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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 5190885, 13 pages https://doi.org/10.1155/2022/5190885 Research Article Weighting Factor-Less Sequential Predictive Control of LC-Filtered Voltage Source Inverters 1 1 2 Changming Zheng , Zheng Gong , and Rongwu Zhu School of Electrical Engineering, China University of Mining and Technology, Xuzhou, China School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, China Correspondence should be addressed to Zheng Gong; zgo@cumt.edu.cn Received 11 January 2022; Accepted 12 March 2022; Published 16 April 2022 Academic Editor: Akshay Kumar Saha Copyright © 2022 Changming Zheng et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To eliminate the weighting factor tuning eﬀort of typical ﬁnite-set model predictive control (FS-MPC), this paper proposes a weighting factor-less sequential model predictive control (SMPC) scheme for LC-ﬁltered voltage source inverters. Two inde- pendent cost functions for minimizing capacitor-voltage and inductor-current tracking errors are deployed in a cascaded structure, eliminating the weighting factor. First, the optimal cascade order of the cost function is selected by the internal relationship of two control variables. (en, a graphical method is proposed to determine the optimal number of candidate voltage vectors selected from the ﬁrst cost function. Moreover, to realize the strict current limitation, the current-constraint term is proved to be included in the voltage-related cost function. Another attractive feature of the proposed SMPC is that a smoother inductor-current starting response can be obtained compared to typical FS-MPC. Simulation and experimental results verify the feasibility of the presented approach. control objectives is balanced by a weighting factor. It is 1. Introduction known that the proper tuning of the weighting factor is very Two-level three-phase voltage source inverters (VSIs) with important for obtaining a satisfying control performance. output LC ﬁlters are critical topologies for renewable energy However, till now, there are no rigorous theoretical systems, including uninterruptible power supplies, distrib- guidelines for optimal selection of the weighting factor. As a uted generation systems, and parallel VSI-based AC consequence, how to select the weighting factors in FS-MPC microgrids [1, 2]. Among the existing control strategies that schemes is still an open issue. aim to obtain a high-quality output voltage, model predictive Several approaches have been proposed to tackle this control, particularly ﬁnite-set model predictive control (FS- issue. In [6], a heuristic “branch and bound” strategy was ﬁrst employed to select the weighting factors, reducing the MPC), has gained increasing attention due to its intuitive concept, inherent fast dynamic response without using number of tuning trials based on the empirical procedures. In [7], algebraic tuning of the weighting factor was proposed modulation, simple handling of system constraints, and multiple control objectives without increasing too much for current distortion minimization. However, it neglects the computational complexity [3–5]. impacts of the varying working conditions. In [8], an online Generally, FS-MPC considers a system model-based cost optimization-based weighting factor selection approach was function (CF) to forecast future system states using online proposed. (e optimal weighting factor is obtained by receding horizon optimization. For control of LC-ﬁltered optimizing the analytical expression of various control VSIs, typical FS-MPC schemes simultaneously include the objectives. Unfortunately, its computational complexity output capacitor voltage and ﬁlter-inductor current control signiﬁcantly increases with the increase of the control ob- objectives in a single CF, and the importance of these two jectives. In [9, 10], metaheuristic optimization-based 2 International Transactions on Electrical Energy Systems priority in the proposed SMPC, an inherent smoother algorithms were proposed for fast selection of weighting factors online. In [11], an artiﬁcial neural network-based current-starting response is achieved with linear loads. (e remainder of this paper is structured as follows. Section 2 approach was employed to select the weighting factor au- tomatically oﬄine. However, a large number of simulations describes the system dynamics and predictive model. Section and experiments are still inevitable for extracting suﬃcient 3 presents the working principle of proposed SMPC for LC- sample data to obtain satisfactory training results. ﬁltered VSIs. Simulation and experimental results are given On the other hand, the weighting factors are eliminated in Section 4, and the work is concluded in Section 5. in FS-MPC schemes, which is a promising strategy to solve the weighting factors’ selection issue [12–20]. In [12], a fuzzy 2. System Description and Predictive Model decision-making-based weighting factor elimination method was proposed, replacing the CF-based expression Figure 1 depicts a two-level three-phase VSI system, which is with a multiobjective optimization algorithm. However, the connected to an output LC ﬁlter. (e system dynamic model additional computational burden is the main drawback of in the α-β reference frame is depicted in Figure 2, which can this method. In [13], a ranking optimization-based strategy be expressed as was proposed. For each control objective, all candidate 1 1 voltage vectors are ranked, and the optimal voltage vector 0 − 0 ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ f ⎥ ⎢ f ⎥ i ⎢ ⎥ i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v (OVV) is selected as the one with the minimum averaged f ⎢ ⎥ f ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ d ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ � ⎢ ⎥ ⎣ ⎦ +⎢ ⎥ L , rank. In [14], the weighting factor of the stator ﬂux linkage ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ f ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (1) dt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v ⎢ ⎥ v ⎢ ⎥ i ⎢ ⎥ ⎢ ⎥ was avoided by converting multiple objectives to a single f ⎢ 1 ⎥ f ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ o ⎣ ⎦ ⎣ ⎦ √ √√ √ √√√√√√ √ √√ √ 0 0 − one. However, this method essentially assigns an equal u(t) x _(t) x(t) C C f f √√√√√√√√ √√√√√√√√ weight for all control objectives. Moreover, not all control A B systems can simply merge diﬀerent control objectives. Re- where L and C are output-ﬁlter inductance and capaci- cently, a simple sequential model predictive control (SMPC) f f scheme was proposed to eliminate the weighting factors for tance. v � v + jv , i � i + ji , v � v + jv , and i � f fα fβ f fα fβ α β o i + ji are the stationary α-β frame-based capacitance stator ﬂux linkage and torque control of induction machines oα oβ [15]. Two optimal candidate voltage vectors (CVVs) that can voltage, inductance current, converter-side voltage, and load minimize the ﬁrst CF are selected and then used to ﬁnd the current. OVV for the second CF. Following the same principle in By utilizing a zero-order hold strategy, the discrete state- [15], the sequential predictive torque control scheme was space predictive model of a second order is formulated as [21] extended for 3L-NPC fed induction machine drives in [16], and a direct SMPC scheme was proposed for grid-tied Φ Φ Γ Γ v f,k+1 11 12 f,k 11 12 k converters for grid-current and DC-voltage control in [17]. ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎣ ⎤ ⎦ ⎢ ⎥ ⎣ ⎦ � + , (2) Moreover, SMPC was introduced to multistep-prediction Φ Φ v Γ Γ i f,k 21 22 21 22 o,k f,k+1 √√√√ √√√√√ √ √√√√√√√√ √√√√ √√√√ √√√√√√ MPC schemes in [18]. Φ x Γ u k k k+1 Nevertheless, the aforementioned SMPC schemes ne- AT Aτ glect the impacts of the cascade order of CFs (COCFs) and s where Φ � e and Γ � e Bdτ. the number of CVVs selected from the ﬁrst CF on control In digital implementations, to compensate the one-step performance. By contrast, in [19], a generalized SMPC was computational delay, a two-step forward prediction ap- proposed, eliminating the limitation of COCFs in [15] using proach is employed, which is implemented by predicting the p p simulation analysis. Moreover, to tackle the priority selec- k + 2 instant values i and v as f,k+2 f,k+2 tion problem of SMPC for torque and ﬂux linkage control, p p p i � Φ i + Φ v + Γ v + Γ i , 11 12 11 k+1 12 o,k+1 an even-handed SMPC was proposed in [20] using an f,k+2 f,k+1 f,k+1 (3) p p p adaptive priority based on cross-error minimization. v � Φ i + Φ v + Γ v + Γ i , 21 22 21 k+1 22 o,k+1 f,k+2 f,k+1 f,k+1 However, this method is based on online sorting, which would increase the computational burden. Besides, all the where i can be substituted by i since the dynamics of o,k+1 o,k aforementioned SMPC schemes do not consider the han- the load current are very slow [22]. dling of system constraints. Most importantly, the deter- mination of CVVs and COCFs of SMPC schemes still lacks 3. Proposed SMPC Scheme for LC-Filtered VSI theoretical analysis, which somewhat hinders the develop- ment of SMPC schemes. (e block diagram of the proposed SMPC is depicted in To this end, a weighting factor-less SMPC scheme for Figure 3. (e core idea of this paper is to conﬁgure the two LC-ﬁltered VSIs is presented in this paper, eliminating the CFs in a sequential structure for separate control of capacitor weighting factor by incorporating two cascaded CFs for voltage and inductor current, which eliminates the weighting separate control of capacitor voltage and inductor current. factor in typical FS-MPC schemes of LC-ﬁltered VSIs. (e selection of COCFs and the number of CVVs is eval- uated by a graphical method. Besides, the current constraint is discussed, and the current-limiting term should be in- 3.1. CFs in a Cascaded Structure. For voltage control of LC- cluded in voltage-related CF to realize a strict current limit. ﬁltered VSIs, a dual-objective CF has been proved to obtain Since the inductor-current control is assigned a higher an improved voltage control performance compared to International Transactions on Electrical Energy Systems 3 2L-VSI where G and G are the voltage and current-related CFs. 1 2 (e voltage and current tracking errors are Δv � v − fx fx p p v and Δi � i − i with x � α or β. f i fx f i fx,k+2 fx fx,k+2 (e basic principle of the proposed SMPC is shown in V Load dc Figure 4. (e essence of the proposed SMPC is to se- v quentially evaluate two separate CFs. To be speciﬁc, CF 1 is v C f f ﬁrst evaluated by enumerating all eight candidate voltage vectors of a VSI. (en, the ﬁrst n (2≤ n≤7) CVVs that obtain the minimum values of CF 1 are preselected for Figure 1: Topology of a two-level three-phase LC-ﬁltered VSI. further evaluation of CF2. Finally, the OVV that can min- imize CF 2 among the preselected n CVVs is selected and applied to the VSI. Since the two CFs are conﬁgured in a i cascaded structure instead of being integrated into a single CF, the weighting factor is eliminated. v 1 i 1 f + sL sC f f 3.2. Determination of the Number of COCFs and CVVs. From Figure 4, it can be observed that the two CFs G and G 1 2 Figure 2: Second-order dynamic model of an LC ﬁlter. in (6) and (7) can be formulated into two diﬀerent orders: one is COCF1 (i.e., CF � G , CF � G ) and the other one is 1 1 2 2 COCF2 (i.e., CF � G , CF � G ). It should be noted that 1 2 2 1 VSI i i fa oa v fa diﬀerent COCFs may aﬀect the control performance since + the preselected CVVs from the ﬁrst CF for each COCF are Load fc diﬀerent. In addition, a diﬀerent number of CVVs can also DC result in diﬀerent control performances since it determines abc v abc the control set and priorities of the two control objectives. αβ αβ a,b,c i v i i 3.2.1. Selection of Optimal Number of CVVs. It is diﬃcult to f,k o,k f,k+2 directly select proper COCFs and the number of CVVs based Predictive v G +G G f,k o,k 1 lim model on (6) and (7). Herein, the control of capacitor voltage and f,k+2 CF CF 2 1 inductor current should be ﬁrst converted to the control of SMPC their separate reference voltage vector, which can be Figure 3: Block diagram of the proposed SMPC for an LC-ﬁltered expressed as VSI. p p ∗ ∗ v � v − Φ i − Φ v − Γ i , (8) v f 21 22 22 o,k f,k+1 f,k+1 conventional single-voltage-objective CF-based FS-MPC schemes [3]. However, the two control objectives are always ∗ ∗ p p v � i − Φ i − Φ v − Γ i . (9) 11 12 12 o,k deployed in a common CF and their importance is balanced i f f,k+1 f,k+1 by a weighting factor � � � � Hence, the control objective of minimizing (6) and (7) � �2 � �2 p p � ∗ � ∗ � � � � ∗ ∗ G � v − v � + λ i − i � , (4) � � becomes to ﬁnd basic voltage vectors closest to v and v , f � f � f,k+2 f,k+2 v i which means G and G can be rewritten as 1 2 with the voltage and current reference as � � � ∗ � � � ∗ ∗ ∗ ∗ ∗ (10) G � v − v , � � 1 v k+1 v � V cos ω kT + jV sin ω kT , ⎧ ⎨ f s s (5) ⎩ ∗ ∗ ∗ � � i � jC ω v + i , � �2 f f f o,k � � (11) G � �v − v . 2 k+1 ∗ ∗ where V , ω , and λ>0 are the reference voltage amplitude, Herein, we propose a graphical method to discuss all reference angular frequency, and the weighting factor. possible OVV distributions based on the distance between Since the selection of weighting factor needs tedious eight basic voltage vectors and the reference voltage vector in eﬀorts, to resolve this issue, the two control objectives above (8) and (9). Figure 5 shows the newly proposed division of 36 are split into two separate CFs below: sectors, and Table 1 lists the optimal candidate OVV dis- � � � �2 � ∗ � 2 2 tributions. For example, sector S1 and its OVV order mean � � G (6) � v − v � Δv + Δv , � � 1 f fα fβ f,k+2 ∗ ∗ that when v or v is located in sector S1, 8 basic voltage i v ∗ ∗ vectors can be sorted by their distance from v or v from � � i v � �2 � ∗ � 2 2 � � (7) G � i − i � Δi + Δi , the nearest to the farthest: v , v , v , v , v , v , v , v (since � � 2 f f,k+2 fα fβ 1 2 0 7 6 3 5 4 v � v , they are in no ﬁxed order). Next, we would try to 0 7 S4'' S10'' S2 S1 S11' S4' S5' S10' S11'' S5'' S12'' 4 International Transactions on Electrical Energy Systems Table 1: Candidate OVV distributions. v v 3 2 8 CVVs dc OVV Sector OVV order Sector OVV order min(CF ) min(CF2) VSI 0,7 v v 4 v …v v1, v2, v0, v7, v6, v3, v5, v4, v0, v7, v5, v3, v6, v2, 1 opt (1) (n) S1 S7’ v4 v1 First CF Second CF v v n 5 6 v2, v1, v0, v7, v3, v6, v4, v5, v0, v7, v4, v6, v3, v1, S2 S8’ Figure 4: Basic principle of the proposed SMPC. v5 v2 v2, v3, v0, v7, v1, v4, v6, v5, v0, v7, v6, v4, v1, v3, S3 S9’ v5 v2 v3, v2, v0, v7, v4, v1, v5, v6, v0, v7, v5, v1, v4, v2, S4 S10’ v6 v3 V V 3 2 v3, v4, v0, v7, v2, v5, v1, v6, v0, v7, v1, v5, v2, v4, S3 S5 S11’ v6 v3 v4, v3, v0, v7, v5, v2, v6, v1, v0, v7, v6, v2, v5, v3, S6 S12’ v1 v4 v,k+1 i,k+1 v4, v5, v0, v7, v3, v6, v2, v0, v7, v1, v2, v6, v3, v5, S7 S1” v1 v4 v5, v4, v0, v7, v6, v3, v1, v0, v7, v2, v1, v3, v6, v4, S6' S6'' V S1'' S1' 1 V S8 S2” 7 θ v2 v5 S7' S12' v5, v6, v0, v7, v4, v1, v3, v0, v7, v2, v3, v1, v4, v6, S9 S3” v2 v5 v6, v5, v0, v7, v1, v4, v2, v0, v7, v3, v2, v4, v1, v5, S10 S4” v3 v6 v6, v1, v0, v7, v5, v2, v4, v0, v7, v3, v4, v2, v5, v1, S11 S5” v3 v6 S9 S10 v1, v6, v0, v7, v2, v5, v3, v0, v7, v4, v3, v5, v2, v6, S12 S6” V V 5 6 v4 v1 v1, v0, v7, v2, v6, v3, v5, v0, v7, v4, v5, v3, v6, v2, Figure 5: Graphical method for selecting the number of CVVs. S1’ S7” v4 v1 v2, v0, v7, v1, v3, v6, v4, v0, v7, v5, v4, v6, v3, v1, S2’ S8” v5 v2 ﬁnd the proper number of CVVs that can guarantee a v2, v0, v7, v3, v1, v4, v6, v0, v7, v5, v6, v4, v1, v3, S3’ S9” satisfying control performance based on Figure 5 and v5 v2 v3, v0, v7, v2, v4, v1, v5, v0, v7, v6, v5, v1, v4, v2, Table 1. S4’ S10” v6 v3 (e number of CVVs should not miss some optimal v3, v0, v7, v4, v2, v5, v1, v0, v7, v6, v1, v5, v2, v4, options [20]. Considering that the ﬁnal OVV (which opti- S5’ S11” v6 v3 mizes the second CF) may cause poor performance of the v4, v0, v7, v3, v5, v2, v6, v0, v7, v1, v6, v2, v5, v3, S6’ S12” ﬁrst CF, it is expected to select an OVV that is located in the v1 v4 ﬁrst four ranks of the OVV order in Table 1 for both CFs. For example, if the CVV number for optimizing the ﬁrst CF is selected as n � 2, it can be deduced from Table 1 that nu- merous cases may cause the ﬁnal OVV to be located in the 3.2.3. Enumeration Veriﬁcation. Further, we evaluate all possible COCFs and the number of CVVs (2≤ n≤7) under last 4 ranks, which cannot guarantee the second CF to obtain a good performance. In contrast, if n � 3, the ﬁnal OVV can various working conditions by simulations with the pa- rameters listed in Table 2. be assured to be located in the ﬁrst 3 ranks for both CFs, since there is an intersection among the ﬁrst three vectors, (e performance indexes for evaluating diﬀerent COCFs and the number of CVVs are phase-voltage total harmonic i.e., v . Since v is always followed by v , selecting four CVVs 0 0 7 can also guarantee the ﬁnal OVV to be located in the ﬁrst 4 distortion (THD) and magnitude of the fundamental component. To emulate the actual implementations, one- ranks. Further, if the CVV number is selected as n≥5, the ﬁrst CF may also obtain poor performance under certain step computational delay and its compensation are also considered. Figure 6 depicts the steady-state voltage control cases. Hence, the optimal CVV number is chosen as n � 3 or performance with a diﬀerent number of CVVs and COCFs 4, which can guarantee a relative optimal performance for under diﬀerent load conditions, including a linear resistive both CFs. load and a nonlinear diode rectiﬁer bridge load. By com- paring Figures 6(a)–6(c), it shows that for COCF1, to obtain 3.2.2. Selection of Optimal COCFs. (en, it is required to a satisfactory control performance, the feasible number of determine the optimal COCFs. According to the dynamic CVVs is n �2 and 3, and the system undergoes large dis- model in Figure 2, it can be deduced that the capacitor tortions or oscillations when n>3. In contrast, the feasible voltage is determined by the inductor current. Hence, to number of CVVs for COCF2 is n � 3 to 7, and most of them obtain an optimal capacitor voltage control performance, it can result in better performance than that using COCF1. (e is necessary to ﬁrst optimize the inductor current control system has large distortions or oscillations when n< 3 since objective, which means the optimal COCF is COCF2. the lack of voltage control causes the oscillations. According S4 S7'' S8'' S2'' S8' S9' S6 S5 S11 S12 S3' S2' S9'' S3'' S8 S7 International Transactions on Electrical Energy Systems 5 Table 2: Nominal parameters of the system. Similarly, the handling of the current constraint is ∗ ∗ discussed for the proposed SMPC. According to our sim- Reference voltage V �300V, ω �100π rad/s DC bus voltage V �700V ulation study, to obtain a strict current-limiting capability, dc Sampling period T �20 µs s G should be included in the voltage-related G as shown in lim 1 LC ﬁlter L �2.4 mH, C �15µF f f Figure 3. For simplicity, considering a particular case that Nominal linear load R �60Ω the number of CVVs is selected as n � 8 in the proposed Nonlinear load R �465Ω, L �1.8 mH, C �2.2mF n n n SMPC, the current control objective G is invalid. In this case, if G is included in G , the current-limiting capability lim 2 will be lost. Hence, to realize a strict current limit under all cases, G in the proposed SMPC should be included in G . to our study, the conclusions above are still valid even with a lim 1 (e ﬂowchart of the proposed SMPC for LC-ﬁltered VSIs is much larger load current (>100 A). (is means COCF2 is depicted in Figure 8. more universal compared to COCF1. Hence, in the case of a wide range of load variations, the relative optimal control performance is obtained by COCF2 with 3 or 4 CVVs. 4. Simulation and Experimental Results It should be mentioned that the analysis above is based on the nominal system parameters. It is also important to Simulation and experimental comparisons between typical evaluate the diﬀerent numbers of CVVs and COCFs under weighting factor-based FS-MPC in (5) and the presented model mismatches. Figure 7 illustrates the typical cases of SMPC are given to validate the feasibility of the proposed model mismatches. It can be seen that 3 or 4 CVVs can oﬀer method. (e weighting factor in typical FS-MPC is set to be a relatively optimal control performance. Moreover, for all λ = 3 using an artiﬁcial-neural-network algorithm [11]. the model mismatch cases, the optimal performance is al- Figures 9(a) and 9(b) depict the basic structure and ex- ways achieved by COCF2 instead of COCF1. Hence, the perimental prototype of the whole system. (e nonlinear optimal COCF is selected as COCF2. load is a diode rectiﬁer bridge load as shown in Figure 9(c). Both control algorithms are implemented in a dSPACE DS1202 PowerPC DualCore 2-GHz processor platform with 3.3. Priority Division of CFs. In eﬀect, the control priorities the system parameters tabulated in Table 2. of capacitor voltage and inductor current are directly de- termined by the CVV number selected from the ﬁrst CF, which can be classiﬁed by the median of basic voltage 4.1. Switching Frequency and Computational Burden. vectors’ number as shown in Table 3. To be speciﬁc, if the Considering that both typical weighting factor-based FS- CVV number n<5, this means there are fewer CVVs that MPC and proposed SMPC schemes generate a variable can be selected from CF for minimizing CF . As a result, the 1 2 switching frequency, it is necessary to evaluate their average ﬁnal selected OVV for CF has a greater probability to 2 switching frequency f . For a fair comparison, the sam- asw achieve a better minimization of CF instead of CF , and 1 2 pling frequency of both methods is set as 50kHz to obtain a thus CF has a higher control priority, and vice versa. 1 similar average switching frequency f for both methods. asw In addition, an extreme case is that if the CVV number Corresponding f of both methods under diﬀerent load asw n � 1 or 8, then the two sequential CFs are degraded into conditions is calculated and depicted in Figure 10. conventional single-voltage or single-current objective CF, It can be observed that the f of typical FS-MPC is asw which means the other control objective is lost. reduced compared to the proposed SMPC under heavy It should be noted that the CVV number in the proposed loads, which indicates that the control performance using SMPC is selected as n � 3 or 4, which indicates that the typical FS-MPC will be degraded since the weighting factor current-related G is assigned a higher control priority. As a may be sensitive to diﬀerent load conditions. Moreover, to result, the inductor current is more tightly controlled to compare the computational burden of the two methods, the track its reference (start from 0 A). Hence, an inherent total turnaround time of each method in one sampling smoother current starting response without overshoot is period, including the algorithm execution time and the A/D obtained using the proposed SMPC compared to typical conversion time, is computed by the dSPACE proﬁler. (e weighting factor-based FS-MPC, which is beneﬁcial to results reﬂect that the turnaround time of both methods is hardware safety. 12μs, verifying that the proposed SMPC method does not increase the computational burden compared to the typical FS-MPC scheme. 3.4. Handling of Current Constraint. To realize the over- current protection, an inductor-current constraint term is usually included in the CF of typical weighting factor-based 4.2.EvaluationofSteady-StatePerformance. Figure 11 shows FS-MPC below: the experimental results of a steady-state voltage tracking � � � � response supplying a nominal linear load (R � 60Ω), where ⎧ ⎪ � � l ∞, if i > I , ⎨ � f,k+2� max v is the phase-voltage tracking error. It can be observed G � � � (12) fe lim ⎪ � � ⎩ � � � � from Figure 11 that voltage RMSE and THD of the proposed 0, if i ≤ I , � � f,k+2 max SMPC are slightly larger than those of typical FS-MPC due where I is the maximum limit of the inductor current. to the reduced number of CVVs. In general, the proposed max 6 International Transactions on Electrical Energy Systems 3 3 2.5 2.5 2.5 2 2 X: 3 1.5 1.5 1.5 Y: 1.199 Suboptimal Optimal Suboptimal 1 1 X: 8 X: 4 Y: 1.211 Optimal Optimal 0.5 0.5 Y: 0.9765 0.5 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 X: 4 Suboptimal Suboptimal Y: 298.1 Suboptimal 300 300 Optimal Optimal X: 3 Optimal Y: 297.7 280 280 260 260 240 240 12345678 12345678 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 (a) (b) (c) Figure 6: Evaluation of steady-state capacitor voltage control performance with diﬀerent CVVs and COCFs under diﬀerent load conditions. (a) Rated linear load: R �60Ω (5A). (b) Heavy linear load: R �5Ω (60A). (c) Rated nonlinear diode bridge rectiﬁer load. 3 3 2.5 2.5 2 2 1.5 1.5 Suboptimal Optimal 2 Optimal Optimal 1 0.5 0.5 12 3456 7 8 12 3456 78 12 3456 7 8 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 Suboptimal Suboptimal 300 300 300 Suboptimal Optimal Optimal Optimal 280 280 280 260 260 260 240 240 240 12345678 12345678 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 (a) (b) (c) Figure 7: Continued. Mag of fundamental [V] Mag of fundamental [V] Phase Voltage THD [%] Phase Voltage THD [%] Mag of fundamental [V] Mag of fundamental [V] Phase Voltage THD [%] Phase Voltage THD [%] Mag of fundamental [V] Mag of fundamental [V] Phase Voltage THD [%] Phase Voltage THD [%] International Transactions on Electrical Energy Systems 7 5 5 5 4 4 4 3 3 3 Suboptimal Suboptimal Optimal 2 2 2 Suboptimal Optimal Optimal 1 1 1 12 3456 7 8 12 3456 78 12 3456 7 8 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 Optimal Suboptimal Suboptimal 300 300 300 Suboptimal Optimal Optimal 280 280 280 260 260 260 240 240 240 12 3456 7 8 12 3456 7 8 12 3456 7 8 Number of CVV Number of CVV Number of CVV COCFs1 COCFs1 COCFs1 COCFs2 COCFs2 COCFs2 (d) (e) (f) Figure 7: Evaluation of steady-state voltage performance with diﬀerent CVVs and COCFs under model mismatches. (a) −50% L . (b) −50% C . (c) +100% L . (d) +100% C . (e) −50% L and +100% C . (f) +100% L and −50% C . f f f f f f f Table 3: Priority of CFs classiﬁed by the CVV number. CVV number: n Priority of CFs COCFs1 (G −G ) COCFs2 (G −G ) 1 2 2 1 Higher priority to G 2, 3, 4 5, 6, 7 Higher priority to G 5, 6, 7 2, 3, 4 Single-voltage CF:G 1 8 Single-current CF:G 8 1 Start for i=0...7 1. Measurement: v , i , i f,k f,k o,k 2. Predict: v v , i v 4. Predict: i (v f,k+1 ( io,k) f,k+1 ( io,k) f,k+2 i,k+1) * * 5. Evalute: min(G ) 3. Voltage, current reference: v , i 2 f f Yes i ≤7 for i=1...3 or 4 No CF 7. Predict: v , i f,k+2 f,k+2 6. Pre-select: v v i,k+1 (1)... i,k+1 (4) 8. Evaluate: G Yes No i ≤ 3 or 4 9. Select optimal: v io,k+1 CF End Figure 8: Flowchart of the proposed SMPC for LC-ﬁltered VSIs. SMPC can provide comparable performance as a typical proposed method are similar to those of typical FS-MPC under a wide range of load variations. However, under heavy weighting factor-based FS-MPC. To evaluate that the weighting factors in typical FS-MPC load conditions (load current>30 A), the voltage tracking are sensitive to diﬀerent load conditions, Figure 12 depicts RMSE of typical FS-MPC signiﬁcantly increases, while that the simulation comparison of steady-state response using of the proposed SMPC is still stable due to the elimination of typical FS-MPC and proposed SMPC under various linear weighting factors. (is conclusion is consistent with that of loads. It can be seen that the voltage RMSE and THD of the Figure 10. Hence, the proposed SMPC is superior to the Mag of fundamental [V] Phase Voltage THD [%] Mag of fundamental [V] Phase Voltage THD [%] Mag of fundamental [V] Phase Voltage THD [%] 8 International Transactions on Electrical Energy Systems DC source LC LEM ControlDesk dSPACE sensor Scope D/A Inte rface dSPACE DS 1202 VSI CPU FPGA v , i , i f f o dc Load PWM A/D A/D Load (a) (b) n R (c) Figure 9: Experimental setup. (a) Basic structure. (b) Laboratory prototype. (c) Diode-rectiﬁer nonlinear load. 9.8 9.6 9.4 9.2 5 20 40 60 80 100 120 140 160 180 R [Ω] Proposed method Typical FS-MPC Figure 10: Average switching frequency of typical FS-MPC and proposed SMPC under diﬀerent load conditions. typical weighting factor-based FS-MPC method under very increase in average switching frequency of both methods heavy loads. under light load as shown in Figure 10. Further, Figure 13 shows the experimental results of steady-state response using two methods with a nonlinear load. Similarly, the voltage THD and tracking RMSE 4.3. Evaluation of Transient Performance. Figure 14 shows using the proposed SMPC are comparable to those using the experimental results of transient performance using two typical FS-MPC. Note that the THD using both methods approaches under a nominal linear load step. It reﬂects that with a nonlinear load is reduced compared to that with a both methods have a similar transient time during the load- linear load in Figure 11. (is is caused by the slight step process. However, the voltage ﬂuctuation with the DC Brake f [kHz] asw International Transactions on Electrical Energy Systems 9 Tek PreVu Tek PreVu v * v * fa fa 250V/div 250V/div 1 1 v v fa fa 2 2 250V/div 250V/div i i oa oa v v fe fe 5A/div 5A/div 4 4 50V/div 50V/div 1 250 V 2 250 V 4.00ms 25.0MS/s 1 21 May 2019 1 250 V 2 250 V 4.00ms 25.0MS/s 1 21 May 2019 3 5.00 A 4 50.0 V 4.56000ms 1M points 0.00 V 16: 47: 52 3 5.00 A 4 50.0 V 4.56000ms 1M points 0.00 V 16: 10: 30 (a) (b) Fundamental (50 Hz) = 297.4 , THD= 1.47% Fundamental (50 Hz) = 296.7 , THD= 1.53% 0.3 0.3 FS-MPC SMPC 0.2 0.2 0.1 0.1 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency (Hz) Frequency (Hz) (c) Figure 11: Experimental results of steady-state response with rated Rl �60Ω. (a) Typical FS-MPC (RMSE �4.600V). (b) Proposed SMPC (RMSE �4.957V). (c) Voltage harmonic spectrum. 4 1.1 3.5 0.9 2.5 0.8 0.7 1.5 0.6 10 20 30 40 50 60 10 20 30 40 50 60 R [Ω] R [Ω] Proposed method Proposed method Typical FS-MPC Typical FS-MPC (a) (b) Figure 12: Simulation results of steady-state response using typical FS-MPC and proposed SMPC with diﬀerent linear loads. (a) RMSE. (b) THD. proposed SMPC is 40V, which is smaller than that with the SMPC, simulation and experimental results are given in typical FS-MPC, 45V. Hence, the robustness against load Figures 15–17. variations is enhanced using the proposed SMPC. From Figures 15(a) and 16, it can be deduced that an inherent smoother current-starting response from 0 A with no overshoot is obtained by the proposed SMPC in com- 4.4. Current Start-Up Response and Current Limitation. parison to typical FS-MPC under linear loads, reducing the To validate that the proposed SMPC with COCF2 and 4 shock to system hardware. (e reason is that the current CVVs can result in a smoother current-starting response control is assigned a higher priority. Figures 15(b), 15(c), and to verify the current-limiting capability of the proposed and 17 reﬂect that during the system starting process with a Voltage tracking RMSE [V] Mag (% of Fundamental) THD [%] 10 International Transactions on Electrical Energy Systems Tek PreVu Tek PreVu v * v * fa fa 250V/div 250V/div 1 1 v v fa fa 2 2 250V/div 250V/div 3 3 i i oa oa v v fe fe 2A/div 2A/div 4 4 50V/div 50V/div 1 250 V 2 250 V 4.00ms 25.0MS/s 3 23 May 2019 1 250 V 2 250 V 4.00ms 25.0MS/s 3 23 May 2019 3 2.00 A 4 50.0 V 8.08000ms 1M points 4.00 A 13: 44: 50 3 2.00 A 4 50.0 V 8.08000ms 1M points 4.00 A 14: 02: 20 (a) (b) Fundamental (50 Hz) = 297.5, THD= 1.21% Fundamental (50 Hz) = 296.8 , THD= 1.29% 0.3 0.3 FS-MPC SMPC 0.2 0.2 0.1 0.1 0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency (Hz) Frequency (Hz) (c) Figure 13: Experimental results of steady-state response with a nonlinear load. (a) Typical FS-MPC (RMSE �3.576V). (b) Proposed SMPC (RMSE �3.531V). (c) Phase-voltage harmonic spectrum. Tek PreVu Tek PreVu v * v * fa fa 250V/div 1 1 fa fa 2 2 250V/div 250V/div 3 3 i i oa oa 45V 40V fe fe 5A/div 5A/div 4 4 50V/div 50V/div 1 250 V 2 250 V 4.00ms 25.0MS/s 3 22 May 2019 1 250 V 2 250 V 4.00ms 25.0MS/s 3 22 May 2019 3 5.00 A 4 50.0 V 8.36000ms 1M points 2.80 A 17: 29: 26 3 5.00 A 4 50.0 V 8.56000ms 1M points 4.00 A 18: 12: 33 (a) (b) Figure 14: Experimental results of transient response with a linear load step (no load to 60 Ω). (a) Typical FS-MPC. (b) Proposed SMPC. nonlinear load, a very large load current is induced when model parameter mismatches, where Δ THD and Δ RMSE Glim is included in G , which means the current-limiting are the deviations of THD and RMSE between typical FS- capability is invalid. In contrast, when G is included in G , MPC and the proposed SMPC. (e results reﬂect that both lim 1 a strict current-limiting capability can be obtained. methods are sensitive to model parameter mismatch to diﬀerent degrees. Nevertheless, in most model mismatch cases, |ΔTHD| < 0.5% and |Δ RMSE| < 3 V (1% of the 4.5.SensitivitytoModelMismatches. To further evaluate the reference voltage). Moreover, Table 4 shows that the sensitivity to model mismatches using proposed SMPC, proposed SMPC is robust to various model mismatches, Figure 18 and Table 4 illustrate the simulation and ex- which is comparable with typical FS-MPC schemes as perimental results using two control schemes under shown in Figure 7. Mag (% of Fundamental) International Transactions on Electrical Energy Systems 11 [A] [V] [V] Overshoot 200 200 v v 0 0 0 f,abc f,abc i i i La Lb Lc -200 -200 –20 FS-MPC 00.2 0.4 00.2 0.4 0 0.01 0.02 0.03 [A] [A] [A] 10 10 20 Current limit > 100 A 0 0 i i i i o,abc La Lb Lc o,abc –20 -10 -10 SMPC 0 0.01 0.02 0.03 00.2 0.4 00.2 0.4 Time [s] Time [s] Time [s] (a) (b) (c) Figure 15: Simulation results of current starting and limiting response. (a) Inductor-current starting response using typical FS-MPC and proposed SMPC. (b) Load current starting response with G included in G using proposed SMPC. (c) Load current starting response with lim 2 G included in G using proposed SMPC. lim 1 Tek PreVu Tek PreVu i i i i i i fa fb fc fa fb fc 1 1 Smooth Overshoot 10A/div 10A/div 1 10.0 A 2 10.0 A 20.0ms 5.00MS/s 3 12 Jun 2019 1 10.0 A 2 10.0 A 20.0ms 5.00MS/s 1 21 May 2019 3 10.0 A 44.8000ms 1M points 4.00 A 17: 31: 25 3 10.0 A 558.000ms 1M points 0.00 A 17: 39: 56 (a) (b) Figure 16: Experimental results of starting response of inductor current with a nominal linear load. (a) Typical FS-MPC. (b) Proposed SMPC. Tek PreVu fa Current Limit: 8 A oa 2 250 V 3 40.0 ms 2.50MS/s 2 Sep 2019 3 5.00 A 117.800ms 1M points 800mA 14: 08: 28 Figure 17: Experimental result of current-limiting capability of proposed SMPC during a starting process with a nonlinear load. 12 International Transactions on Electrical Energy Systems 2 0.25 2 1.5 1.5 0.25 0.5 -1 1 1 0.75 -2 -1 -3 0.5 0.5 0.5 1 1.5 2 0.5 1 1.5 2 L /L ∆ THD [%] L /L ∆ RMSE [V] f fn f fn (a) (b) Figure 18: Simulation results of steady-state phase-voltage THD and tracking RMSE deviation between typical FS-MPC and proposed SMPC with model mismatches. (a) ∆ THD. (b) ∆ RMSE. Table 4: Experimental results using two control schemes under model parameter mismatches. Typical FS-MPC Proposed SMPC Model mismatches RMSE (V) THD (%) RMSE (V) THD (%) L , C 4.60 1.47 4.96 1.53 f f L , 50% C 6.43 1.31 6.74 1.42 f f 50% L , C 8.32 2.38 6.20 1.90 f f L , 200% C 9.14 2.07 6.41 2.12 f f 200% L , C 7.44 1.97 7.93 2.02 f f 5. Conclusion λ: Weighting factor G : Voltage-related cost function (is paper presents a weighting factor-less sequential G : Current-related cost function model predictive control scheme for LC-ﬁltered VSIs. A Δv : Voltage tracking error fx graphical method is ﬁrst proposed to determine the optimal Δi : Current tracking error fx candidate voltage vector number. (e cascade order of the v : Vector reference for capacitor-voltage tracking cost function is also obtained by the internal relationship of v : Vector reference for inductor-current tracking the state variables. (en, the current constraint is con- G : Inductor-current constraint term lim sidered and included in the voltage-related CF, resulting in I : Maximum limit of the inductor current max a strict current-limiting capability. Additionally, an in- VSI: Voltage source inverter herent smooth current-starting response is achieved. FS-MPC: Finite-set model predictive control Simulation and quantitative experimental results reveal CF: Cost function that the presented approach can avoid the complex OVV: Optimal voltage vector weighting factor tuning eﬀorts without sacriﬁcing the SMPC: Sequential model predictive control transient and steady-state performance compared to typical CVVs: Candidate voltage vectors FS-MPC. COCFs: Cascade order of CFs RMSE: Root mean square error Abbreviations: THD: Total harmonic distortion. L : Filter inductance Data Availability C : Filter capacitance v : α-β frame capacitance voltage (e data that support the ﬁndings of this study are available i : α-β frame inductance current from the corresponding author upon reasonable request. v: α-β frame converter-side voltage i : α-β frame load current v : Capacitor-voltage reference Conflicts of Interest i : Inductor-current reference V : Reference voltage amplitude (e authors declare that there are no conﬂicts of interest ω : Reference angular frequency regarding the publication of this paper. C /C f fn C /C f fn International Transactions on Electrical Energy Systems 13 Transactions on Industrial Electronics, vol. 60, no. 2, Acknowledgments pp. 589–599, 2013. [13] C. A. Rojas, J. Rodriguez, F. Villarroel, J. R. Espinoza, (is study was supported in part by the Natural Science C. A. Silva, and M. 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International Transactions on Electrical Energy Systems – Hindawi Publishing Corporation
Published: Apr 16, 2022
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