Finite-state model predictive control of NPC inverter using multi-criteria fuzzy decision-making

Finite-state model predictive control of NPC inverter usingmulti-criteria fuzzy decision-making

Vikas Kumar*,†, Prerna Gaur and Alok Prakash Mittal

Division of Instrumentation and Control Engineering, Netaji Subhas Institute of Technology, New Delhi, India

SUMMARY

In this paper, the multiple objectives optimization problem in a finite-state model predictive control (FS-MPC)is formulated using fuzzy multi-criteria decision-making. Conventionally, to optimize the multiple objectivesin a FS-MPC, the aggregate objective function or a single cost function is constructed, which requires theweighting factor tuning to select the appropriate switching state. Determination of these weighting factorsfor a particular objective function is a complex and time-consuming task as no systematic procedure isavailable in literature. The main aim of this paper is to replace the time-consuming task of weighting factortuning by a simple and systematic procedure, which relies on the relative importance of the individual objec-tive functions. The relative importance of objective functions derived from the expert’s knowledge and thedesired control objectives is used for appropriate switching state selection. The method is validated with thehelp of simulation results of a neutral point clamped inverter for the multiple objectives viz. reference currenttracking, capacitor voltage balance and switching frequency minimization. The result outcomes of theproposed methodology are compared with the conventional FS-MPC and space vector pulse width modulationbased current control schemes. Copyright © 2014 John Wiley & Sons, Ltd.

key words: neutral point clamped inverter; predictive control; fuzzy decision functions; multi-criteriadecision-making and multi-objective optimization

1. INTRODUCTION

The multi-level inverters (MLI) are becoming increasingly popular for medium voltage and highpower applications as they can offer several advantages such as high output voltage with very lowdistortion, low electromagnetic interference, low dv/dt stresses, low switching frequency and higherefficiency with good power quality over two-level counterparts [1,2]. The quality of the converteroutput depends upon the control technique used such as the hysteresis-based control, linear controlcombined with a modulation stage and space vector pulse width modulation (SVPWM). SVPWM isthe most popular and well-documented control technique for power converters [3–7]. Recently, theadvances in the processing capabilities of the digital signal processors has enabled the implementationof the complex control techniques such as Fuzzy logic, neural networks, sliding mode control andpredictive control, which were difficult to implement earlier [8–11]. The finite-state model predictivecontrol (FS-MPC) is emerged as a powerful alternative to the control of power converters and electricaldrives due to its fast dynamic response, easy inclusion of system constraints and flexibility to includemultiple objectives [12–15]. The FS-MPC allows the greater flexibility and achieves the multipleobjective optimizations by forming the single cost function or aggregate objective function (AOF).Among the three well-established topologies available for MLI, the diode-clamped MLI (DC-MLI),

flying capacitor MLI and cascaded H-bridge MLI, the DC-MLI is known as the most popular MLItopology in the application areas of medium to high power variable speed drives [16–18] due to its

*Correspondence to: Vikas Kumar, Division of Instrumentation and Control Engineering, Netaji Subhas Institute ofTechnology, New Delhi, India.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMSInt. Trans. Electr. Energ. Syst. (2014)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1880

compactness, efficiency and good performance. Traditionally, to achieve multi-objective optimizationin a DC-MLI by using FS-MPC, AOF, which represents the desired behaviour of the system, is createdusing weighting factors as a linear combination of each objective function. The value of a weightingfactor depends upon the absolute importance of that particular objective. Determination of theseweighting factors for each objective function is a complex and time-consuming task, as theseweighting factors are determined analytically, and the methods available for tuning are not systematic.The tuning of weighting factors is still an open area of research, and some design guidelines have beenfound in [19,20]. Cortes et al. proposed some guidelines for weighting factor determination, andbranch and bound algorithm is used effectively to determine the weighting factors for objectivefunctions having unequal importance [19]. However, the branch and bound algorithm is an offlineprocedure, and the computational complexity increases manifolds when more than two objective func-tions are considered. Rojas et al [20] proposed a ranking-based method for weighting factor determi-nation for objective functions having equal importance, and branch and bound algorithm may becompletely avoided. A number of approaches are available for multi-objective optimization [21–27].The analytical hierarchical process proposed by Saaty, depends on the crisp value and is used by manyresearchers [21]. The determination of crisp value for each objective function is not straight forward.Fuzzy representation of the relative importance of objective function is a better alternative, as it useslinguistic-based variables to describe the relative importance [22]. The alternative approach to theempirical and offline procedures for weighting factor selection can be the fuzzy multi-criteriadecision-making (FMCDM). The FMCDM approach is applied successfully in various fields foroptimizing multiple and conflicting objectives [25–28]. FMCDM for optimization in MPC isimplemented by Villarroel et al in a matrix converter [29]. For equally important objective function,weighting factor selection is avoided, but weighting factor tuning guidelines for the objective functionhaving unequal importance still remains unaddressed.In this paper, the weighting factor tuning for multiple objectives having unequal importance in a FS-MPC

using FMCDM is proposed. The proposed algorithm requires only the relative importance of each objectivefunction over one another, which can be easily determined using decision maker’s knowledge and thecontrol objectives. After the relative importance of each objective function is determined a pairwise com-parison matrix is constructed, and then, the priority matrix is calculated offline, so as to relieve the onlinecomputation burden. The fuzzy decisionmatrix calculated on line for all possible switching states togetherwith the priority matrix is used to determine the optimized decision, and hence, the weighting factortuning is completely avoided. The feasibility of the proposed approach is evaluated by performing thecomputer simulation for the neutral point clamped (NPC) inverter for multiple objectives optimization.The performance of the proposed approach is analysed for two cases such as (a) three objective functionsviz. reference current tracking, capacitor voltage balance and switching frequency minimization areconsidered; and (b) only two objective functions viz. reference current tracking, capacitor voltage balanceare considered. The comparative analysis of the FMCDM approach with respect to AOF approach ispresented for both the cases considered previously. The performance of the proposed approach isevaluated for load power factor and DC link capacitor variations. The simulation results obtained areagain compared with the SVPWM-based current control approach.This paper is organized in six sections. In Section 2, the discrete time modelling of the DC-MLI

topology and the load is presented. In Section 3, the MPC is discussed. In Section 4, the weightingfactor selection and mathematical details of the proposed methodology is presented and simulationresults are presented in Section 5. The conclusions are presented in Section 6.

2. DISCRETE TIME MODELLING OF DIODE-CLAMPED MLI AND LOAD

The DC-MLI topology is shown in Figure 1(a). The DC link capacitor has been split to create a neutralpoint. A pair of switching devices is connected in series, each with an additional diode between the neutralpoint and the centre of the pair. The schematic of the DC-MLI for a single phase operation is shown inFigure 1(b). insulated-gate bipolar transistors (IGBTs) Q1 and Q4 function as the main pair and IGBTsQ2 and Q3 function as the auxiliary pair, which help to clamp the output potential to the neutral point withthe help of diodes D1 andD4. The NPC inverter can produce three voltage levels on the output: the DC bus

V. KUMAR ET AL.

Copyright © 2014 John Wiley & Sons, Ltd. Int. Trans. Electr. Energ. Syst. (2014)DOI: 10.1002/etep

positive voltage, zero voltage and DC bus negative voltage. For a one-phase operation, when IGBTs Q1

and Q2 are turned on with first switching state as shown in Table I, the output is connected to +Vp; whenIGBTs Q2 and Q3 are on with second switching state as shown in Table I, the output is connected to Va0;and when IGBTs Q3 and Q4 are on with third switching state as shown in Table I, the output is connectedto�Vn. Hence, the three voltage levels on the output waveform are formed. Switching states for the fourswitches are listed in Table I. For a three-phase operation total, 27 switching states are generated as shownin Figure 2, which produces 19 distinct voltage vectors. The mathematical modelling of the three-phasethree-level NPC inverter follows in the subsequent section [30].

(a) (b)

Figure 1. Three-level neutral point clamped inverter: (a) three-phase operation and (b) single-phase operation.

Table I. Switching table for single phase operation.

SwitchV0 = 0.5 ×Vdc

Sx= 1V0 = 0Sx= 0

V0 =�0.5 ×VdcSx=�1

Q1 On Off OffQ2 On On OffQ3 Off On OnQ4 Off Off On

Figure 2. Switching states of three levels neutral point clamped inverter.

FS-MPC OF NPC INVERTER USING FMCDM


The space vector output voltage of the inverter can be written as

v ¼ 23

va0 þ avb0 þ a2vc0� �

; a ¼ e j2π=3 (1)

The DC link capacitor voltage is given by

dV c

dt¼ 1

CiC1 V c ¼ DC link capacitor voltage (2)

By using forward Euler’s approximation, Equation (2) can be approximated as

dV c

dt≈

V c k þ 1ð Þ � V c kð ÞT s

; T s is sampling time (3)

By using Equations (2) and (3), the predicted voltage across C1 and C2 can be rewritten in discretetime form as [30]

VPC1

k þ 1ð Þ ¼ VC1 kð Þ þ 1CiC1 kð ÞT s (4)

VPC2

k þ 1ð Þ ¼ VC2 kð Þ þ 1CiC2 kð ÞT s (5)

whereVPC1

k þ 1ð Þ and VPC2

k þ 1ð Þ are the predicted, and VC1 kð Þ andVC2 kð Þ are the measured DC linkvoltages of capacitor C1 and C2, respectively.The DC link capacitor currents iC1 kð Þ and iC2 kð Þ depend on the switching state of the inverter and

using Figure 1(b) can be rewritten as [30]

iC1 kð Þ ¼ idc kð Þ � S1aia kð Þ � S1bib kð Þ � S1cic kð Þ (6)

iC2 kð Þ ¼ idc kð Þ þ S2aia kð Þ þ S2bib kð Þ þ S2cic kð Þ (7)

idc (k) is the current supplied by the DC voltage source. S1x and S2x represent the switching states of thedifferent switches as shown in Table I and are defined as follows:

S1x ¼1 if Sx ¼ 1

0 Otherwise

(

S2x ¼1 if Sx ¼ �1

0 Otherwise

(9>>>>>=>>>>>;

x ¼ a; b; c: (8)

2.1. Dynamics of load

The three phase RLE type load is connected to the NPC inverter. The dynamics of the load is describedby the differential equation [31]

v ¼ Riþ Ldidt

þ e (9)

The load model can be expressed using vectorial transformation as shown in Equation (10):

α

β

� �¼ 2=3 � 1=3 � 1=3

0ffiffiffi3

p=3 � ffiffiffi

3p

=3

� � a

b

c

264

375 (10)

where a, b and c are the phase variables of voltage or current, and α, β are the vectorial variables. Byusing Equation (9), Equation (10) can be written as

vα;β ¼ Riα;β þ Ldiα;βdt

þ eα;β (11)

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vα,β is the inverter voltage vector, iα,β is the real and imaginary parts of the load current vector, and eα,βis the real and imaginary parts of the back electromotive force (emf).By using forward Euler’s approximation, di/dt can be written as

diα;βdt

≈iα;β k þ 1ð Þ � iα;β kð Þ

T s(12)

By using Equations (11) and (12), the predicted load current can be rewritten as

iPα;β k þ 1ð Þ ¼ 1� RT s

L

� �iα;β kð Þ� �þ T s

Lvα;β kð Þ � eα;β kð Þ� �

(13)

eα,β(k) is the estimated back-emf, iPα k þ 1ð Þ; iPβ k þ 1ð Þ is the predicted real and imaginary part of loadcurrent vector. By using Equation (13), the load current can be predicted. The real and imaginary partsof the estimated back-emf can be predicted using the following Equation (14), which can be derivedusing Equation (13).

eα;β k � 1ð Þ ¼ vα;β k � 1ð Þ � L

Tsiα;β kð Þ� �� R� L

Ts

� �iα;β k � 1ð Þ� �

(14)

3. PREDICTIVE CONTROL IMPLEMENTATION AND CONTROL REQUIREMENTS

The implementation of the predictive control algorithm based on Equations (1)–(14) and the controlobjectives of NPC inverter are defined in the following section.

3.1. Predictive control implementation

The control problem for power converter can be defined as the selection of the proper control action,that is, switching state, so that the output variable follows the reference variable. As a power convertercontains finite number of switching states, the control objective is to select the switching state so thatthe overall objective function is minimized. For the diode-clamped inverter, the predictive controlstrategy is shown in Figure 3. The control objective is the selection of a switching state out of 27possible switching states, so that the output current follows the reference trajectory (Ir(k)) as shownin Figure 3.The reference current, the measured load current and the load voltage, together with the model of the

load are used to predict the future values of load current as given in Equations (13) and (14). The costfunction as given in Equation (15) is evaluated for all the 27 possible switching states, and theswitching state, which minimizes the cost function is selected and applied in the subsequent state oftime. The overall cost for two objective functions can be defined as follows:

g kð Þ ¼ g1 kð Þ þ λg2 kð Þ (15)

Where g1 and g2 are the cost function, and λ is the weighting factor associated with the costfunction g2.

Figure 3. Predictive control strategy for three-level neutral point clamped inverter.



3.2. Multi-objective predictive control

The control objectives of NPC inverter are the following:

(1) reference current tracking;(2) DC link capacitor voltage balance; and(3) minimization of switching frequency.

In order to meet the mentioned objectives, a single cost function is defined later in Equation (16) isevaluated for each of the 27 switching states. The state which minimizes the cost function g is appliedfor the next sampling instant [31].

g k þ 1ð Þ ¼ irα k þ 1ð Þ � ipα k þ 1ð Þ þ jirβ kþ 1ð Þ � ipβ k þ 1ð Þþ λcjV c1 k þ 1ð Þ � V c2 k þ 1ð Þj þ λsNs

(16)

where irα; irβ are the real and imaginary component of the reference current vector, ipα; i

pβ are the real and

imaginary components of the predicted current vector, Vc1 and Vc2 are the DC link capacitor voltages,and Ns represents the number of switchings involved when changing from present to the future state. λcand λs are the weighting factors, used for the relative importance of the capacitor voltage balance andthe switching frequency. The more the value of the weighting factors, more will be the importance ofthe particular objective. The value of these weighting factors is determined through a non-trivialprocedure, which is very complex and time consuming task to perform. In order to alleviate theproblem of weighting factor tuning, FMCDM is presented in the subsequent section.

4. MULTI-OBJECTIVE OPTIMIZATION

The multi-objective optimization in a FS-MPC by using AOF approach and FMCDM is discussed inthe following section.

4.1. Branch and bound approach

The AOF as given in Equation (15) is used for optimization. The weighting factors are calculated usingbranch and bound algorithm. The branch and bound algorithm for two objective functions g1 and g2 isshown in Figure 4. The algorithm starts with the four different values of the weighting factors λ1 = 0,λ2 = 1, λ3 = 10 and λ4 = 100. The objective functions given in Equation (15) is evaluated and for thementioned values of λ. Let for λ1 and λ2, we obtain the small numerical value of the cost function gcompared with λ3 and λ4. The λ is modified as the average of λ1 and λ2 that is 0.5. The cost functionis again evaluated for λ1, λ2 and λ. Let this time the minimum value lies between λ1 and λ, that is,0≤ λ≤ 0.5, the λ is modified to 0.25. The procedure is repeated till we obtain the minimum valueof the cost function [31]. The algorithm using AOF approach is shown in Figure 5(b).

Figure 4. Weighting factors calculation using branch and bound algorithm.

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4.2. Fuzzy multi-objective optimization

The cost function defined in Equation (16) contains three objective functions. Each objectivefunction is of varying degree of importance. The determination of weighting factors requiresthe absolute importance of each objective function, the FMCDM on the other hand relies onthe relative importance of the each objective function over one another, and fuzzy multi-criteriaoptimization is particularly suitable for choosing, on the basis of the subjective and qualitativeknowledge provided by the decision makers. Individual objective function for the referencecurrent tracking, DC link capacitor voltage balance and switching frequency minimization usingEquation (16) can be written as

g1 k þ 1ð Þ ¼ irα k þ 1ð Þ � ipα k þ 1ð Þ þ jirβ k þ 1ð Þ � ipβ k þ 1ð Þjg2 k þ 1ð Þ ¼ V c1 k þ 1ð Þ � Vc2 k þ 1ð Þj jg3 k þ 1ð Þ ¼ Ns

9>=>; (17)

The FMCDM technique consists of the following seven steps:

(a) (b)

Figure 5. (a) fuzzy multi-criteria decision-making finite state model predictive control algorithm with delaycompensation and (b) aggregate objective function approach.



Step 1: Depending upon the desired objective, the relative importance is assigned to each objectivefunction by different decision makers. For each decision maker’s decision perform thefollowing steps:

Step 2: Construct the pairwise comparison matrix Γ : Compare the n objective functionscriteria with each other and construct the pairwise comparison matrix Γ based onTable II [21,24]. The reference current tracking is given the highest importance, andthe switching frequency reduction is given the least importance. For three objectivefunctions, the individual elements of pairwise comparison matrix Γ3 × 3 are calculatedas follows:Γij = 1 if objective functions i and j have equal importance, Γij = 9 if the objectivefunctions i has absolute importance over objective function j, and if the objectivefunctions j has absolute importance over objective function i, in this case the elementΓij = 1/9. Rest of the elements are calculated in the same fashion.

Step 3: Determine the eigenvector associated to the maximum eigenvalue of the comparisonmatrix: Calculate the eigenvalues of the comparison matrix calculated in Step 2. Letthe eigenvalues of the comparison matrix are given by: [λ1, λ2, ……….., λr], wherer is the rank of the matrix Γ. Corresponding to the maximum eigenvalue λmax,calculate the eigenvector γmax = [γmax1………….γmaxn] . Compute the priority vectorR as follows [19]:

R ¼ R1…… :Rn½ �T ¼ γmaxj

∑nj¼1γmaxj

(18)

The relative importance of each objective function is calculated by taking the mean of the eachelement of the priority matrix as calculated for individual decision maker’s decision. Eachelement of the priority matrix [R1……. Rn]

T represents the relative importance of respectiveobjective function.

Step 4: Determine the fuzzy decision matrix: Calculate the value of each cost function usingEquation (17) for all possible switching states. For 27 switching states and three costfunctions, calculate the decision matrix is [d]27X3.

The fuzzified decision matrix [DM]27X3 is calculated for three objective functions (n = 3) and 27switching states (ns = 27) using linear membership functions as given in Equation (19),

Table II. Table for relative importance [18].

Relativeimportance Definitions Explanation

1 Equal importance Two activities contribute equally to theobjective

3 Moderate importance Experience and judgement slightly favourone activity over another

5 Strong importance Experience and judgement strongly favourone activity over another

7 Very strong importance Experience and judgement very stronglyfavour one activity over another

9 Absolute importance The evidence favouring one activity overanother is of the highest possible order ofaffirmation

2,4,6,8 Intermediate values between two adjacentscale judgements

Reciprocalsof above

If activity i has one of the above non-zeronumbers assigned to it when compared withactivity j, then j has the reciprocal valuewhen compared with i.

If activity i has one of the above non-zeronumbers assigned to it when compared withactivity j, then j has the reciprocal valuewhen compared with i.

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DM ¼ dij ¼1 if dij < dminj

dmaxj � dijdmaxj � dminj

if dminj < dij < dmaxj

0 if dij < dmaxj

8>>><>>>:

(19)

where i = 1……….ns and j = 1……….n

Step 5: Combine the fuzzy decision matrix and the priority matrix: Calculate the following matrix Vby raising the criterion priority as calculated in Step 3 to the corresponding row of the fuzzydecision matrix [26], given by Equation (19):

Vj ¼ V1j……… :Vnsj

�T ¼ d1j� �Rj……… dnsj

� �Rjh iT

; j ¼ 1…… ::n (20)

where Rj is the relative importance of the objective functions as calculated in Step 3.

Step 6: Determine the decision model: For each state, calculate the matrix P, which is given by theminimum value of the each row of matrix V calculated in Step 5.

Pi ¼ min V1j;…… ::V1n� �

; i ¼ 1……ns; j ¼ 1… ::n (21)

Step 7: Determine the optimized state: The optimized state X corresponds to the maximum value ofthe matrix P.

i:e: X ¼ max P1j;…… ::P1ns� �

(22)

4.3. Fuzzy multi-objective optimization algorithm

Fuzzy multi-objective optimization algorithm is shown in Figure 5(a), and the algorithm calculationexample is presented in the following Section 4.4.

4.4. Algorithm calculation example

The step by step algorithm calculation example is given to illustrate the procedure. The procedureconsists of 27 switching states and three objective functions. First, with the help of three decisionmakers imprecise knowledge, the relative importance of each objective function is determined. Fromthe NPC inverter control point of view, reference current tracking has the highest importance, thecapacitor voltage balance is given the medium importance and switching frequency minimization isgiven the least importance. The pairwise comparison matrix depending upon the relative importanceof each objective function is computed as explained in Table II and is given as follows:

Γ ¼1 7 8

1=7 1 4

1=8 1=4 1

264

375 (23)

The eigenvector corresponding to the maximum value of the eigenvalue (3.1769) is calculated as

γ ¼ 0:97 0:22 0:05½ � (24)

The computed priority matrix is given as

R ¼ 0: 77 0:20 0:03½ � (25)

For a given objective function, the priority matrix remains same throughout the entire operation, sothe priority matrix is calculated offline so as to reduce the computational burden. The value of



individual cost functions g1, g2 and g3 given by Equation (17) is determined for 27 switching statesusing the procedure explained in Section 3. The value of g1, g2 and g3 is fuzzified using linear mem-bership function as given in Equation (19) to obtain d1, d2 and d3. V1, V2 and V3 are calculated usingEquation (20). V1, V2 and V3 as given in Table III form the matrix V. Max–Min inference mechanism isused to calculate the optimized state. The matrix P is the minimizing decision function. The final state ischosen using maximizing decision function. The optimized state is 21 corresponding to 0.99, that is,maximum value of variable P. The calculation example of matrix d, V and P is presented in Table III.

4.5. SVPWM approach

The SVPWM is a popular technique for MLIs as it offers lower switching losses and improvedharmonic contents. For implementation of SVPWM, the whole vector area is divided into six mainsectors, and the reference vector is relocated in one of these sectors, and that main sector is againdivided into six subsectors. Then using calculation method, the dwell time of the voltage vector iscalculated. SVPWM-based current control scheme is implemented for a three-level NPC inverter.The simulation parameters are kept same as given in Appendix. The details of the implementationmay be seen in [3–7] and [32].

5. RESULTS OF VARIOUS CONTROL SCHEMES

The proposed FMCDM method is implemented by means of simulation. The MATLAB/SIMULINK (TheMathWorks, Inc., Natick, Massachusetts, USA) environment is used for the simulation. The simulationparameters, the parameters of the three-phase load and the inverter are given in the Appendix. The algo-rithm shown in Figure 5(a) is implemented for NPC inverter to meet the control objectives viz. the refer-ence current tracking, the DC link capacitor voltage balance and the switching frequency minimization.To investigate the effectiveness of the proposed approach, a comparison of the results is also presented

Table III. Calculated values of matrix d, V and P.

State g1 g2 g3 d1 d2 d3 V1 V2 V3 P

1 0.71 11.11 2 0.58 0.5 0.6 0.65 0.87 0.99 0.652 0.71 11.11 1 0.58 0.5 0.8 0.65 0.87 1 0.653 0.71 11.11 5 0.58 0.5 0 0.65 0.87 0 04 0.95 10.72 3 0.37 0.65 0.4 0.46 0.92 0.98 0.465 0.95 11.49 2 0.37 0.35 0.6 0.46 0.81 0.99 0.466 0.62 9.81 4 0.65 1 0.2 0.72 1 0.97 0.727 0.62 12.41 1 0.65 0 0.8 0.72 0 1 08 0.37 10.19 3 0.86 0.85 0.4 0.89 0.99 0.98 0.899 0.37 12.02 0 0.86 0.15 1 0.89 0.68 1 0.6810 0.46 11.49 4 0.79 0.35 0.2 0.83 0.81 0.97 0.8111 0.46 10.72 1 0.79 0.65 0.8 0.83 0.92 1 0.8312 0.8 12.41 3 0.5 0 0.4 0.58 0 0.98 013 0.8 9.81 2 0.5 1 0.6 0.58 1 0.99 0.5814 1.04 12.02 4 0.29 0.15 0.2 0.38 0.68 0.97 0.3815 1.04 10.19 3 0.29 0.85 0.4 0.38 0.99 0.98 0.3816 1.2 11.11 3 0.15 0.5 0.4 0.23 0.87 0.98 0.2317 0.86 12.02 2 0.44 0.15 0.6 0.53 0.68 0.99 0.5318 0.63 11.11 3 0.65 0.5 0.4 0.71 0.87 0.98 0.7119 0.38 11.49 2 0.86 0.35 0.6 0.89 0.81 0.99 0.8120 0.13 11.11 1 1.07 0.5 0.8 1.05 0.87 1 0.8721 0.21 9.81 2 1 1 0.6 1 1 0.99 0.9922 0.55 11.11 3 0.71 0.5 0.4 0.77 0.87 0.98 0.7723 0.64 12.02 2 0.63 0.15 0.6 0.7 0.68 0.99 0.6824 0.89 11.11 3 0.42 0.5 0.4 0.51 0.87 0.98 0.5125 1.14 11.49 4 0.21 0.35 0.2 0.3 0.81 0.97 0.326 1.38 11.11 5 0 0.5 0 0 0.87 0 027 0.21 9.81 4 1 1 0.2 1 1 0.97 0.97

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for NPC inverter controlled using AOF approach as shown in Figure 5(b) and SVPWM approach.The weighting factors for AOF approach are determined using branch and bound algorithm asshown in Figure 4.

5.1. FMCDM results

The results of the NPC inverter with three-phase load for the FMCDM approach are shown in Figure 6–8.As shown in Figure 6(a), the load current followed the reference current with a mean square error (MSE)of 0.0930. The d–q components of load currents are shown in Figure 6(b). With the proposed approach,the DC link capacitor voltage balance maintained as shown in Figure 6(c). The three-phase load current isshown in Figure 6(d). The load phase voltage and its spectrum analysis are shown in Figure 7(a) and (b),respectively. The calculated total harmonic distortion (THD) for the load phase voltage is26.42%. The load current and the THD calculated for load current is 1.11% as shown in Figure 8.The average switching frequency per semiconductor (Fs) is calculated by taking the average ofthe total number of switching that takes place in a known time for all 12 semiconductor devicesas given later in Equation (26):

Fs ¼ Total number of switchings for 12 semiconductor switches per secondð Þ=12 (26)

The average switching frequency per semiconductor for the proposed approach is 700Hz.

5.1.1. Performance analysis under load power factor and DC link capacitor variations. Theperformance of the proposed methodology is analysed under DC link capacitor and load power factorvariations. The following two cases are considered for the performance analysis:

Case 1: The simulation results for power factor ≈1 and C= 750μF are shown in Figure 9. The cur-rent tracking and capacitor voltage balance is maintained for said conditions. The loadpower is kept at close to 1, and the value of DC link capacitor is varied as (i) C = 50μF,

Figure 6. Performance of neutral point clamped inverter using fuzzy multi-criteria decision-makingtechnique: (a) reference current tracking, (b) d–q component of load current, (c) capacitor voltage balance

and (d) three-phase load current.



Figure 8. Performance of neutral point clamped inverter using fuzzy multi-criteria decision-makingtechnique: (a) load current (b) spectrum analysis. THD, total harmonic distortion.

Figure 7. Performance of neutral point clamped inverter using fuzzy multi-criteria decision-makingtechnique: (a) load voltage and (b) spectrum analysis.

Figure 9. Performance of neutral point clamped inverter using fuzzy multi-criteria decision-makingtechnique for load power factor ≈1: (a) reference current tracking, (b) load component of d–q current, (c)

capacitor voltage balance and (d) three-phase load current.

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(ii) C = 400 μF, (iii) C= 500μF and (iv) C= 600μF. The simulation results are shown inFigure 10. A large unbalance is seen in DC link capacitor voltage at C= 50μF as shownin Figure 10(a) and least variations at C= 600μF as shown in Figure 10(d).

Case 2: The simulation results for power factor ≈0 and C = 750 μF are shown in Figure 11. Thecurrent tracking and capacitor voltage balance is maintained for said conditions. As shownin Figure 11(a) and (c). The load power is kept at close to zero, and the value of DC linkcapacitor is varied as (i) C= 50μF, (ii) C= 400μF, (iii) C= 500μF and (iv) C= 900μF.The variation of DC link capacitor voltage for different values of capacitor is shown inFigure 12. A large unbalance in DC link capacitor voltage at C= 50μF as shown inFigure 12(a) as compared with Figure 10(a) where the load power factor is ≈1.

Figure 10. Capacitor voltage Vs time for load power factor ≈1 for (a) C1 =C2 = 50μF, (b) C1 =C2 = 400μF,(c) C1 =C2 = 500μF and (d) C1 =C2 = 600μF.

Figure 11. Results obtained for neutral point clamped inverter using fuzzy multi-criteria decision-makingtechnique for load power factor ≈0: (a) reference current tracking, (b) d–q component of load current, (c)

capacitor voltage balance and (d) three-phase load current.



5.2. AOF approach results

The results of the NPC inverter with three-phase load for the AOF approach are shown in Figures 13–15.The load current follows the reference current as shown in Figure 13(a) with a MSE of 0.0907. The d–qcomponents of load currents are shown in Figure 13(b). The AOF approach is able to maintain capacitorvoltage balance as shown in Figure 13(c); the three-phase load current is shown in Figure 13(d). The load

Figure 12. Direct current link capacitor voltage Vs time for load power factor ≈0 for (a) C1 =C2 = 50μF, (b)C1 =C2 = 400μF, (c) C1 =C2 = 500μF and (d) C1 =C2 = 600μF.

Figure 13. Performance of neutral point clamped inverter using aggregate objective function approach: (a)reference current tracking, (b) d–q component of load current, (c) capacitor voltage balance and (d) three-

phase load current.

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phase voltage and its spectrum analysis are as shown in Figure 14(a) and (b), respectively. The calculatedTHD for load phase voltage is 27.05%, and the calculated THD for the load current is 1.13% as shown inFigure 15(a) and (b), respectively. The average switching frequency per semiconductor as calculatedusing Equation (26) for the AOF approach is 725Hz.

5.3. SVPWM results

The results of the NPC inverter with three-phase load are shown in Figure 16–18. The load currentfollowed the reference current with a MSE of 0.0838 as shown in Figure 16(a). The d–q componentsof load currents are shown in Figure 16(b). With the proposed approach, the DC link capacitor volt-age balance maintained as shown in Figure 16(c). There is a large variations in the DC link capacitorvoltage are seen. The three-phase load current is shown in Figure 16(d). The load phase voltage andits spectrum analysis are shown in Figure 17(a) and (b), respectively. The calculated THD for theload phase voltage is 34.62%. The load current and the THD calculated for load current is 1.33%as shown in Figure 18(b). The average switching frequency per semiconductor (Fs) as calculatedusing Equation (26) is 1272Hz.

Figure 14. Performance of neutral point clamped inverter using aggregate objective function approach: (a)load voltage and (b) spectrum analysis. THD, total harmonic distortion.

Figure 15. Performance of neutral point clamped inverter using aggregate objective function approach: (a)load current and (b) spectrum analysis. THD, total harmonic distortion.



Figure 16. Performance of neutral point clamped inverter using space vector pulse width modulationapproach: (a) reference current tracking, (b) d–q component of load current, (c) capacitor voltage balance

and (d) three-phase load current. DC, direct current.

Figure 17. Performance of neutral point clamped inverter using space vector pulse width modulationapproach: (a) load voltage and (b) spectrum analysis. THD, total harmonic distortion.

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5.4. Dynamic performance evaluation

The dynamic performance of the scheme is evaluated by reducing the amplitude of the reference cur-rent irq to �5A from �10A at time t= 0.155 s, whereas the magnitude of the reference current ird is keptfixed. The dynamic response of the proposed control scheme is fast as shown in Figure 19 as comparedwith AOF and SVPWM approach as shown in Figures 20 and 21, respectively. With the proposedFMCDM scheme, the load current reaches the reference trajectory in time 0.0025 s; whereas withthe AOF approach, it took around 0.003 s; and with SVPWM approach, it took around 0.025 s to trackthe reference trajectory, while an inherent decoupling is maintained between the two current compo-nents of load currents for all the implemented techniques. The comparison of the results is shown inTable IV, and the summary of the results is presented in the following section.The simulation results for the implemented strategies demonstrated the effectiveness of the proposed

approach as the FMCDM approach offered the better response as compared with the AOF approach

Figure 18. Performance of neutral point clamped inverter using space vector pulse width modulationapproach: (a) load current and (b) spectrum analysis. THD, total harmonic distortion.

Figure 19. Performance of neutral point clamped inverter using fuzzy multi-criteria decision-makingapproach: (a) Iq reduced to �5A and (b) Id is decoupled from Iq.



and the well-established SVPWM approach for capacitor voltage balance, switching frequencyminimization and the THD of the load phase voltage and load current. The MSE for the referencecurrent tracking is slightly higher than the AOF approach as the AOF approach relies on the absolute

Figure 20. Performance of neutral point clamped inverter using aggregate objective function approach fordynamic performance evaluation: (a) Iq reduced to �5A and (b) Id is decoupled from Iq.

Figure 21. Performance of neutral point clamped inverter using space vector pulse width modulationapproach for dynamic performance evaluation: (a) Iq is reduced to �5A and (b) Id is decoupled from Iq.

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importance of each objective function and is least for SVPWM approach. The weighting factordetermination using branch and bound and/or trial and error method is complex and time consuming,and the computational complexity increases manifolds when more than two objective functions areconsidered. FMCDM on the other hand relies on the relative importance of each objective function,which is easily determined using the knowledge of the decision maker and the desired controlobjectives. The SVPWM approach is also computationally complex and the THD for load current;load voltage are found to be higher as compared with the AOF and the proposed FMCDM approach.The proposed FMCDM approach is found to be the computationally demanding, and it took around65μs to run the entire algorithm on a dSPACE DS-1104 as shown in Figure 22, as compared withthe AOF approach, which took around 47μs to run on the same platform; FMCDM can beimplemented using high speed DSP’s and FPGA’s.

6. CONCLUSION

The determination of weighting factors is a very complex and time-consuming task when more thantwo objective functions are considered or objective functions are of varying degree of importance.The FMCDM approach is presented as an alternative to the AOF approach for multi-objectiveoptimization problems in FS-MPC. The proposed approach is very systematic and is based on therelative importance of the multiple objectives and the fuzzy decision-making.The simulation results have shown that with the proposed approach good transient and steady state

performance is obtained under load power factor variations and DC link capacitor variations. Theproposed algorithm may be effectively for multi-objective optimization in a FS-MPC such as DTCof PMSM and IM drives for multiple objective functions and multi-objective switching state selectionin a matrix converter. The algorithm can be easily modified for back-emf estimation.

Table IV. Summary of results.

AOFapproach

FMCDMapproach

AOFapproach

FMCDMapproach

Parameters

Cost function consists ofreference current tracking, DCcapacitor voltage balance and

switching frequencyminimization

Cost function consists ofreference current tracking andDC capacitor voltage balance

SVPWMapproach

Current MSE 0.0907 0.0930 0.1010 0.0838 0.0782Switchingfrequency (Hz)

725 700 1111 1073 1272

THD loadvoltage (%)

27.05 26.42 28.15 28.51 45.46

THD loadcurrent (%)

1.13 1.10 0.82 0.84 1.33

DC link voltage DC link capacitor voltage balance is maintained at 260 ± 4V for FMCDM approach, at260 ± 6V for AOF approach and at 260 ± 15V for SVPWM approach.

Bold data signifies the minimun and the desired value.AOF, aggregate objective function; FMCDM, fuzzy multi-criteria decision-making; SVPWM, space vector pulse width modulation;DC, direct current; MSE, mean square error; THD, total harmonic distortion.

Figure 22. Total execution time for fuzzy multi-criteria decision-making algorithm is 6.526e-05, calculatedusing dSPACE DS-1104 in real time.



7. LIST OF SYMBOLS AND ABBREVIATIONS

7.1. Symbols

R Load resistanceL Load inductanceeα,β Real and imaginary parts of load back-emfv Space vector output voltage of the inverterg Cost functionIr Reference current trajectoryC DC link capacitorVC1;2 DC link voltage of capacitor C1 and C2

iC1;2 DC link currents of capacitor C1 and C2

idc Current supplied by the DC voltage sourceVPC1;2

Predicted voltage of DC link capacitor C1 and C2

Id,q d and q components of load currentNs Average switching frequency per semiconductorR Priority matrixΓ Pairwise comparison matrixd Fuzzified decision matrixγ Eigenvector matrix corresponding to maximum eigenvalue of matrix Γλ Weighting factorIpα;β Real and imaginary components of predicted load currentIrα;β Real and imaginary components of reference current vectorQx,14 Four IGBT switches for each phase, x = a, b and cSx Switching states of various switches for x= a, b and c phasesTs Sampling time

7.2. Abbreviations

FMCDM Fuzzy multi-criteria and decision-makingFS-MPC Finite-state model predictive controlAOF Aggregate objective functionMPC Model predictive controlDTC Direct torque controlPMSM Permanent magnet synchronous machineIM Induction machineSVPWM Space vector pulse width modulationDSP Digital Signal ProcessorsFPGA Field Programmable Gate ArrayDM Decision MatrixTHD Total harmonic distortionMSE Mean square errorMLI Multi-level inverter

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APPENDIX

Simulation parameters for MATLAB/SIMULINK

Parameter Value

Start time (s) 0.00.15Stop time

Solver type Fixed stepFixed step size 1e-6Solver Ode-5

Inverter and three-phase load parameters usedfor simulation

Parameter Value

DC link Voltage 520VDC link Capacitor 750μFLoad R 20ΩLoad L 50mH

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Finite-state model predictive control of NPC inverter using multi-criteria fuzzy decision-making