modiﬁed model predictive control of a bidirectional ac–dc ... · energy sources to ensure...

704 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 2, FEBRUARY 2016

Modified Model Predictive Control of aBidirectional AC–DC Converter Based

on Lyapunov Function for EnergyStorage Systems

Md. Parvez Akter, Saad Mekhilef, Senior Member, IEEE ,Nadia Mei Lin Tan, Member, IEEE , and Hirofumi Akagi, Fellow, IEEE

Abstract—Energy storage systems have been widely ap-plied in power distribution sectors as well as in renewableenergy sources to ensure uninterruptible power supply.This paper proposes a modified model predictive control(MMPC) method based on the Lyapunov function to im-prove the performance of a bidirectional ac–dc converter,which is used in an energy storage system for bidirec-tional power transfer between the three-phase ac voltagesupply and energy storage devices. The proposed controltechnique utilizes the discrete behavior of the converter,considering the unavoidable quantization errors betweenthe controller and the control actions selected from thefinite control set of the bidirectional ac–dc converter. Theproposed control method reduces the execution time delayby 18% compared with the conventional model predictivecontrol. Moreover, the nonlinear system stability of the pro-posed MMPC technique is ensured by the direct Lyapunovmethod and a nonlinear experimental system model. De-tailed experimental results with a 2.5-kW downscaledhardware prototype are provided to show the efficacy of theproposed control system.

Index Terms—Bidirectional ac–dc power conversion, en-ergy storage system, Lyapunov methods, modified modelpredictive control (MMPC), stability analysis.

I. INTRODUCTION

ENERGY storage systems play an important role in utilityand transport applications as well as in renewable en-

Manuscript received January 14, 2015; revised April 21, 2015and June 22, 2015; accepted August 5, 2015. Date of publicationSeptember 14, 2015; date of current version January 8, 2016. This workwas supported in part by the High Impact Research–Ministry of HigherEducation under Project UM.C/HIR/MOHE/ENG/24 and in part by theUniversity of Malaya Research Grant under Project RP006E-13ICT.

M. Parvez Akter and S. Mekhilef are with the Power Electronics andRenewable Energy Research Laboratory (PEARL), Department of Elec-trical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia(e-mail: [email protected]; [email protected]).

N. M. L. Tan is with the Department of Electrical Power Engineering,Universiti Tenaga Nasional, 43000 Kajang, Malaysia (e-mail: [email protected]).

H. Akagi is with the Department of Electrical and Electronic Engi-neering, Tokyo Institute of Technology, Tokyo 152-8552, Japan (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2015.2478752

ergy sources to ensure power reliability, active power control,load leveling, and frequency control [1], [2]. Generally, theenergy storage system uses static storage devices such as anelectric double-layer capacitor, Li-ion battery, lead–acid bat-tery, and nickel–metal hydride battery [3]. These static storagedevices contain high power and energy density but requireproper operation such as low ripple current and voltage at thedc side.

The bidirectional ac–dc converter is an essential part of theenergy storage system due to its bidirectional power flow, gridsynchronization, and dc power management capabilities [4].The control algorithm of this ac–dc converter should be highlystable and efficient as it needs to prevent the problems of poorpower quality due to high total harmonic distortion, low powerfactor, ac voltage distortion, and ripple in the dc current andvoltage [5], [6]. Therefore, several research studies have beencarried out to improve the efficiency and performance of thisbidirectional ac–dc converter. The classical control techniquesare based on voltage-oriented control [7], virtual-flux-orientedcontrol [8], and direct power control (DPC) [9] schemes, whichutilize the proportional–integral (PI) controllers. The major lim-itation of these control schemes is tuning the PI controllers thatfurther affect the coordinate transform accuracy. To overcomethis limitation of PI controllers, a fuzzy-logic-based switchingstate selection criterion has been presented in [10] by avoidinga predefined switching table. Although the active and reactivepower values are smoothed in a fuzzy-logic-based DPC algo-rithm compared with classical DPC, its sampling frequency ishigh. Therefore, a sliding mode nonlinear [11] and artificialneural network [12] control approach has been investigated foractive and reactive power regulation of a grid-connected dc–acconverter, which is dependent on control variables.

The working principle of the model predictive control (MPC)scheme is to predict the future behavior of the control variables.This MPC algorithm has become an attractive mode of controlfor the bidirectional ac–dc converters, compared with the clas-sical solutions, due to its simple and intuitive concept with nopulsewidth modulation blocks [13]. Moreover, the MPC algo-rithm is very easy to configure with constraints and nonlinearityand also for practical implementation [14]–[16]. Due to theseadvantages, the MPC algorithm has been extensively applied

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PARVEZ AKTER et al.: MMPC OF A BIDIRECTIONAL AC–DC CONVERTER BASED ON LYAPUNOV FUNCTION 705

in an active front-end rectifier [17], [18], an indirect matrixconverter [19], a three-level converter [20], a voltage sourceinverter [21], and a neutral-point-clamped converter [22]. Inspite of these boundless advantages, MPC faces computationalburden due to solving the underlying optimization problem ofthe discrete manipulated variables [23], [24]. Hence, computa-tional issues have become very important for long predictionhorizons. As a result, the modification in the existing MPCtechniques is required for minimizing the amount of calcula-tions as well as the computational time. Although a Luenbergobserver and a Runge–Kutta fourth-order method-based pre-dictive control has been proposed for delay compensationand convergence in the digital implementation of the virtualmultilevel inverter obtained in [25] and [26], the robustnessanalysis against parameter uncertainties is missing in thesepapers. Furthermore, a complete research on the bidirectionalac–dc converter, considering practical nonlinearity issues andexplicitly addressing the stability issues in the MPC algorithm,is missing in the present state of the art.

This paper proposes a modified model predictive control(MMPC) algorithm based on the Lyapunov function that isapplied in a bidirectional ac–dc converter for an energy storagesystem to reduce the computational time, ensure stability androbustness, and increase the system performance. The systemconfiguration and working principle of the bidirectional ac–dcconverter are elaborately described in Section II. The formu-lation of the conventional MPC method with a discrete-timemodel and the cost function is reviewed in Section III. The de-termination of the proposed Lyapunov-function-based MMPCalgorithm and its practical implementation into the bidirec-tional ac–dc converter are elaborately discussed in Section IV.In Section V, the performance of the proposed Lyapunov-function-based MMPC method for the bidirectional ac–dc con-verter is investigated with a 2.5-kW experimental system, andthe experimental results are presented. Section VI presents thenonlinear system stability of the proposed MMPC controllerwith the direct Lyapunov method and the nonlinear experimen-tal system model, as well as the robustness analysis againstparameter uncertainties. Moreover, the comparative evaluationof the proposed Lyapunov-function-based MMPC techniquewith the conventional MPC method in terms of execution timeis presented in Section VII. Finally, the conclusions are drawnin Section VIII.

II. BIDIRECTIONAL AC–DC CONVERTER TOPOLOGY

A. System Configuration

Fig. 1 shows the three-phase bidirectional ac–dc convertertopology that transfers power between the three-phase ac volt-age supply and the dc voltage bus. The bidirectional ac–dcconverter consists of six insulated-gate bipolar transistor–diodeswitches (S1−S6), which is connected with three-phase acvoltage supply through series filter inductance (Ls) and re-sistance (Rs). A dc capacitor (Cdc) is connected across dcvoltage bus to keep the dc bus voltage (Vdc) constant. Thebidirectional ac–dc converter operates in two modes. The firstmode is the rectifier mode, in which the bidirectional ac–dc

Fig. 1. Three-phase bidirectional ac–dc converter topology.

converter operates as a front-end rectifier and allows powertransfer from the three-phase ac voltage end to the dc voltagebus. The second mode is the inverter mode, where the powerflows from the dc voltage bus to the ac voltage end, and theconverter acts as a voltage source inverter.

B. Working Principle

The power circuit of the three-phase bidirectional ac–dc con-verter converts the electrical power between the ac and dc form,utilizing the electrical scheme shown in Fig. 1. To avoid shortcircuit, the two switches in each leg of the bidirectional ac–dcconverter should be operated in a complementary mode. Hence,the gating signals Sa, Sb, and Sc determine the switching statesof the three-phase bidirectional ac–dc converter as follows:

Sa =

{1, S1 is on and S2 is off

0, S1 is off and S2 is on(1)

Sb =


0, S3 is off and S4 is on(2)

Sc =


0, S5 is off and S6 is on.(3)

Therefore, the switching function vector (�S) of the bidirectionalac–dc converter can be expressed as

�S =2

3(Sa + �ωSb + �ω2Sc) (4)

where �ω = ej2π/3 = −(1/2) + j√3/2 is a unitary vector,

which represents the 120◦ phase displacement between thephases.

The output voltage space vector (�vconv) of the bidirectionalac–dc converter for both the rectifier and inverter modes can bepresented with phase-to-neutral voltages (vao, vbo, and vco) as

�vconv =2

3(vao + �ωvbo + �ω2vco). (5)

The output voltage space vector (�vconv) can also be relatedto the dc bus voltage (Vdc) and the switching function vector(�S) as

�vconv = �S × Vdc. (6)


TABLE IVOLTAGE SPACE VECTORS OF THE BIDIRECTIONAL

AC–DC CONVERTER

There are eight possible voltage vectors that can be obtainedfrom the eight consequence switching states of the switchingsignals Sa, Sb, and Sc. These eight voltage space vectors arelisted in Table I.

The energy storage system allows bidirectional power trans-fer between the three-phase ac voltage side and the energy stor-age device through the bidirectional ac–dc converter. Hence,the bidirectional ac–dc converter needs to be operated in twomodes, which are specified as the rectifier mode and the in-verter mode. The operating principle of the bidirectional ac–dcconverter for both the rectifier and inverter modes is elaboratelydescribed in the following section.

1) Rectifier Mode of Operation: During the rectifier modeof operation, the bidirectional ac–dc converter acts as a front-end rectifier that is connected to the three-phase ac voltagesource through the input filter inductance Ls and resistance Rs,as shown in Fig. 1. By applying Kirchhoff’s voltage law at theac side of the rectifier, the relationship between the three-phaseac voltage and rectifier input voltage vectors is

�vs = Lsd�is_rec

dt+Rs

�is_rec +2

3(vao + �ωvbo + �ω2vco)

− 2

3(vno + �ωvno + �ω2vno). (7)

The space vector model of three-phase ac voltage (�vs) andcurrent (�is) can be derived from phase voltage and current as

�vs =2

3(vsa + �ωvsb + �ω2vsc) (8)

�is =2

3(isa + �ωisb + �ω2isc) (9)

where vsa, vsb, and vsc are phase voltages, and isa, isb, and iscare phase currents of the three-phase ac voltage source.

Note that the last term of (7) is equal to zero as

2

3(vno + �ωvno + �ω2vno) =

2

3vno(1 + �ω + �ω2) = 0. (10)

Therefore, the relationship between the three-phase ac voltageand rectifier input voltage vectors can be rewritten from (5), (7),and (10) as [27]

�vs = Lsd�is_rec

dt+Rs

�is_rec + �vconv. (11)

Hence, the input current dynamics of the bidirectional ac–dcconverter during the rectifier mode of operation is

d�is_rec

dt=

Rs

Ls

�is_rec +1

Ls�vs −

1

Ls�vconv. (12)

2) Inverter Mode of Operation: The bidirectional ac–dcconverter works as a voltage source inverter during the invertermode, which allows power transfer from the dc voltage busto the three-phase ac voltage end. Hence, the load current is180◦ out-of-phase with respect to the load voltage. Therefore,the load current dynamics of the bidirectional ac–dc converterduring the inverter mode of operation can be presented as

d�is_inv

dt= −Rs

Ls

�is_inv +1

Ls�vconv −

1

Ls�vs. (13)

III. CONVENTIONAL MPC METHOD

The formulation of the conventional MPC algorithm forthe three-phase bidirectional ac–dc converter is described inthe following section. The MPC controller is formulated in thediscrete-time domain. Therefore, it is necessary to transform thedynamic system of the bidirectional ac–dc converter for boththe rectifier and inverter modes of operation represented in (12)and (13), respectively, into a discrete-time model at a specificsampling time Ts.

A. Discrete-Time Model for Prediction Horizon

A discrete-time model is used to predict the future values ofcurrents and voltages in the next sampling interval (k), fromthe measured currents and voltages at the (k − 1)th samplinginstant. The system model derivative dx/dt from Euler approx-imation can be expressed as

dx

dt≈ x(k)− x(k − 1)

Ts. (14)

Using one step advance of the above approximation, thediscrete-time model of predictive currents and voltages forthe next (k + 1) sampling instant of the bidirectional ac–dcconverter in the rectifier and inverter modes can be derived.

The discrete-time model of predictive input currents at thenext sampling instant (k + 1) for the rectifier mode of thebidirectional ac–dc converter can be evaluated from (12) withthe help of Euler approximation as

�is_rec(k + 1)

=1

RsTs+Ls

{Ls�is_rec(k)+Ts[�vs(k+1)−�vconv_rec(k+1)]

}.

(15)


Again, the discrete-time model of predictive input currents atthe next sampling instant (k + 1) for the inverter mode of thebidirectional ac–dc converter can also be evaluated from (13) as

�is_inv(k + 1)

=1

RsTs+Ls

{Ls�is_inv(k)+Ts[�vconv_inv(k + 1)−�vs(k + 1)]

}.

(16)

B. Cost Function

The main objective of the MPC algorithm is to minimizethe error with fast dynamic response between the predictedand reference values of the discrete variables. To achieve thisobjective, an appropriate cost function (e) is defined with ameasurement of predicted input error. Hence, the cost functionfor the rectifier and inverter modes can be expressed with theabsolute error between the predictive and reference values ofinput and load currents for both the rectifier and inverter modesof operation as follows:

�e =∣∣∣�iref(k + 1)−�ip(k + 1)

∣∣∣ (17)

where e is the cost function. The reference input and predictedcurrent are �iref_rec(k+1) and �ip_rec(k+1) and �iref_inv(k+1)

and �ip_inv(k + 1) for the rectifier and inverter modes,respectively.

The operating mode of the bidirectional ac–dc converter isfirst selected depending on the charging state of the energystorage device, which is determined by the dc bus voltage (Vdc).If the charging state (determined by dc voltage) is less thanthe threshold level, then it is operated in the rectifier mode;otherwise, it is operated in the inverter mode.

IV. PROPOSED MMPC TECHNIQUE BASED

ON LYAPUNOV FUNCTION

A. Determination of the Lyapunov-Function-BasedMMPC Algorithm

The proposed MMPC method directly applies the voltagevector constrained in the finite set. Hence, the future voltagevector (�v(k + 1)) is expressed with the voltage vector gen-erated by the converter (�vconv(k + 1)) and the unavoidablequantization error vectors as

�v(k + 1) = �vconv(k + 1) + δ(k + 1) (18)

where δ(k + 1) is the quantization error vector that satisfies‖δ(k + 1)‖ ≤ ϕ with a constant ϕ > 0. The future voltagevector (�v(k + 1)) is bounded in the finite set, as mentioned inTable I, and the quantization error δ(k + 1) vector ensures thestate where �vconv(k + 1) is bounded.

To determine the Lyapunov-function-based control algorithmfor the modification of the conventional MPC method, it isimportant to analyze the bidirectional ac–dc converter systemfrom the control point of view. Therefore, the future current

error vector of the input current dynamics (15) during therectifier mode can be rewritten as

�ierror_rec(k + 1)

=�is_rec(k + 1)−�iref_rec(k + 1)

=1

RsTs + Ls

{Ls�is_rec(k)+Ts [�vs(k + 1)− �vrec(k + 1)]

}−�iref_rec(k + 1). (19)

On the other hand, during the inverter mode of operation, thefuture current error vector can also be evaluated from (16) as

�ierror_inv(k + 1)

=�is_inv(k + 1)−�irefinv(k + 1)

=1

RsTs + Ls

{Ls�is_inv(k)+Ts [�vinv(k + 1)− �vs(k + 1)]

}−�iref_inv(k + 1). (20)

An effective control algorithm is essential for the bidirectionalac–dc converter so that the current (�is) tracks the referencevalue (�iref) for both the rectifier and inverter modes of oper-ation. Therefore, it is necessary to find a control function suchthat the current tracking error (�ierror) asymptotically convergesto zero. The Lyapunov direct method is used for the specificapplication.

The discrete Lyapunov function (L) is taken as

L(k) =1

2

[�ierror(k)

]T [�ierror(k)

]. (21)

From (19) and (21), the rate of change of the Lyapunov functioncan be expressed for the rectifier mode as

ΔLrec(k)

= L(�ierror_rec(k + 1)

)− L

(�ierror_rec(k)

)=

1

2

[1

RsTs + Ls

×(Ls�is_rec(k)+Ts (�vs(k + 1)− �vconv_rec(k + 1)

− δ(k+1)))−�iref_rec(k+1)

]T×[

1

RsTs + Ls

×(Ls�is_rec(k)+Ts (�vs(k +1)− �vconv_rec(k + 1)

− δ(k + 1)))−�iref_rec(k+1)

]− 1

2

[�ierror_rec(k)

]T [�ierror_rec(k)

]. (22)


On the other hand, the rate of change of the Lyapunov functionfor the inverter mode of operation can be written from (20) and(21) as

ΔLinv(k)

= L(�ierror_inv(k + 1)

)− L

(�ierror_inv(k)

)=

1

2

[1

RsTs + Ls

×(Ls�is_inv(k)+Ts(�vconv_inv(k + 1) + δ(k + 1)

−�vs(k + 1)))−�iref_inv(k+1)

]T×[

1

RsTs + Ls

×(Ls�is_inv(k)+Ts(�vconv_inv(k + 1) + δ(k + 1)

−�vs(k + 1)))−�iref_inv(k + 1)

]− 1

2

[�ierror_inv(k)

]T [�ierror_inv(k)

]. (23)

To make an effective control algorithm for converging thetracking error (�ierror) to zero, the rate of change of theLyapunov function (ΔL) always needs to be negative. There-fore, the discrete voltage vector during the rectifier mode at thenext sampling instant (�vrec(k + 1) = �vconv_rec(k + 1) + δ(k +1)), which assures that the rate of change of the Lyapunovfunction (22) is negative, which is written as

�vrec(k + 1) =Ls

Ts

�is_rec(k) + �vs(k + 1)

− RsTs + Ls

Ts

�iref_rec(k + 1). (24)

On the other hand, the discrete voltage vector during theinverter mode at the next sampling instant (�vinv(k + 1) =�vconv_inv(k + 1) + δ(k + 1)), which assures that the rate ofchange of the Lyapunov function (23) is negative, which is bewritten as

�vinv(k + 1) = −Ls

Ts

�is_inv(k) + �vs(k + 1)

+RsTs + Ls

Ts

�iref_inv(k + 1). (25)

B. Implementation of the Lyapunov-Function-BasedMMPC Algorithm for Bidirectional AC–DC Converter

The conventional MPC is generally based on current-oriented control techniques, in which the current tracks the ref-erence value by utilizing the discrete behavior of the converter.In this conventional MPC method, the future current of theconverter is calculated for each of the eight possible switching

Fig. 2. Proposed MMPC control technique based on the Lyapunovfunction.

states, and the state that minimizes the cost function is selectedfor firing the power switches. One of the major limitationsof this conventional MPC algorithm is high execution timedelay caused by the large number of calculation for the currentprediction. To reduce the execution time, the conventional MPCalgorithm is modified based on the Lyapunov function. Theproposed MMPC is based on voltage-oriented control, in whichthe optimum possible future voltage vector (�vopt(k + 1)) of theconverter is directly selected to track the calculated referencevoltage vector (�v(k + 1)) of (24) and (25), for both the rectifierand inverter modes by utilizing the modified cost function as

g(k + 1) = |�v(k + 1)− �vopt(k + 1)| . (26)

Fig. 2 shows the proposed control strategy of the MMPCalgorithm, which is a one-step modification of the conventionalfinite control set (FCS)-MPC control scheme [13], [28]. Thereference voltage vector (�v(k + 1)) of the converter is calcu-lated from the measured current ac current (�is(k)) and thereference current (�iref(k)) by utilizing (24) and (25) for therectifier and inverter modes, respectively. The overall controltechnique for the Lyapunov-function-based MMPC has beendescribed with an execution time diagram and is presented inFig. 3. The proposed Lyapunov-function-based MMPC methodsatisfies the following steps.

Step 1: Measurement of the ac currents (�is(k)) and calcula-tion of the reference current (�iref(k)).

Step 2: Calculation of the future reference voltage vector(�vref(k + 1)) from the measured current ac current(�is(k)) and the reference current (�iref) by utilizing(24) and (25).


Fig. 3. Execution time diagram of the proposed Lyapunov-function-based MMPC.

Step 3: Estimation of the initial cost function value as(ginit(k + 1) = 1e10) from the calculation in the kthsampling period.

Step 4: Prediction of the optimum possible future voltagevector (�vopt(k + 1)) from Table I.

Step 5: Evaluation of the cost function (g(k+1))by using (26).Step 6: Optimization of the cost function and selection of the

appropriate switching state that minimizes the costfunction.

Step 7: Application of the selected switching state of thebidirectional ac–dc converter for firing the switches.

V. EXPERIMENTAL RESULTS

A 2.5-kW downscaled laboratory prototype of the bidirec-tional ac–dc converter is developed for the verification ofthe proposed Lyapunov-function-based MMPC algorithm. Theschematic layout of the experimental system is presented inFig. 4. The parameters presented in Table II are used in theexperimentation. The experimental verification of the proposedMMPC bidirectional ac–dc converter is carried out by usingthe rapid prototyping and real-time interface system dSPACE

Fig. 4. Experimental system of the bidirectional ac–dc converter withMMPC.

TABLE IISIMULATION AND EXPERIMENTAL PARAMETERS

with a DS1104 control card that consist of a Texas InstrumentsTMS320F240 subprocessor and the Power PC 603e/250-MHzmain processor. This dSPACE control desk works together withMath-work MATLAB/Simulink-R2013a real-time workshopand real-time interface control cards to implement the proposedMMPC algorithm. The voltage is measured with a differentialprobe [PINTEK DP-25], and the current is measured with acurrent transducer [LEM LA 25-NP].

Fig. 5(a) shows that the ac phase voltage and current areexactly in phase during the rectifier operation mode, whichensures a unity power factor. Again, in the inverter mode,the converter allows power transfer with a unity power factorfrom the dc voltage bus to the ac voltage end by keeping thephase voltage and current with 180◦ phase shift, as presentedin Fig. 5(c). The dc bus voltage and current for both operatingmodes are depicted in Fig. 5(b) and (d). The results illustratethat the dc bus voltage ripple and the pulsation in dc current arevery low during both operating modes.

During the rectifier mode, the dc bus reference voltage(Vdc_ref) is varied from 270 to 320 V to check the stabilityand transient responsiveness of the MMPC control algorithm.


Fig. 5. Experimental results. (a) AC phase voltage and current atrectifier mode. (b) DC terminal voltage and current at rectifier mode.(c) AC phase voltage and current at inverter mode. (d) DC terminalvoltage and current at inverter mode of operation.

Fig. 5(a) and (b) also shows that when the reference voltagechanges from 270 to 320 V, the steady state of the ac wave-forms is reached within a very short time (less than 10 ms).This rapid step change confirms the fast response of the pro-posed Lyapunov-function-based MMPC method. The steady-state output of the dc-link voltage and current in Fig. 5(b)remains linear in a wide range of time with very low voltage andcurrent ripple, which ensure the stability and good performanceof the proposed MMPC algorithm. Similarly, the stability andresponsiveness of the MMPC method for the bidirectionalac–dc converter in the inverter mode of operation are shown inFig. 5(c) and (d), by varying the ac reference current (�iref_inv).

VI. STABILITY ANALYSIS

The stability issue of the proposed Lyapunov-function-basedMMPC algorithm is very important for the high performance ofthe bidirectional ac–dc converter in an energy storage system.This section shows the stability analysis of the proposed MMPCwith the direct Lyapunov method and nonlinear stability inves-tigation. Moreover, the robustness of the proposed Lyapunov-function-based MMPC algorithm against parameter uncertaintyhas also been analyzed in this section.

A. Stability Analysis With Direct Lyapunov Method

The stability of the proposed MMPC technique for thebidirectional ac–dc converter is investigated with the directLyapunov method. The direct Lyapunov method gives the fol-lowing stability criteria for a continuous function L(�ierror(k)),in which the solutions of current dynamics in the rectifier mode(12) and the inverter mode (13) are uniformly and ultimatelybounded [29], i.e.,

L(�ierror(k)

)≥ c1

∣∣∣�ierror(k)∣∣∣l , ∀�ierror(k) ∈ G

L(�ierror(k)

)≤ c2

∣∣∣�ierror(k)∣∣∣l , ∀�ierror(k) ∈ Γ

L(�ierror(k + 1)

)− L

(�ierror(k)

)< −c3

∣∣∣�ierror(k)∣∣∣l + c4

(27)

where c1, c2, c3, and c4 are positive constants, l ≥ 1, G ⊆ Rn

is a positive control invariant set, and Γ ⊂ G is a compact set.By applying the value of future voltage vector (�v(k + 1)) of theconverter for the rectifier (24) and inverter (25) modes, the rateof change of the Lyapunov function can be written as

ΔL(k)≤−1

2

[�ierror(k)

]T[�ierror(k)

]+1

2

(Ts

(RsTs + Ls)

)2

ϕ2.

(28)

Therefore, the stability condition (27) is satisfied by the con-stant values as

c1 = c2 = 1; c3 =1

2; c4 =

1

2

(Ts

(RsTs + L)

)2

ϕ2. (29)

As a result, all signals in the Lyapunov-function-basedclosed-loop MMPC bidirectional ac–dc converter system are


ultimately uniformly bounded. Then, the rate of change of theLyapunov function in (28) is

ΔL(k) ≤ −2c3L(�ierror_inv(k)

)+ c4. (30)

This inequality implies that, as time increases, the currentcontrol error vectors converge to the compact set as

A =

⌊‖�ierror‖‖�ierror‖ ≤

√c4c3

⌋. (31)

Thus, all signals of the proposed MMPC control method in (24)and (25) are uniformly and ultimately bounded.

B. Stability Analysis With Nonlinear Model

The stability analysis of the proposed MMPC-controlledbidirectional ac–dc converter is performed with the directLyapunov method in the previous section by neglecting thenonlinear criteria such as the unsymmetrical three-phase sup-ply, existence of higher harmonics in the converter input volt-age, cross-coupling effect, and time delay variation. Therefore,the nonlinear system stability of the Lyapunov-function-basedMMPC algorithm for the bidirectional ac–dc converter is an-alyzed with the laboratory experimental prototype. The sameparameters as in Table II are employed.

During the rectifier mode of operation, the dc bus referencevoltage (Vdc_ref) is varied from 270 to 320 V to check thenonlinear stability of the proposed MMPC control algorithm.The three-phase ac current variation drawn by the bidirectionalac–dc converter in the rectifier mode due to the dc bus referencevoltage (Vdc_ref) variation is presented in Fig. 6(a). The result inFig. 6(a) shows that the output phase current is accurately track-ing the reference value and reached its steady-state conditionwith very fast dynamic response, which verifies the stability andeffectiveness of the MMPC controller for the ac–dc converter.Moreover, the stability of the proposed control algorithm canbe further confirmed with the variation of the dc-link voltageand current, as shown in Fig. 6(b). The dc-link voltage andthe dc current vary with respect to the dc bus reference voltage(Vdc_ref) variation as the load is constant.

On the other hand, during the inverter mode of operation, theac load reference current (�iref_inv) is varied from 5 to 10 A peri-odically to check the stability of the proposed MMPC controllerfor bidirectional ac–dc converters. The result in Fig. 6(c) showsthat the output three-phase ac current is accurately trackingthe reference value and reached its steady-state condition withvery fast dynamic response, which verifies the stability andeffectiveness of the MMPC controller in case of output acreference current variation. Again, the stability of the proposedcontrol algorithm can be further confirmed with the variation ofthe dc-link voltage and current, as shown in Fig. 6(d). The dccurrent varies with respect to the ac reference current variationas the dc voltage and load are constant.

Fig. 6. Stability investigation via an experimental system. (a) Three-phase ac current (isa) in the rectifier mode. (b) DC bus voltage (Vdc)versus current (Idc) with the dc-link reference voltage (Vdc_ref ) changein the rectifier mode. (c) Three-phase ac current (isa) in the invertermode. (d) DC bus voltage (Vdc) versus current (Idc) with the acreference current (iref_inv) change in the inverter mode.


C. Robustness Analysis

The control variables of nonlinear practical power convertersare always affected by state measurement errors. Generally,these state measurement errors are caused by the filter param-eter variation in practical power converters. Therefore, it isimportant that the closed-loop control techniques of nonlinearpower converters should be robust with parameter variation[30]–[32].

To analyze the robustness of the proposed Lyapunov-function-based MMPC algorithm, the discrete voltage vector(24) and cost function (26) can be simplified from the controlpoint of view, as follows [31]:

x(k + 1) = �Ay(k) + �Bu(k + 1) + �D (32)

g(k + 1) = |x(k + 1)− xref(k + 1)| (33)

x(k + 1) ∈ Xv(k + 1) or g(k + 1) < g(k) (34)

where x = �vrec, y =�is_rec, u =�iref_rec, �A = Ls/Ts, �B =

((RsTs + Ls)/Ts), �D = �vs, and Xv is the voltage set region.The robustness of the proposed Lyapunov-function-based

MMPC algorithm for the bidirectional ac–dc converter againstparameter uncertainties has been analyzed by considering theaddition of uncertain values of the filter resistance (R̃) andinductance (L̃). These uncertain values lead to the parameteruncertainties of the model as

�̃A =L̃s

Ts, �̃B =

R̃sTs + L̃s

Ts. (35)

Therefore, the control state equation (32) of the proposed MMPCalgorithm can be written with parameter uncertainties as

xunc(k + 1) = ( �A+ �̃A)y(k) + ( �B + �̃B)u(k + 1) + �D. (36)

Substituting (35) into |xunc(k + 1)− xref(k + 1)| leads to

|xunc(k + 1) = xref(k + 1)| ≤ |x(k + 1)− xref(k + 1)|

+

∣∣∣∣ �̃Ay(k) + �̃Bu(k + 1)

∣∣∣∣ . (37)

The deviation from the reference at (k + 1) is thus bounded bythe sum of two terms—the nominal response and an uncertaintyterm, which is a function of the state vector and the input.On the uncertainty term, the upper bound can be presented asfollows: ∣∣∣∣ �̃Ay(k) + �̃Bu(k + 1)

∣∣∣∣ ≤ p1 |y(k)|+ p2. (38)

To ensure (robust) convergence in the presence of additionalparameter uncertainties, the right-hand side of (37) has to be

strictly less than |x(k) − xref(k)|, i.e.,

|x(k + 1)− xref(k + 1)|+ p1 |y(k)|+ p2 < |x(k)− xref(k)|.(39)

This is equivalent to

g(k + 1) + p1 |y(k)|+ p2 < g(k). (40)

To ensure the robustness of the controller, when the con-trolled variable is outside of the bound, the constraint on theright-hand side in (34) is to be replaced by (40). This ensuresthat only voltage vectors are selected, which point with a certainminimum tracking toward the current reference.

Accordingly, when the control variable is within its bound,the constraint on left-hand side of (34) needs to be modified.Specifically, the constraint at k + 1 is replaced by x(k + 1) ∈X̃v, where X̃v uses the radius (τ̃ ), which can be presented as

τ̃ = τv − p1 |y(k)| − p2. (41)

As a result, X̃v is a subset of Xv. This ensures the robustness ofthe proposed MMPC algorithm against parameter uncertaintiesonce the controlled variables are within the bounds.

Similarly, the robustness of the proposed MMPC algorithmfor the bidirectional ac–dc converter with parameter uncertain-ties can also be analyzed during the inverter mode of operation.

The theoretical robustness analysis of the proposed MMPCmethod with additive uncertain values of the filter resistance(R̃) and inductance (L̃) in (35) has been verified with a nonlin-ear MATLAB/Simulink model and visualized in Fig. 7, whichshows the variation of current error (�ierror =�is −�iref) againstthe variation of the filter parameter. The value of filter resistance(R̃) and inductance (Ls) in (35) has been varied up to 200%(nominal filter resistance, Rs = 0.1 Ω and inductance, Ls =5 mH) to verify the robustness against parameter uncertainties.

The results in Fig. 7 confirm that the variation of currenterror is bounded within 1 A (less than 10%) for differentvalues of the filter parameter, which ensures the robustnessof the proposed control algorithm. Moreover, the percentagedeviation of current error (�ierror) with different values of filterparameters for both the rectifier and inverter modes of operationare compared and presented in Fig. 7(g), which shows that themaximum deviation of current tracking error of about 8% isnegligible. Although the current tracking error is decreasingwith the increase in the filter inductor value, the power lossincreases with the increase in inductor. The main focus of thispaper is to reduce the execution time with MMPC, and thissection shows the robustness with parameter variation. Hence,filter value selection and further discussion on power losseshave been omitted. The percentage of current error is calculatedfrom the reference (�iref) and measured (�is) current as

�ierror =�is −�iref�iref

× 100%. (42)


Fig. 7. Robustness analysis of the proposed MMPC algorithm againstparameter variation. (a) Reference current (isa_ref ) and measured cur-rent (isa_meas) of phase A. Current error (isa_error) of phase A with(b) Rs = 0.1 Ω, Ls = 5 mH (nominal), (c) Rs = 0.15 Ω, Ls = 7 mH,(d) Rs = 0.20 Ω, Ls = 10 mH, (e) Rs = 0.25 Ω, Ls = 12 mH and(f) Rs = 0.30 Ω, Ls = 15 mH filter parameter value. (g) Percentagecurrent error.

Fig. 8. Control algorithm of (a) the conventional MPC and (b) theproposed Lyapunov-function-based MMPC.

Therefore, it is possible to demonstrate that the proposedcontrol strategy is robust and can efficiently operate even underfilter parameter variations.

VII. COMPARATIVE EVALUATION OF THE PROPOSED

MMPC WITH CONVENTIONAL MPC IN TERMS

OF EXECUTION TIME

One of the major advantages of the proposed Lyapunov-function-based MMPC algorithm is its very low execution time.The proposed MMPC algorithm reduces the amount of calcula-tions required to predict a future variable by half compared withthe conventional MPC method, which is elaborately presentedin Fig 8.

In the experimental system, the execution time requiredin completing the algorithms of the conventional MPC andthe proposed Lyapunov-function-based MMPC techniques arecalculated by measuring the calculation cycles of the DS1104control card of the dSPACE real-time prototype. The totalexecution times of the conventional MPC method are 3.95 and3.8 μs for the rectifier and inverter modes, respectively. Oth-erwise, the total execution times of the proposed Lyapunov-function-based MMPC method are about 3.24 μs during therectifier mode and 3.12 μs during the inverter mode. Theexecution time of the rectifier mode is higher than that ofthe inverter mode due the usage of the PI controller. Hence, thetotal execution time is reduced by 17.89% with the proposedMMPC method.

Fig. 9(a) presents the experimental results, which show thestep change of the high dc bus voltage and current with stepchange of the dc bus reference voltage (Vdc_ref). The high dcbus voltage and current reached its steady-state level within


Fig. 9. Experimental results with the MMPC scheme. (a) DC bus voltageand current at the rectifier mode. (b) Three-phase ac current and costfunction at the inverter mode.

3.24 μs, which confirms the low execution time of the proposedLyapunov-function-based MMPC technique during the rectifiermode of operation. On the other hand, Fig. 9(b) shows theexperimental wave shapes of the three-phase ac currents andthe cost function of the proposed MMPC method during theinverter mode of operation. The three-phase ac currentcontrolled by the proposed MMPC technique follows thereferences accurately with very fast dynamic response.Moreover, the Lyapunov-function-based cost function pre-sented in Fig. 9(b) can be referred to the ac current error,which is bounded for its transient stability conditions withquantization error resulting from the finite number of possiblevoltage vectors of the bidirectional ac–dc converter.

A comparative evaluation between the conventional MPCand the proposed Lyapunov-function-based MMPC has beenperformed and summarized in Table III. The proposed MMPCalgorithm operates as a voltage mode control technique, whichreduces the execution time by minimizing the amount of cal-culation compared with the conventional current-mode MPCalgorithm. Moreover, the average switching frequency of theMMPC algorithm is also less than that of the conventionalMPC method. Finally, the proposed MMPC ensures the systemstability with the direct Lyapunov stability analysis method,which is impossible for the conventional MPC.

VIII. CONCLUSION

In this paper, a Lyapunov-function-based MMPC algorithmhas been proposed to control the bidirectional ac–dc converter,

TABLE IIICOMPARISON BETWEEN CONVENTIONAL MPC AND PROPOSED

LYAPUNOV-FUNCTION-BASED MMPC

which is applied in an energy storage system to transfer powerbetween the three-phase ac voltage source and the dc volt-age bus. The Lyapunov-function-based MMPC is a powerfulcontrol algorithm in the field of bidirectional ac–dc powerconverters, which provides bidirectional power flow with in-stantaneous mode-changing capability, fast dynamic response,and nonlinear system stability. The stability analysis of thiscontrol method is performed with the direct Lyapunov methodand nonlinear model analysis. The result confirms that theMMPC system is stable for the operation of the bidirectionalac–dc converter. Moreover, the most important advantage of theproposed Lyapunov-function-based MMPC algorithm is that ithas very low execution time. The total execution time of theproposed MMPC algorithm is about 18% lower than that ofthe conventional MPC algorithm. The results associated in thisinvestigation are very encouraging and will continue to play astrategic role in the improvement of modern high-performancebidirectional ac–dc converters.

REFERENCES

[1] J. M. Carrasco et al., “Power-electronic systems for the grid integration ofrenewable energy sources: A survey,” IEEE Trans. Ind. Electron., vol. 53,no. 4, pp. 1002–1016, Jun. 2006.

[2] S. Vazquez, S. M. Lukic, E. Galvan, L. G. Franquelo, and J. M. Carrasco,“Energy storage systems for transport and grid applications,” IEEE Trans.Ind. Electron., vol. 57, no. 12, pp. 3881–3895, Dec. 2010.

[3] N. M. L. Tan, T. Abe, and H. Akagi, “Design and performance of a bidi-rectional Isolated DC–DC converter for a battery energy storage system,”IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1237–1248, Mar. 2012.

[4] J. R. Rodríguez, L. Dixon, J. R. Espinoza, J. Pontt, and P. Lezana, “PWMregenerative rectifiers: State of the art,” IEEE Trans. Ind. Electron.,vol. 52, no. 1, pp. 5–22, Feb. 2005.

[5] P. Akter, M. Uddin, S. Mekhilef, N. M. L. Tan, and H. Akagi, “Modelpredictive control of bidirectional isolated DC–DC converter for energyconversion system,” Int. J. Electron., vol. 102, no. 8, pp. 1407–1427,2015.

[6] B. Singh et al., “A review of three-phase improved power quality AC–DCconverters,” IEEE Trans. Ind. Electron., vol. 51, no. 3, pp. 641–660,Jun. 2004.

[7] J. Dannehl, C. Wessels, and F. W. Fuchs, “Limitations of voltage-orientedPI current control of grid-connected PWM rectifiers with LCL filters,”IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 380–388, Feb. 2009.

[8] M. Malinowski, M. P. Kazmierkowski, and A. M. Trzynadlowski, “Acomparative study of control techniques for PWM rectifiers in AC ad-justable speed drives,” IEEE Trans. Power Electron., vol. 18, no. 6,pp. 1390–1396, Nov. 2003.

[9] D. Zhi, L. Xu, and B. W. Williams, “Improved direct power control ofgrid-connected DC/AC converters,” IEEE Trans. Power Electron., vol. 24,no. 5, pp. 1280–1292, May 2009.


[10] A. Bouafia, F. Krim, and J.-P. Gaubert, “Fuzzy-logic-based switching stateselection for direct power control of three-phase PWM rectifier,” IEEETrans. Ind. Electron., vol. 56, no. 6, pp. 1984–1992, Jun. 2009.

[11] J. Hu, L. Shang, Y. He, and Z. Zhu, “Direct active and reactive powerregulation of grid-connected DC/AC converters using sliding mode con-trol approach,” IEEE Trans. Power Electron., vol. 26, no. 1, pp. 210–222,Jan. 2011.

[12] C. Reza, M. D. Islam, and S. Mekhilef, “Modeling and experimentalverification of ANN based online stator resistance estimation in DTC-IMdrive,” J. Elect. Eng. Technol., vol. 9, no. 2, pp. 550–558, Mar. 2014.

[13] J. Rodriguez et al., “State of the art of finite control set model predictivecontrol in power electronics,” IEEE Trans. Ind. Informat., vol. 9, no. 2,pp. 1003–1016, May 2013.

[14] M. P. Akter, S. Mekhilef, N. M. L. Tan, and H. Akagi, “Model predictivecontrol of bidirectional AC–DC converter for energy storage system,”J. Elect. Eng. Technol., vol. 10, no. 1, pp. 165–175, Jan. 2015.

[15] M. P. Akter, S. Mekhilef, N. M. L. Tan, and H. Akagi, “Stability andperformance investigations of model predictive controlled active-front-end (AFE) rectifiers for energy storage systems,” J. Power Electron.,vol. 15, no. 1, pp. 202–215, Jan. 2015.

[16] P. Cortés, M. P. Kazmierkowski, R. M. Kennel, D. E. Quevedo, andJ. Rodríguez, “Predictive control in power electronics and drives,” IEEETrans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, Dec. 2008.

[17] M. Parvez, S. Mekhilef, N. M. L. Tan, and H. Akagi, “Model predictivecontrol of a bidirectional AC–DC converter for V2G and G2V applica-tions in electric vehicle battery charger,” in Proc. ITEC, 2014, pp. 1–6.

[18] S. Muslem Uddin et al., “Model predictive control of an active front endrectifier with unity displacement factor,” in Proc. IEEE ICCAS, 2013,pp. 81–85.

[19] M. Uddin, S. Mekhilef, M. Mubin, M. Rivera, and J. Rodriguez, “Modelpredictive torque ripple reduction with weighting factor optimization fedby an indirect matrix converter,” Electr. Power Compon. Syst., vol. 42,no. 10, pp. 1059–1069, 2014.

[20] R. Vargas, P. Cortés, U. Ammann, J. Rodríguez, and J. Pontt, “Predictivecontrol of a three-phase neutral-point-clamped inverter,” IEEE Trans. Ind.Electron., vol. 54, no. 5, pp. 2697–2705, Oct. 2007.

[21] S. Kwak and S.-K. Mun, “Model predictive control methods to reducecommon-mode voltage for three-phase voltage source inverters,” IEEETrans. Power Electron., vol. 30, no. 9, pp. 5019–5035, Sep. 2015.

[22] J. Scoltock, T. Geyer, and U. Madawala, “Model predictive directpower control for grid-connected neutral-point-clamped converters,”IEEE Trans. Ind. Electron., vol. 62, no. 9, pp. 5319–5328, Sep. 2015.

[23] P. Cortes, J. Rodriguez, C. Silva, and A. Flores, “Delay compensation inmodel predictive current control of a three-phase inverter,” IEEE Trans.Ind. Electron., vol. 59, no. 2, pp. 1323–1325, Feb. 2012.

[24] D. Quevedo, R. Aguilera, and T. Geyer, “Predictive control in powerelectronics and drives: Basic concepts, theory, and methods,” in Ad-vanced and Intelligent Control in Power Electronics and Drives, vol. 531,T. Orłowska-Kowalska, F. Blaabjerg, and J. Rodríguez, Eds. Cham,Switzerland: Springer-Verlag, 2014, pp. 181–226.

[25] M. Pozo-Palma and M. Pacas, “Predictive torque control for an inductionmachine with a virtual multilevel inverter,” in Proc. IEEE IECON, 2013,pp. 5848–5853.

[26] M. Pozo-Palma and M. Pacas, “Predictive control for a PMSM withLC-filter and a virtual multilevel inverter,” in Proc. ICEM, 2014,pp. 877–883.

[27] J. Rodriguez and P. Cortes, Predictive Control of Power Converters andElectrical Drives, vol. 40. New York, NY, USA: Wiley, 2012.

[28] S. Vazquez et al., “Model predictive control: A review of its applicationsin power electronics,” IEEE Ind. Electron. Mag., vol. 8, no. 1, pp. 16–31,Mar. 2014.

[29] J.-J. E. Slotine and W. Li, Applied Nonlinear Control, vol. 199.Englewood Cliffs, NJ, USA: Prentice-Hall, 1991.

[30] S. Dasgupta, S. N. Mohan, S. K. Sahoo, and S. K. Panda, “Lyapunovfunction-based current controller to control active and reactive powerflow from a renewable energy source to a generalized three-phase mi-crogrid system,” IEEE Trans. Ind. Electron., vol. 60, no. 2, pp. 799–813,Feb. 2013.

[31] T. Geyer, R. P. Aguilera, and D. E. Quevedo, “On the stability and ro-bustness of model predictive direct current control,” in Proc. IEEE ICIT ,2013, pp. 374–379.

[32] M. Rivera, V. Yaramasu, J. Rodriguez, and B. Wu, “Model predictivecurrent control of two-level four-leg inverters—Part II: Experimentalimplementation and validation,” IEEE Trans. Power Electron., vol. 28,no. 7, pp. 3469–3478, Jul. 2013.

Md. Parvez Akter was born in Pabna,Bangladesh. He received the B.Sc.Engg. de-gree from Chittagong University of Engineer-ing and Technology, Chittagong, Bangladesh,in 2011 and the M.Eng.Sc. degree from theUniversity of Malaya, Kuala Lumpur, Malaysia,in 2015.

He is currently a Research Assistant withthe Power Electronics and Renewable EnergyResearch Laboratory, Department of ElectricalEngineering, University of Malaya. His research

interests include power converters and electrical drives, bidirectionalpower conversion techniques, predictive and digital control, renewableenergy, smart grids, and wireless power transfer.

Saad Mekhilef (M’01–SM’12) received theB.Eng. degree in electrical engineering fromthe University of Setif, Setif, Algeria, in 1995and the M.Eng.Sc. and Ph.D. degrees from theUniversity of Malaya, Kuala Lumpur, Malaysia,in 1998 and 2003, respectively.

He is currently a Professor with the De-partment of Electrical Engineering, Universityof Malaya. He is the author or coauthor ofmore than 250 publications in international jour-nals and conference proceedings. He is actively

involved in industrial consultancy for major corporations in power elec-tronics projects. His research interests include power conversion tech-niques, control of power converters, renewable energy, wireless powertransfer, and energy efficiency.

Nadia Mei Lin Tan (S’07–M’10) was bornin Kuala Lumpur, Malaysia. She received theB.Eng. degree (Hons.) from The University ofSheffield, Sheffield, U.K., in 2002, the M.Eng.degree from the Universiti Tenaga Nasional,Kajang, Malaysia, in 2007, and the Ph.D. de-gree from Tokyo Institute of Technology, Tokyo,Japan, in 2010, all in electrical engineering.

Since October 2010, she has been a Se-nior Lecturer with the Department of ElectricalPower Engineering, Universiti Tenaga Nasional.

Her current research interests include power conversion systems andbidirectional isolated dc–dc converters.

Dr. Tan is a Graduate Member of the Institution of Engineers Malaysiaand a member of the Institution of Engineering and Technology, U.K.

Hirofumi Akagi (M’87–SM’94–F’96) was bornin Okayama, Japan, in 1951. He received thePh.D. degree in electrical engineering fromTokyo Institute of Technology, Tokyo, Japan,in 1979.

Since 2000, he has been a Professor withthe Department of Electrical and ElectronicEngineering, Tokyo Institute of Technology. Priorto this, he was with Nagaoka University of Tech-nology, Nagaoka, Japan, and Okayama Univer-sity, Okayama. He has authored or coauthored

some 120 IEEE TRANSACTIONS papers. His research interests includepower conversion systems and their applications to industry, transporta-tion, and utilities.

Dr. Akagi received six IEEE TRANSACTIONS Prize Paper Awards and14 IEEE Industry Applications Society (IAS) Committee Prize PaperAwards. He was a recipient of the 2001 IEEE William E. Newell PowerElectronics Award, the 2004 IEEE IAS Outstanding Achievement Award,the 2008 IEEE Richard H. Kaufmann Technical Field Award, the 2012IEEE PES Nari Hingorani Custom Power Award, and the 2014 EPEOutstanding Service Award. During 2007–2008, he was the Presidentof the IEEE Power Electronics Society. Since 2015, he has been servingas the IEEE Division II Director.

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