Modified Model Predictive Control of a Bidirectional AC-DC Converter ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2015.2478752, IEEE Transactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Modified Model Predictive Control of a Bidirectional AC-DC Converter Based on Lyapunov Function for Energy Storage 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 applied in power distribution sectors as well as in renewable energy sources to ensure uninterruptible power supply. This paper proposes a modified model predictive control (MMPC) method based on Lyapunov function to improve the performance of a bidirectional ac-dc converter, which is used in an energy storage system for bidirectional power transfer between the three-phase ac voltage supply and energy storage devices. The proposed control technique utilizes discrete behavior of the converter considering the unavoidable quantization errors between the controller and the control actions selected from the finite control set of the bidirectional ac-dc converter. The proposed control method reduces the execution time delay by 18% compared to conventional MPC. Moreover, the nonlinear system stability of the proposed MMPC technique is ensured by the direct Lyapunov method and a nonlinear experimental system model. Detailed experimental results with a 2.5-kW downscaled hardware prototype are provided to show the efficacy of the proposed control system. Index Terms— Modified model predictive control (MMPC), Lyapunov methods, Stability analysis, Bidirectional ac-dc power conversion, Energy storage system.

I. INTRODUCTION

E

NERGY storage system plays an important role in utility and transport applications as well as in renewable energy sources to ensure power reliability, active power control, load leveling and frequency control [1, 2]. Generally, the energy Manuscript received January 14, 2015; revised April 21, 2015 and June 22, 2015; accepted August 5, 2015. Copyright ©2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported by the High Impact Research–Ministry of Higher Education under Project UM.C/ HIR/MOHE/ENG/24 and UMRG project No.RP006E-13ICT. M. Parvez and S. Mekhilef are with the Power Electronics and Renewable Energy Research Laboratory (PEARL), Department of Electrical 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, Kajang 43000, Malaysia (e-mail: [email protected]). H. Akagi is with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo 152-8552 Japan (e-mail: [email protected]).

storage system uses static storage devices such as electric double-layer capacitor, Li-ion battery, lead-acid battery and nickel metal-hydride battery [3]. These static storage devices contain high power and energy density but require proper operation such as low ripple current and voltage at the dc side. The bidirectional ac-dc converter is an essential part of the energy storage system due to its bidirectional-power-flow, grid synchronization and dc power management capabilities [4]. The control algorithm of this ac-dc converter should be highly stable and efficient as it needs to prevent the problems of poor power quality due to high total harmonic distortion (THD), low power factor, ac voltage distortion, and ripple in the dc current and voltage [5, 6]. Therefore, several researches have been carried out to improve the efficiency and performance of this bidirectional ac-dc converter. The classical control techniques are based on voltage-oriented control (VOC) [7], virtual-flux-oriented control [8], and direct power control (DPC) [9], schemes, which utilize the PI controllers. The major limitation of these control schemes is tuning the PI controllers that further affect the co-ordinate transform accuracy. In order to overcome this limitation of PI controllers, a fuzzy-logic based switching state selection criteria has been presented in [10] by avoiding predefined switching table. Although the active and reactive powers are smoothed in fuzzy-logic based DPC algorithm compared with classical DPC, its sampling frequency is high. Therefore, a sliding mode nonlinear [11] and artificial neural network [12] control approach has been investigated for active and reactive power regulation of grid connected dc-ac converter, which is dependent on control variables. The principle feature 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 control for the bidirectional ac-dc converters comparing with the classical solutions due to its simple and intuitive concept with no PWM blocks [13]. Moreover, MPC algorithm is very easy to configure with constraints and non-linearity and also for practical implementation [14-16]. Due to these advantages, the MPC algorithm has been extensively applied in active front-end rectifier [17, 18], indirect matrix converter [19], three-level converter [20], voltage source inverter [21], and neutral-point-clamped converter [22] etc. In spite of these

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS boundless advantages, MPC faces computational burden due to solving the underlying optimization problem of the discrete manipulated variables [23, 24]. Hence, computational issues have become very important for long prediction horizons. As a result, the modification in the existing MPC techniques is required for minimizing the amount of calculations as well as the computational time. Although, a Luenberg observer and a Runge-Kutta fourth-order method based predictive control has been proposed for delay compensation and obtain a convergence in digital implementation of the virtual multilevel inverter in [25, 26], the robustness analysis against parameter uncertainties is missing in these papers. Furthermore, a complete research on bidirectional ac-dc converter, considering practical nonlinearity issues with explicit addressing of stability issues in MPC algorithm, is missing in the present state-of-art. This paper proposes a modified model predictive control (MMPC) algorithm based on Lyapunov function that is applied in a bidirectional ac-dc converter for energy storage system to reduce the computational time, ensure stability and robustness, and increase the system performance. The system configuration and working principle of the bidirectional ac-dc converter are elaborately described in section II. The formulation of the conventional MPC method with discrete time model and the cost function is reviewed in section III. The determination of the proposed Lyapunov-function-based MMPC Algorithm and its practical implementation into bidirectional ac-dc converter are elaborately discussed in section IV. The performance of the proposed Lyapunovfunction-based MMPC method for the bidirectional ac-dc converter is investigated with a 2.5 kW experimental system and the experimental results are presented in section V. Section VI presents the nonlinear system stability of the proposed MMPC controller with the direct Lyapunov method and nonlinear experimental system model, and also the robustness analysis against parameter uncertainties. Moreover, the comparative evaluation of proposed Lyapunov-functionbased MMPC technique with conventional MPC method in terms of execution time is presented in section VII. Finally the conclusions are drawn in section VIII. II. BIDIRECTIONAL AC-DC CONVERTER TOPOLOGY A. System Configuration Fig. 1 shows the three-phase bidirectional ac-dc converter topology which transfers power between the three-phase ac voltage supply and the dc voltage bus. The bidirectional ac-dc converter consists of six IGBT-Diode switches (S1-S6), which is connected with three-phase ac voltage supply through series filter inductance (Ls) and resistance (Rs). A dc capacitor (Cdc) is connected across dc voltage bus to keep the dc bus voltage (Vdc) constant. The bidirectional ac-dc converter operates in two modes. The first mode is rectifier mode, in which the bidirectional ac-dc converter operates as a front-end-rectifier and allows power transfer from the three-phase ac voltage end to the dc voltage bus. The second mode is inverter mode,

dc Voltage Bus Three Phase ac Voltage vsa Rs

Idc S1 Ls vao

isa vsb n vsc

S5

S3

Rs

Ls vbo

isb Rs

Vdc

Cdc

Ls vco

isc S2

S6

S4

o

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

where the power flows from dc voltage bus to ac voltage end and the converter acts as a voltage source inverter. B. Working Principle The power circuit of the three-phase bidirectional ac-dc converter converts the electrical power between the ac and dc form utilizing the electrical scheme shown in Fig. 1. In order to avoid short circuit, the two switches in each leg of the bidirectional ac-dc converter should be operated in a complementary mode. Hence, the gating signals Sa, Sb and Sc determine the switching states of the three-phase bidirectional ac-dc converter as follows:

 1, S1 is on and S2 is off Sa    0, S1 is off and S2 is on

(1)

 1, S3 is on and S 4 is off Sb    0, S3 is off and S 4 is on

(2)

 1, S5 is on and S6 is off Sc    0, S5 is off and S6 is on

(3)

 Therefore, the switching function vector ( S ) of the bidirectional ac-dc converter can be expressed as: TABLE I VOLTAGE SPACE VECTORS OF THE BIDIRECTIONAL AC-DC CONVERTER Voltage Space Vector Switching States Sa

Sb

Sc

0

0

0

0

0

1

0

1

0

0

1

1

1

0

0

1

0

1

1

1

0

1

1

1

 vconv  v1  0  v 2   1 / 3 Vdc  j  v3   1 / 3 Vdc  j

     v 4  2 / 3Vdc  v5  2 / 3Vdc  

 

 v6  1 / 3 Vdc  j  v7  1 / 3 Vdc  j  v8  0


 3 / 3Vdc  3 / 3Vdc

 

 3 / 3Vdc

3 / 3 Vdc



S

2 3

2 ( S a  S b   S c )

(4)

1  j 2 / 3    j 3 / 2 is a unitary vector, which where,   e 2 represents the 120o phase displacement between the phases.  The output voltage space vector ( vconv ) of the bidirectional ac-dc converter for both the rectifier and inverter mode can be presented with phase to neutral voltages (vao, vbo and vco) as: v conv 

2 3

2 (vao  vbo   vco ) .

(5)

 The output voltage space vector ( vconv ) can also be related to  the dc bus voltage (Vdc) and the switching function vector ( S ) as:  (6) v conv  S  V . dc There are eight possible voltage vectors that can be obtained from the eight consequence switching states of the switching signals Sa, Sb and Sc. These eight voltage space vectors are listed in Table 1. The energy storage system allows bidirectional power transfer between three-phase ac voltage side and energy storage device through the bidirectional ac-dc converter. Hence, the bidirectional ac-dc converter needs to be operated in two modes, which are specified as rectifier mode and inverter mode. The operating principle of the bidirectional ac-dc converter for both the rectifier and inverter modes are elaborately described in the following subsection. 1) Rectifier Mode of Operation: During rectifier mode of operation, the bidirectional ac-dc converter acts as a front-end-rectifier that is connected to the three-phase ac voltage source through the input filter inductance Ls and resistance Rs as shown in Fig. 1. By applying Kirchhoff’s voltage law at the ac side of the rectifier, the relationship between the three-phase ac voltage and rectifier input voltage vectors are,  dis_rec  2 2   vs  Ls  Rs is_rec   vao  vbo   vco  dt 3  2

   vno  vno   vno  3 

(7)

2

 The space-vector model of three-phase ac voltage ( vs ) and  current ( is ) can be derived from phase voltage & current as: 2 2  vs  (vsa  vsb   vsc ) 3

(8)

and

is 

2 3

2 (isa  isb   isc ) ,

(9)

where vsa, vsb, and vsc are phase voltages; isa, isb, and isc are phase currents of three-phase ac voltage source. Note that the last term of (7) is equal to zero as: 2

2 2  2   vno  vno   vno   vno 1       0 3  3  

(10)

Therefore, the relationship between the three-phase ac voltage and rectifier input voltage vectors can be rewritten from (5), (7) and (10) as [27]:  dis_rec   (11) vs  Ls  Rs is_rec  v conv dt Hence, the input current dynamics of the bidirectional ac-dc converter during rectifier mode operation is:  R  dis_rec 1  1    s is_rec  vs  vconv . (12) L dt Ls Ls s 2) Inverter Mode of Operation: The bidirectional ac-dc converter works as a voltage source inverter during the inverter mode, which allows the power transfer from the dc voltage bus to the three-phase ac voltage end. Hence, the load current is 180o out-of-phase with respect to the load voltage. Therefore, the load current dynamics of the bidirectional ac-dc converter during inverter mode of operation can be presented as:  dis_inv R  1  1    s is_inv  vconv  vs . (13) L dt Ls Ls s III. CONVENTIONAL MPC METHOD The formulation of conventional model predictive control (MPC) algorithm for three-phase bidirectional ac-dc converter is described in the following section. The MPC controller is formulated in discrete time domain. Therefore, it is necessary to transform the dynamic system of bidirectional ac-dc converter for both rectifier and inverter mode of operation represented in (12) and (13) respectively, into discrete time model at a specific sampling time Ts. A. Discrete Time Model for Prediction Horizon A discrete time model is used to predict the future values of currents and voltages in the next sampling interval (k), from the measured currents and voltages at the (k-1)th sampling instant. The system model derivative dx / dt from Euler approximation can be expressed as: dx xk   xk  1  (14) dt T s Using one step advance of the above approximation, the discrete time model of predictive currents and voltages for the next (k+1) sampling instant of the bidirectional ac-dc converter in rectifier and inverter mode can be derived. The discrete time model of predictive input currents at the next sampling instant (k+1) for the rectifier mode of the



IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS bidirectional ac-dc converter can be evaluated from (12) with the help of Euler approximation as,  Ls is_rec k   



  (15) 1 is_rec k  1    R T  L T v k  1  v   k  1  s s s  s s conv_rec    

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

Ls is_inv k       (16) 1 is_inv k  1     R T  L T v  s s s  s  conv_inv k  1  vs k  1     

B. Cost Function The main objective of model predictive control algorithm is to minimize the error with fast dynamic response between the predicted and reference values of the discrete variables. To achieve this objective, an appropriate cost function (e) is defined with a measurement of predicted input error. Hence, the cost function for the rectifier and inverter can be expressed with the absolute error between the predictive and reference values of input and load current for both the rectifier and inverter mode of operation as below:    (17) e  iref ( k  1)  ip ( k  1) where, e is the cost function. The reference input and   predicted current 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 is first selected depending on charging state of the energy storage device, which is determined by the dc bus voltage (Vdc). If the charging state (determined by dc voltage) is less than threshold level, then it is operated in rectifier mode, otherwise it is operated in inverter mode. IV. PROPOSED MMPC TECHNIQUE BASED ON LYAPUNOV FUNCTION A. Determination of the Lyapunov-Function-Based MMPC Algorithm The proposed MMPC method directly applies the voltage vector constrained in the finite set. Hence, the future voltage  vector ( v (k  1) ) is expressed with the voltage vector  ( k  1) ) and the generated by the converter ( v conv unavoidable quantization error vectors as:   v (k  1)  v (k  1)   (k  1) (18) conv where,  (k  1) is the quantization error vector which satisfies

 (k  1)   with a constant   0 . The future voltage

 vector ( v (k  1) ) is bounded in finite set as mentioned in Table I and the quantization error  (k  1) vector ensures the  state where v ( k  1) is bounded. conv To determine the Lyapunov-function-based control algorithm for modification of conventional MPC method, it is important to analyze the bidirectional ac-dc converter system with control point of view. Therefore, the future-current error vector of the input current dynamics (15) during rectifier mode can be rewritten as:    k  1  is_rec (k  1)  iref_rec (k  1) i error_rec  . (19) Ls is_rec ( k )   1   i ( k  1 )   ref_rec R T  L  T v ( k  1)  v ( k  1)  s s s rec  s s





On the other hand, during the inverter mode of operation, the future current error vector can also be evaluated from (16) as:    k  1  is_inv (k  1)  iref_inv (k  1) i error_inv  (20) Ls is_inv (k )   1   i ( k  1 )   ref_inv R T  L  T v ( k  1)  v ( k  1)  s s s s  s inv





An effective control algorithm is essential for bidirectional ac dc converter so that the current ( i ) tracks the reference value s  ( i ) for both the rectifier and inverter mode of operation. ref Therefore, it is necessary to find a control function such that  the current tracking error ( i ) asymptotically converges to error zero. Lyapunov direct method is used for the specific application. The discrete Lyapunov function (L) is taken as: 1  T  Lk   [ierror ( k )] [ierror ( k )] . (21) 2 From (19) and (21) the rate of change of the Lyapunov function can be expressed for the rectifier mode as:   L ( k )  L i ( k  1)   L i ( k )  rec  error_rec   error_rec  T    v ( k  1)     s           1  R T  L  Ls is_rec ( k )  Ts   vconv_rec ( k  1)       s 1 s s     ( k  1)          2   i  ( k  1)  ref_rec     v ( k  1)     s           1 L i ( k )  T  v ( k  1 )     R T  L  s s_rec s conv_rec     s s s     ( k  1)           i  ( k  1)  ref_rec  

1  T  [i ( k )] [ierror_rec ( k )] 2 error_rec


(22)


IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS On the other hand, the rate of change of the Lyapunov function for the inverter mode of operation can be written from (20) and (21) as,   L ( k )  L i ( k  1)   L i ( k )  inv  error_inv   error_inv  T    v ( k  1)        conv_inv   1     R T  L  Ls is_inv ( k )  Ts      ( k  1)  v ( k  1)     s s s s   1       2  i  ( k  1)  ref_inv     v ( k  1)        conv_inv   1  L i ( k )  T     R T  L  s s_inv  s   ( k  1)  v ( k  1)     s s s  s       i  ( k  1)  ref_inv  

Bidirectional ac-dc Converter Three-Phase ac supply

S1

S3

S5 idc

Ls

Rs R

Cdc

Y B S2

Voltage Measurement

 is

Sa

(23)

1  T  [ierror_inv ( k )] [ierror_inv ( k )] 2

In order to make an effective control algorithm for  converging the tracking error ( i ) to zero; the rate of error change of the Lyapunov function ( L ) always needs to be negative. Therefore, the discrete voltage vector during the rectifier mode at next sampling instant   ( v ( k  1)  v ( k  1)   ( k  1) ), which assures the rec conv_rec rate of change of the Lyapunov function (22) is negative, which is written as: L    v ( k  1)  s is_rec ( k )  vs ( k  1) rec T s (24) RsTs  Ls   iref_rec ( k  1) Ts On the other hand, the discrete voltage vector during the inverter mode at next sampling instant   ( v (k  1)  v (k  1)   (k  1) ), which assures inv conv_inv the rate of change of the Lyapunov function (23) is negative, which is be written as: L    v ( k  1)   s is_inv ( k )  vs ( k  1) inv T s (25) RsTs  Ls   iref_inv ( k  1) Ts B. Implementation of the Lyapunov-Function-Based MMPC Algorithm for Bidirectional ac-dc Converter The conventional MPC is generally based on current oriented control techniques, in which the current tracks the reference value by utilizing the discrete behavior of the converter. In this conventional MPC method, the future current of the converter is calculated for each of eight possible switching states and the state that minimizes the cost function

EV Battery

S6

Current Measurement

 vs

Reference Load Current for Inverter Mode

Lyapunov  i ref_inv Function-

×



iref_rec Reference input Current Calculation for Rectifier Mode Vref_dc +

PI -

Vdc

Voltage Measurement

Sc

 vmeas Predictive Control Algorithm

Based Controller

Ø

Sb

Switching State Selection

 is

 vs

S4

Vdc

Vdc

 vref Operating Mode Selection

Vdc

Modified Model Predictive Control Desk

Fig. 2. Proposed MMPC control technique based on Lyapunov function.

is selected for firing the power switches. One of the major limitations of this conventional MPC algorithm is high execution time delay caused by the large number of calculation for the current prediction. In order to reduce the execution time, the conventional MPC algorithm is modified based on the Lyapunov function. The proposed modified model predictive control (MMPC) is based on voltage oriented control, in which the optimum possible future voltage vector  ( v ( k  1) ) of the converter is directly selected to track the opt  calculated reference voltage vector ( v ( k  1) ) of (24) and (25), for both the rectifier and inverter modes by utilizing the modified cost function as:   g ( k  1)  v ( k  1)  v ( k  1) . (26) opt Fig. 2 shows the proposed control strategy of the MMPC algorithm which is a one-step modification of the conventional FCS-MPC control scheme [28, 29]. The reference voltage  vector ( v ( k  1) ) of the converter is calculated from the  measured current ac current ( is ( k ) ) and the reference current  ( iref ( k ) ) by utilizing (24) and (25) for rectifier and inverter modes, respectively. The overall control technique for Lyapunov-function-based MMPC has been described with an execution time diagram and is presented in Fig. 3. The proposed Lyapunov-function-based MMPC method satisfies the following steps:  Step 1: Measurement of the ac currents ( i ( k ) ) and s  calculation of the reference current ( i ( k ) ). ref Step 2: Calculation of the future reference voltage vector  ( vref (k  1) ) from the measured current ac current




  ( is ( k ) ) and the reference current ( i ) by utilizing ref (24) and (25). Step 3: Estimation of the initial cost function value as (ginit(k+1)=1e10) from the calculation in the kth sampling period. Step 4: Prediction of the optimum possible future voltage  vector ( v ( k  1) ) from Table I. opt

k

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 cost function. Step 7: Application of the selected switching state of bidirectional ac-dc converter for firing the switches.

k+1

V. EXPERIMENTAL RESULTS A 2.5-kW downscaled laboratory prototype of the bidirectional ac-dc converter that is developed for the verification of the proposed Lyapunov-function-based MMPC algorithm. The schematic layout of the experimental system is presented in Fig. 4. The parameters presented in Table II, are used in the experimentation. The experimental verification of the proposed modified model predictive controlled (MMPC) bidirectional ac-dc converter is carried out by using the rapid prototyping and real-time interface system dSPACE with DS1104 control card which consist of Texas Instruments TMS320F240 sub-processor and the Power PC 603e/250 MHz main processor. This dSPACE control desk works together with Math-work MATLAB/Simulink-R2013a real-time workshop and real-time interface (RTI) control cards to implement the proposed MMPC algorithm. The voltage is measured with differential probe [PINTEK DP-25] and the current with current transducer [LEM LA 25-NP]. Fig. 5(a) shows that the ac phase voltage and current are exactly in phase during the rectifier operation mode, which ensures unity power factor. Again in the inverter mode, the converter allows power transfer with unity power factor from

Step 1:

Measurement of the ac Currents (is(k)) and Calculation of the Reference Current (iref(k))

Step 2:

Calculation of the Future Reference Voltage (vref(k+1)) from (24) & (25).

Step 3:

Estimation of the Initial Cost Function Value as (ginit(k+1)=1e10)

Step 4:

Prediction of the Possible Optimum Future Voltage Vector (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 Switching State Selection

Step 7:

Application of the Selected Switching State

k+2

Fig. 3. The execution time diagram of the proposed Lyapunov-functionbased MMPC. Three-Phase Portable Power Supply (vs) Filter Inductor (Ls)

3

 vs

Load AFE Rectifier

Capacitor (Cdc)

3

 is

S a Sb Sc

Vdc Host PC DS 1104

TMS320F240

TABLE II SIMULATION AND EXPERIMENTAL PARAMETERS Parameters values and unit Variables and Parameters

dSPACE Control Desk

Fig. 4. Experimental system of the bidirectional ac-dc converter with modified model predictive control.

Symbols

Values

Unit

Power rating

P

2.5

kW

Three-phase supply voltage

vs

110 (rms)

V

Grid frequency

fs

50

Hz

dc bus voltage

Vdc

320

V

Input filter inductance

Ls

5

mH

Input filter resistance

Rs

0.1

Ω

Sampling time

Ts

50

µS

Load resistance

RLoad

50

Ω

Capacitor value

C

1000

µF

the dc voltage bus to ac voltage end by keeping the phase voltage and current with 180o phase shift, as presented in Fig. 5(c). The dc bus voltage and current for both operating modes are depicted in Figs. 5(b) and 5(d). The results illustrate that the dc bus voltage ripple and the pulsation in dc current are very low during both operating modes. During rectifier mode, the dc bus reference voltage (Vdc_ref) is varied from 270 V to 320 V to check the stability and transient responsiveness of the MMPC control algorithm. Figs. 5(a) and 5(b) also show that when the reference voltage changes from 270 V to 320 V, the steady state of the ac waveforms is reached within a very short time (less than 10 ms). This rapid step change confirms the fast response of the



IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS vsa [50 V/div]

Current (isa) change due to change of Vdc_ref in rectifier mode

VI. STABILITY ANALYSIS

A. Stability Analysis with Direct Lyapunov Method The stability of the proposed MMPC technique for the bidirectional ac-dc converter is investigated with direct Lyapunov method. The direct Lyapunov method gives the following stability criteria for a continuous function  Li ( k ) , in which the solutions of current dynamics in error the rectifier mode (12) and inverter mode (13) are uniformly and ultimately bounded [30]: l    Li (k )  c i (k ) , ierror ( k )  G error 1 error

(a)

Voltage (Vdc1) [V] and current (Idc1) [A]

The stability issue of the proposed Lyapunov-functionbased MMPC algorithm is very important for the high performance of bidirectional ac-dc converter in energy storage system. This section shows the stability analysis of the proposed MMPC with direct Lyapunov method and nonlinear stability investigation. Moreover, the robustness of the proposed Lyapunov-function-based MMPC algorithm against parameter uncertainty has also been analyzed in this section.



(27)

n where, c1, c2, c3 and c4 are positive constants, l  1 , G  R is positive control invariant set and   G is a compact set.  By applying the value of future voltage vector ( v (k  1) ) of the converter for rectifier (24) and inverter (25) modes, the rate of change of the Lyapunov function can be written as: 1  1 T  2 2 (28) L(k )   [ierror (k )] [ierror (k )]  T / R T  L  s s s s 2 2 Therefore, the stability condition (27) is satisfied by the constant values as: 1 1 2 2 c  c  1; c  ; c  T / R T  L  (29) s s 1 2 3 2 4 2 s

 

 





As a result, all signals in the Lyapunov-function-based close loop modified model predictive controlled bidirectional ac-dc converter system are ultimately uniformly bounded. Then, the rate of change of the Lyapunov function in (28) is,

Idc1 = 6.76 A

Idc1 = 5.05 A

Step change of dc voltage (Vdc1) and current (Idc1) due to change in Vdc_ref during rectifier mode

vsa [50 V/div]

Voltage (vsa) [V] and current (isa) [A]

  l    L i (k )   c i (k ) , ierror ( k )   error 2 error l    L i ( k  1)   L i ( k )   c i (k )  c error error 3 error 4

Vdc1 = 320 V

Vdc1 = 270 V

(b) isa [5 A/div]

Current (isa) change due to reference current (iref) change in inverter mode (c) Vdc1 = 270 V




isa [20 A/div]

Voltage (vsa) [V] and current (isa) [A]

proposed Lyapunov-function-based MMPC method. The steady state output of dc-link voltage and current in Fig. 5(b) remain linear in wide range of time with very low voltage and current ripple, which ensure the stability and good performance of the proposed MMPC algorithm. Similarly, the stability and responsiveness of the MMPC method for bidirectional ac-dc converter in the inverter mode of operation are shown in Figs. 5(c) and 5(d), by varying the ac reference  current ( iref_inv ).

Idc1 = – 1.80 A Idc1 = – 5.45 A

dc current (Idc1) change due to reference current change in inverter mode

(d) Fig. 5. Experimental results: (a) ac phase voltage and current at rectifier mode, (b) dc terminal voltage and current at rectifier mode, (c) ac phase voltage and current at inverter mode, and (d) dc terminal voltage and current at inverter mode of operation.






(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 proposed MMPC controlled bidirectional ac-dc converter is performed with direct Lyapunov method in previous section by neglecting the nonlinear criteria such as: the unsymmetrical three-phase supply, existence of higher harmonics in converter input voltage, cross-coupling effect and time delay variation. Therefore, the nonlinear system stability of the Lyapunovfunction-based MMPC algorithm for bidirectional ac-dc is analyzed with laboratory experimental prototype. The same parameters as in Table II, are employed. During the rectifier mode of operation, the dc bus reference voltage (Vdc_ref) is varied from 270 V to 320 V to check the nonlinear stability of the proposed MMPC control algorithm. The three-phase ac current variation drawn by the bidirectional ac-dc converter in rectifier mode due to the dc bus reference voltage (Vdc_ref) variation is presented in Fig. 6(a). The result in Fig. 6(a) shows that, the output phase current is accurately tracking the reference value and reached its steady state condition with very fast dynamic response, which verifies the stability and effectiveness of the MMPC controller for ac-dc converter. Moreover, the stability of the proposed control algorithm can be further confirmed with the variation of the dc-link voltage and current as shown in Fig. 6(b). The dc-link voltage and the 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,  the ac load reference current ( iref_inv ) is varied from 5A to 10A periodically to check the stability of the proposed MMPC controller for bidirectional ac-dc converters. The result in Fig. 6(c) shows that, the output three-phase ac current is accurately tracking the reference value and reached its steady state condition with very fast dynamic response, which verifies the stability and effectiveness of the MMPC controller in case of output ac reference current variation. Again, the stability of the proposed control algorithm can be further confirmed with the variation of the dc-link voltage and current as shown in Fig. 6(d). The dc current varies with respect to the ac reference current variation as the dc voltage and load are constant. C. Robustness Analysis The control variables of nonlinear practical power converters are always affected by state measurement errors. Generally, these state measurement errors are caused by the filter parameter variation in practical power converters. Therefore, it is important that the closed-loop control

isb

isa

isc

(a) Vdc1 = 320 V

Vdc1 = 270 V Voltage (Vdc1) [V] and current (Idc1) [A]



  A i i  c /c error error 4 3

Idc1 = 6.76 A

Idc1 = 5.05 A

(b)

isb Three-Phase ac Current [A]

This inequality implies that, as time increases, the current control error vectors converge to the compact set as:

Three-Phase ac Current [A]

(30)

isc

isa

(b) Voltage (Vdc1) [V] and current (Idc1) [A]

 L( k )  2c L i ( k )   c . 3  error_inv  4

Vdc1 = 270 V

Idc1 = – 1.80 A Idc1 = – 5.45 A

(d)

Fig. 6. The stability investigation via experimental system: (a) Three-Phase ac Current (isa) in rectifier mode; (b) dc Bus Voltage (Vdc) vs Current (Idc) with dc-link reference voltage (Vdc_ref) change in rectifier mode; (c) ThreePhase ac Current (isa) in inverter mode; and (d) dc Bus Voltage (Vdc) vs Current (Idc) with ac reference current (iref_inv) change in inverter mode.




( k  1)  x (k  1) leads to Substituting (35) into x unc ref

Reference current (isa_ref) [A] and measured current (isa) [A]

5 0 -5

Rectifier Mode

-15 1

Inverter Mode (a)

Current error (isa_error) [A] with Rs=0.1Ω and Ls=5 mH

0.8 0.6 0.4 0.2 0

-0.2 -0.4 -0.6 -0.8

Rectifier Mode

Inverter Mode

-1

(b)

0.6 0.4 0.2 0

-0.2 -0.4

Rectifier Mode

Inverter Mode (c)

0.6 0.4 0.2 0

-0.2 -0.4

Rectifier Mode

Inverter Mode (d)

0.4 0.2 0

-0.2

(37)

The deviation from the reference at (k+1) is thus bounded by the sum of two terms—the nominal response and an uncertainty term, which is a function of the state vector and the input. On the uncertainty term the upper bound can be presented as follows: ~ ~ (38) Ay (k )  Bu (k  1)  p y (k )  p 1 2 To ensure (robust) convergence in the presence of additional parameter uncertainties, the right hand side of (37) has to be strictly less than x ( k )  x ( k ) , i.e. ref

x(k  1)  x (k  1)  p y (k )  p  x(k )  x (k ) (39) ref ref 1 2 This is equivalent to g (k  1)  p y (k )  p  g (k ) (40) 1 2 To ensure the robustness of the controller, when the controlled variable is outside of the bound, the constraint on

Rectifier Mode

-0.4

Inverter Mode (e)

0.4 0.2 0

-0.2

Rectifier Mode

-0.4

Inverter Mode (f)

9 8

Percentage Current error (isa_error) [%]

x ( k  1)  x ( k  1)  x ( k  1)  x ( k  1) unc ref ref ~ ~  Ay ( k )  Bu (k  1)

10

Current error (isa_error) [A] with Rs=0.15Ω and Ls=7 mH

g ( k  1)  x( k  1)  x ( k  1) (33) ref and (34) x(k  1)  X (k  1) or g (k  1)  g (k ) v     where, x  v , y  is_rec , u  iref_rec , A  Ls / Ts , rec  R T  Ls   , D  vs and X is the voltage set region. B s s v Ts The robustness of the proposed Lyapunov-function-based MMPC algorithm for bidirectional ac-dc converter against parameter uncertainties has been analyzed by considering the ~ addition of uncertain values of the filter resistance ( R ) and ~ inductance ( L ). These uncertain values lead to the parameter uncertainties of the model as: ~ ~ ~ ~ Ls ~ R Ts  Ls s A , B . (35) T Ts s Therefore, the control state equation (32) of the proposed MMPC algorithm can be written with parameter uncertainties as:  ~  ~  (36) x ( k  1)  ( A  A) y ( k )  ( B  B )u ( k  1)  D unc

15

-10

Current error (isa_error) [A] with Current error (isa_error) [A] with Current error (isa_error) [A] with Rs=0.30Ω and Ls=15 mH Rs=0.25Ω and Ls=12 mH Rs=0.20Ω and Ls=10 mH

techniques of nonlinear power converters should be robust with parameter variation [31-33]. In order to analyze the robustness of the proposed Lyapunov-function-based MMPC algorithm, the discrete voltage vector (24) and cost function (26) can be simplified with control point of view as [32]:    (32) x(k  1)  Ay (k )  Bu (k  1)  D

7 6

R e c tifie r M o d e

5

In ve r te r M o d e

4 3

2 1 0

Rs=0.1Ω Ls=5 mH

Rs=0.15Ω Ls=7 mH

Rs=0.20Ω Ls=10 mH (g)

Rs=0.25Ω Ls=12 mH

Rs=0.30Ω Ls=15 mH

Fig. 7. Robustness analysis of proposed MMPC algorithm against parameter variation: (a) reference current (isa_ref) and measured current (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; and (g) percentage current error.



IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS the right hand side in (34) is to be replaced by (40). This ensures that only voltage vectors are selected that point with a certain minimum tracking towards the current reference. Accordingly, when the control variable is within its bound, the constraint at left hand side of (34) needs to be modified. Specifically, the constraint at k+1 is replaced by ~ ~ x(k  1)  X , where, X uses the radius ( ~ ) which can be v v presented as: (41) ~    p y (k )  p v 1 2 ~ As a result, X is a subset of X . This ensures the robustness v v of the proposed MMPC algorithm against parameter uncertainties once the controlled variables are within the bounds. Similarly, the robustness of the proposed MMPC algorithm for bidirectional ac-dc converter with parameter uncertainties can also be analyzed during the inverter mode operation. The theoretical robustness analysis of the proposed MMPC method with additive uncertain values of the filter resistance ~ ~ ( R ) and inductance ( L ) in (35) has been verified with nonlinear MATLAB/Simulink model and visualized in Fig. 7,    which shows the variation of current error ( i  i i ) error s ref against the variation of filter parameter. The value of filter ~ resistance ( R ) and inductance (Ls) in (35) has been varied up to 200% (nominal filter resistance, Rs=0.1Ω and inductance, Ls=5mH) to verify the robustness against parameter uncertainties. The results in Fig. 7 confirm that the variation of current error is bounded within 1 A (less than 10%) for different values of filter parameter, which ensures the robustness of the proposed control algorithm. Moreover, the percentage  deviation of current error ( i ) with different values of error filter parameters for both the rectifier and inverter mode of operation are compared, and presented in Fig. 7(g), which shows that the maximum deviation of current tracking error of about 8% is negligible. Although, the current tracking error is decreasing with the increase of filter inductor value, but the power loss increases with the increase of inductor. The main focus of this paper is to reduce the execution time with MMPC and this section shows the robustness with parameter variation. Hence, the filter value selection and further discussion with power losses has been omitted. The percentage of current error   is calculated from the reference ( i ) and measured ( i ) ref s current as:   i i  i  s  ref  100% (42) error i ref Therefore, it is possible to demonstrate that the proposed control strategy is robust and can operate efficiently even under filter parameter variations.

VII. COMPARATIVE EVALUATION OF PROPOSED MMPC WITH CONVENTIONAL MPC IN TERMS OF EXECUTION TIME One of the major advantages of the proposed Lyapunovfunction-based MMPC algorithm is its very low execution time. The proposed MMPC algorithm reduces the amount of calculations required to predict a future variable by half compared with the conventional MPC method, which is elaborately presented in Fig 8. In the experimental system, the execution time required in completing the algorithms of the conventional MPC and the proposed Lyapunov-function-based MMPC techniques are calculated by measuring the calculation cycles of the DS1104 control card of the dSPACE real time prototype. The total execution times of the conventional MPC method are 3.95 µs and 3.8 µs for the rectifier and inverter modes, respectively. Otherwise the total execution times of the proposed Lyapunovfunction-based MMPC method are about 3.24 µs during the rectifier mode and 3.12 µs during the inverter mode. The execution time of rectifier mode is higher than the inverter mode due the usage of PI controller. Hence, the total execution time is reduced up to 17.89% with the proposed MMPC method. Fig. 9(a) presents the experimental results, which show the step change of the high dc bus voltage and current with step change of the dc bus reference voltage (Vdc_ref). The high dc bus voltage and current reached its steady state level within 3.24 µs which confirms the low execution time of the proposed Lyapunov-function-based MMPC technique during the rectifier mode of operation. On the other hand, Fig. 9(b) shows the experimental wave shapes of the three-phase ac currents and the cost function of the proposed MMPC method during the inverter mode of operation. The three-phase ac current controlled by the proposed MMPC technique follow the references accurately with very fast dynamic response. Moreover, the Lyapunov-function-based cost function Start

Start

Sampling of Measured Currents (iss(k+1)) and Calculation of Reference Current (is_ref s_ref(k+1))

Sampling of Measured Currents (iss) and Calculation of Reference Current (is_ref s_ref)

Initialization of Cost Function (e) for i=1:8,

Calculation of Future Reference Voltage (Vconv conv(k+1)) from (22) & (23) Initialization of Cost Function (g)

Prediction of Future Voltage Vector [V11(0,0,0); V22(0,0,1); V33(0,1,0); V44(0,1,1); V55(1,0,0); V66(1,0,1); V77(1,1,0); V88(1,1,1)]

for i=1:8,

Calculation of Future Current (is(k+1)) [is1(V1); is2(V2); is3(V3); is4(V4); is5(V5); is6(V6); is7(V7); is8(V8)]

Prediction of Future Voltage Vector [V11(0,0,0); V22(0,0,1); V33(0,1,0); V44(0,1,1); V55(1,0,0); V66(1,0,1); V77(1,1,0); V88(1,1,1)]

Calculation of Cost Function (e)

Calculation of Cost Function (g)

i=8

No

Yes

i=8

No

Yes

Minimization of Cost Function (e) and Switching State Selection

Minimization of Cost Function (e) and Switching State Selection

Apply Selected Switching State (a)

Apply Selected Switching State (b)

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



IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS TABLE III COMPARISON BETWEEN CONVENTIONAL MPC AND PROPOSED LYAPUNOVFUNCTION-BASED MMPC


Vdc1[100 v/div]

Item description

Idc1[5 A/div]

Control mode Switching frequency

Execution time (3.24 µs)

Execution time Stability

Current (isa) [A], current (isb) [A], current (isc) [A] and cost function (g)

(a) Current (isc) Current (isa)

Current (isb)

Cost function (g)

Execution time (3.12 µs)

(b)

Fig. 9. Experimental results with MMPC scheme; (a) dc bus voltage and current at the rectifier mode and (b) three-phase ac current and cost function at the inverter mode.

presented in Fig. 9(b) can be referred to the ac current error which is bounded for its transient stability conditions with quantization error resulting from the finite number of possible voltage vectors of the bidirectional ac-dc converter. A comparative evaluation between conventional MPC and proposed Lyapunov-function-based MMPC has been performed and summarized in Table III. The proposed MMPC algorithm operates as a voltage mode control technique which reduces the execution time by minimizing the amount of calculation compared to conventional current mode MPC algorithm. Moreover, the average switching frequency of the MMPC algorithm is also less than the conventional MPC method. Finally, the proposed MMPC ensures the system stability with direct Lyapunov stability analysis method which is impossible for conventional MPC. VIII. CONCLUSION In this paper, a Lyapunov-function-based modified model predictive control (MMPC) algorithm is proposed to control the bidirectional ac-dc converter which is applied in energy storage system to transfer power between three-phase ac voltage source and dc voltage bus. Lyapunov-function-based MMPC is a powerful control algorithm in the field of bidirectional ac-dc power converters which provides bidirectional power flow with instantaneous mode changing capability, fast dynamic response and nonlinear system stability. The stability analysis of this control method is performed with direct Lyapunov method and nonlinear model analysis. The result confirms that, the modified model predictive control (MMPC) system is stable for the operation

Lyapunov-function-based MMPC

Conventional MPC

Current mode control

Voltage mode control

Variable switching frequency 3.95 µs and 3.8µs for rectifier and inverter modes, respectively Impossible to analyze the system stability with direct Lyapunov method

Average switching frequency is less than MPC 3.24 µs and 3.12µs for rectifier and inverter modes, respectively Ensure the system stability with direct Lyapunov method

of the bidirectional ac-dc converter. Moreover, the most important advantage of the proposed Lyapunov-function-based MMPC algorithm is that it has very low execution time. The total execution time of the proposed MMPC algorithm is about 18% lower than the conventional MPC algorithm. The results associated in this investigation are very encouraging and will continue to play a strategic role in the improvement of modern high performance bidirectional ac-dc converter. ACKNOWLEDGMENT The authors would like to thank the Ministry of Higher Education and University of Malaya for providing financial support through HIR-MOHE project UM.C/HIR/MOHE/ ENG/24 and UMRG project No.RP006E-13ICT. REFERENCES [1] J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz, E. Galván, R. P. Guisado, M. A. Prats, et al., "Power-electronic systems for the grid integration of renewable energy sources: A survey," IEEE Trans. Ind. Electron., vol. 53, pp. 1002-1016, 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, pp. 3881-3895, 2010. [3] N. M. L. Tan, T. Abe, and H. Akagi, "Design and performance of a bidirectional Isolated DC–DC converter for a battery energy storage system," IEEE Trans. Power Electron., vol. 27, pp. 1237-1248, 2012. [4] J. R. Rodríguez, L. Dixon, J. R. Espinoza, J. Pontt, and P. Lezana, "PWM regenerative rectifiers: state of the art," IEEE Trans. Ind. Electron., vol. 52, pp. 5-22, 2005. [5] P. Akter, M. Uddin, S. Mekhilef, N. M. L. Tan, and H. Akagi, "Model predictive control of bidirectional isolated DC–DC converter for energy conversion system," International Journal of Electronics, pp. 1-21, 2015. [6] B. Singh, B. N. Singh, A. Chandra, K. Al-Haddad, A. Pandey, and D. P. Kothari, "A review of three-phase improved power quality AC-DC converters," IEEE Trans. Ind. Electron., vol. 51, pp. 641-660, 2004. [7] J. Dannehl, C. Wessels, and F. W. Fuchs, "Limitations of VoltageOriented PI Current Control of Grid-Connected PWM Rectifiers With LCL Filters," IEEE Trans. Ind. Electron., vol. 56, 2009. [8] M. Malinowski, M. P. Kazmierkowski, and A. M. Trzynadlowski, "A comparative study of control techniques for PWM rectifiers in AC adjustable speed drives," IEEE Trans. Power Electron., vol. 18, pp. 1390-1396, 2003. [9] D. Zhi, L. Xu, and B. W. Williams, "Improved direct power control of grid-connected DC/AC converters," IEEE Trans. Power Electron., vol. 24, pp. 1280-1292, 2009. [10] A. Bouafia, F. Krim, and J.-P. Gaubert, "Fuzzy-logic-based switching state selection for direct power control of three-phase PWM rectifier," IEEE Trans. Ind. Electron., vol. 56, pp. 1984-1992, 2009. [11] J. Hu, L. Shang, Y. He, and Z. Zhu, "Direct active and reactive power regulation of grid-connected DC/AC converters using sliding mode



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[32] T. Geyer, R. P. Aguilera, and D. E. Quevedo, "On the stability and robustness of model predictive direct current control," in Proc. IEEE Int. Conf. Industrial Technology (ICIT), 2013, 2013, pp. 374-379. [33] M. Rivera, V. Yaramasu, J. Rodriguez, and B. Wu, "Model predictive current control of two-level four-leg inverters—Part II: experimental implementation and validation," IEEE Trans. Power Electron., vol. 28, pp. 3469-3478, 2013. Md. Parvez Akter was born in Pabna, Bangladesh. He received the B.Sc.Engg. degree from Chittagong University of Engineering and Technology (CUET), Chittagong, Bangladesh, in 2011, and the Master of Engineering Science (M.Eng.Sc) degree from the University of Malaya, Kuala Lumpur, Malaysia, in 2015. Currently he is working as a research assistant with the Power Electronics and Renewable Energy Research Laboratory (PEARL), Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia. His research interest is on power converters and electrical drives, bidirectional power conversion techniques, predictive and digital control, renewable energy, smart grid, and wireless power transfer. Saad Mekhilef (SM’12) received the B.Eng. degree in Electrical Engineering from the University of Setif, Setif, Algeria, in 1995, and the Master of Engineering Science and Ph.D. degrees from the University of Malaya, Kuala Lumpur, Malaysia, in 1998 and 2003, respectively. He is currently a Professor at the Department of Electrical Engineering, University of Malaya, Kuala Lumpur. He is the author or coauthor of more than 250 publications in international journals and proceedings. He is a Senior Member of the IEEE. He is actively involved in industrial consultancy, for major corporations in the power electronics projects. His research interests include power conversion techniques, control of power converters, renewable energy, wireless power transfer, and energy efficiency. Nadia Mei Lin Tan (M’10) was born in Kuala Lumpur, Malaysia. She received the B.Eng. (Hons.) degree from the University of Sheffield, Sheffield, U.K., in 2002, the M. Eng. degree from Universiti Tenaga Nasional, Kajang, Malaysia, in 2007, and the Ph.D. degree from Tokyo Institute of Technology, Tokyo, Japan, in 2010, all in electrical engineering. Since October 2010, she has been a Senior Lecturer in the Department of Electrical Power Engineering, Universiti Tenaga Nasional. Her current research interests include power conversion systems and bidirectional isolated dc–dc converters. Dr. Tan is a Graduate Member of the Institution of Engineers Malaysia (IEM) and a Member of the Institution of Engineering and Technology (IET). Hirofumi Akagi (F’96) was born in Okayama, Japan, in 1951. He received the Ph. D. degree in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1979. Since 2000, he has been Professor in the department of electrical and electronic engineering at the Tokyo Institute of Technology. Prior to it, he was with Nagaoka University of Technology, Nagaoka, Japan, and Okayama University, Okayama, Japan. His research interests include power conversion systems and its applications to industry, transportation, and utility. He has authored and coauthored some 120 IEEE Transactions papers. Dr. Akagi has received six IEEE Transactions Prize Paper Awards and 14 IEEE Industry Applications Society (IAS) Committee Prize Paper Awards. He is the recipient of the 2001 IEEE William E. Newell Power Electronics Award, the 2004 IEEE IAS Outstanding Achievement Award, the 2008 IEEE Richard H. Kaufmann Technical Field Award, the 2012 IEEE PES Nari Hingorani Custom Power Award, and the 2014 EPE Outstanding Service Award. Dr. Akagi served as the President of the IEEE Power Electronics Society during 2007- 2008. Since 2015, he has been serving as the IEEE Division II Director.
