1016/j.chaos.2015.12.021
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