This is the author’s version of a work that was submitted/accepted for pub-lication in the following source:

Neath, M.J., Swain, A.K., Madawala, U.K., Thrimawithana, D.J., & Vi-lathgamuwa, D.M.(2012)Controller synthesis of a bidirectional inductive power interface for electricvehicles. InProceedings of IEEE Third International Conference on Sustainable En-ergy Technologies (ICSET), IEEE, Soltee Crowne Plaza, Kathmandu, pp.60-65.

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https://doi.org/10.1109/ICSET.2012.6357376

Abstract—Bidirectional Inductive Power Transfer (IPT) systems

are preferred for Vehicle-to-Grid (V2G) applications. Typically,

bidirectional IPT systems are high order resonant circuits, and

therefore, the control of bidirectional IPT systems has always been

a difficulty. To date several different controllers have been

reported, but these have been designed using steady-state models,

which invariably, are incapable of providing an accurate insight

into the dynamic behaviour of the system. A dynamic state-space

model of a bidirectional IPT system has been reported. However,

to date this model has not been used to optimise the design of

controllers. Therefore, this paper proposes an optimised

controller based on the dynamic model. To verify the operation of

the proposed controller simulated results of the optimised

controller and simulated results of another controller are

compared. Results indicate that the proposed controller is capable

of accurately and stably controlling the power flow in a

bidirectional IPT system.

I. INTRODUCTION

Inductive Power Transfer (IPT) systems are now recognised

as efficient and effective methods for transferring power

wirelessly. IPT systems are widely used in a number of

different areas ranging from low power biomedical implants to

high power material handling systems [1–5]. IPT technology

has gained popularity in these areas due to the high efficiencies

possible, typically 85 – 90 %, and its ability to operate in hostile

environments, being unaffected by dirt and moisture [6]. Due

to these benefits, the current focus of much of the research on

IPT technology is on developing IPT systems that can be used

to wirelessly charge Electric Vehicles (EVs) [3], [7]. More

recently, several bidirectional IPT systems have been proposed

and developed for applications in which power needs to be

transferred wirelessly in either direction, such as a wireless

Vehicle-to-Grid (V2G) systems [8–10].

The power handling capability of IPT systems is typically

improved through either series or parallel compensation. As a

consequence, these systems become high order resonant

networks, which are complex in nature and thus, difficult to

design and analyse. Especially, when considering the high

operating frequencies of IPT systems, which typically are in the

rage of 10 – 50 kHz [11–14]. Numerous unidirectional IPT

systems have been designed and analysed using relatively

simple steady-state models [9], [15], [16]. However, steady-

state models are incapable of providing an accurate insight into

the dynamic behaviour of the system and, as such, cannot be

regarded as a tool that facilitates proper controller synthesis,

without which the control of the system cannot be optimised.

Furthermore, the control of unidirectional IPT systems is

inherently simpler than the control of bidirectional IPT systems.

However, a dynamic model of a bidirectional IPT system has

been described, but to date this model has only been used in the

design of relatively simple controllers without much

consideration for the overall optimisation of the system [17].

To address this need, this paper proposes a decentralised

Proportional Integral Derivative (PID) controller that controls

the power flow in a bidirectional IPT system. The controller is

designed such that the closed loop transfer function of the

system and the controller match a reference transfer function.

The reference transfer function is obtained by determining the

required rise time, settling time and percentage overshoot.

Once the reference transfer function has been found, the steady-

state model is used to determine the transfer function of the

controller, thus giving the desired closed loop response. To

verify the performance of the controller, the response of the

optimised controller was compared to the response of a

controller designed using the Ziegler-Nichols tuning method.

Finally, the overall response and the applicability of the

controller is discussed.

This paper is organised as follows. Section III, briefly

describes the operation of a bidirectional IPT system. The

design of the optimised controller is presented in Section IV,

along with a brief description of the dynamic model. In Section

V, the response of the system is analysed and compared with

the response obtained from another controller and finally

Section VI presents the conclusions.

II. BIDIRECTIONAL IPT SYSTEMS

A typical bidirectional IPT system is shown in Fig. 1, which

M. J. Neath*, A. K. Swain*, U. K. Madawala*, D. J. Thrimawithana* and D. M. Vilathgamuwa+

*The Department of Electrical and Computer Engineering, The University of Auckland, New Zealand +Division of Power Engineering, School of Electrical & Electronic Engineering, Nanyang Technological University

Email: mnea014@aucklanduni.ac.nz, a.swain@auckland.ac.nz, u.madawala@auckland.ac.nz, duleepajt@gmail.com and

emahinda@ntu.edu.sg

Controller Synthesis of a Bidirectional

Inductive Power Interface for Electric Vehicles

comprises a primary and a pick-up converter, resonant network

and controller. In a wireless V2G system, such as the one

described in [10], the primary converter is connected to a dc bus

fed from the grid by an active rectifier, which is schematically

represented as a dc source, to either supply or absorb power

from the pick-up. The output of the pick-up side converter is

connected to the EVs batteries, which again is schematically

represented as a dc supply, to either absorb or deliver power.

As analogous with unidirectional IPT systems, the primary

power converter is controlled to produce a constant high

frequency sinusoidal current in the track winding . A

pick-up coil is placed in the magnetic field produced by the

primary track current, and by using virtually identical

electronics as on the primary side of the system the pick-up can

deliver or source power to/from the primary converter.

When power is flowing in the forward direction, from the

primary to the pick-up, the primary converter operates as an

inverter and the pick-up operates as a controlled rectifier.

Conversely, in the reverse direction, the pick-up converter

operates as an inverter and the primary converter operates as a

controlled rectifier. Details on controlling the direction and

amount of power flow are given below.

Assume that the primary side converter of the bidirectional

IPT system, shown in Fig. 1, produces a reference sinusoidal

voltage at an angular frequency of , and the track

current, is held constant by the primary controller.

At steady state, the voltage induced in the pick-up coil

due to the track current can be given by

(1)

where M represents the magnetic coupling between the

primary and the pick-up coil inductances.

The pick-up may be operated either as a source or a sink by

the controller and, despite the mode of operation, the voltage

reflected back on to the track can be expressed by

(2)

If the inductor-capacitor-inductor (LCL) circuits on both the

primary and pick-up sides are tuned to the angular frequency ω,

and and then

(3)

Under these conditions, it can be shown that the input

current of the pick-up can be given by

(4)

Solving for using (4) and (1)

(5)

If the equivalent ac voltage of the voltage applied to the

input of the pick-up side resonant network is given by ,

then the output power of the pick-up can be given by

M

Pick-upcontroller

switching signals

Lsi

Lst

vsr Cst~

isi ist

Isin

vstvsi Vsin

switching signals

Lpi

Lpt

vprCpt ~

ipi ipt

Ipin

vptvpiVpin

Primarycontroller

Primary Pick-up

Sp1+

Sp1-

Sp2+

Sp2-

Ss2+

Ss2- Ss1-

Ss1+

Fig. 1 A typical bidirectional IPT system

| |

| | ( )

(6)

It is evident from (6) that the maximum power transfer takes

place when the phase angle θ is ± 90 degrees. A leading phase

angle constitutes power transfer from the pick-up to the

primary, while a lagging phase angle enables power transfer

from the primary to the pick-up. Thus, for any given primary

and pick-up voltages, both the amount and direction of power

flow between the primary and pick-up can be regulated by

controlling both the magnitude and relative phase angle of the

voltages generated by the converters [9].

A. Controller

A simplified diagram showing the operation of the pick-up

side controller is shown in Fig. 2. The input voltage and current

from the pick-up side converter are sampled and converted into

digital signals by the ADC. These signals are multiplied in the

micro-controller to give the power output of the converter. The

error between the output power and the reference power is

calculated, and this error is fed into a controller, typically a PI

or PID. The reference power level is set by the user through a

communication protocol. The output of the controller , is

used to adjust the duty cycle of the pick-up converter. The duty

cycle of the converter varies between ± 90 degrees, where – 90

degrees corresponds to the maximum power in the forward

direction and conversely, + 90 degrees corresponds to the

maximum power in the reverse direction. The output of the

controller is divided in half and each leg of the converter is

shifted by half of the duty cycle period. The left hand leg Ss1+ is

delayed/shifted to the right by αs/2, and the right hand leg Ss1+ is

advanced/shifted to the left, by αs/2. Each leg of the inverter

needs to be shifted like this to ensure that the fundamental

frequency of the voltage has a constant phase angle when

compared with .

Timer Count up

Mode synchronized

with primary

ControllerADC

Vsin

isin

Psin

Compare

Toggle

on match

Power

Reference

Communication

Protocol

Out Ss1+

Out Ss2+

To pick-up

converter

Pick-up

Output V

and I

User

input

αs

1/2Compare

Toggle

on match

αs/2+

+

+

-

Compare

value

Phase shifter

αs/2

Fig. 2 Simplified pick-up control strategy for a bidirectional IPT system

III. CONTROLLER DESIGN

A dynamic state-space model of a bidirectional IPT system

was derived and experimentally verified [17]. The model can

be expressed in the state-space form as

(7)

(8)

where A is the system matrix, B is the input matrix, C is the

output matrix, x is the state matrix, y is the output vector and u

is the input vector.

The inputs to the Multi-Input Multi-Output (MIMO) state-

space system model are the output voltages of the primary

converter, and pick-up converter, . The outputs of the

model are the primary track current, and the pick-up input

current, . Therefore the input and output vectors can be given by

[ ]

(9)

[ ] (10)

The Relative Gain Array (RGA) is a heuristic method to

predict the degree of coupling or interaction between the inputs

and outputs in a MIMO system [18]. The RGA matrix of the

bidirectional IPT system was calculated at 20 kHz and is given

by

[

] (11)

The RGA elements λ11 and λ22 at 20 kHz are close to 1. This

indicates that there exists a strong interaction between and

and between and . Therefore, the input current to the

pick-up side converter, can easily be controlled by . Similarly, the output should be controlled or paired with the

input . Since the other RGA elements λ12 and λ21 are

negative, this implies that the inputs and should not be

paired or controlled with and respectively. From the

RGA analysis it can be observed that a controller can be

designed using a decentralised approach to achieve the desired

performance. Further details about interpreting the RGA

elements can be found in [18].

A. Ziegler-Nichols PI Tuning Method

Initially, a PI controller was designed using the Ziegler-

Nichols method, which requires knowledge of the ultimate gain

and ultimate frequency of the system [19]. From the Bode plot

of the input and the output the gain and ultimate

period can be determined. From these values and the

Ziegler-Nichols tuning rules, the proportional gain and the

integral time can be calculated by

(12)

For the bidirectional IPT system considered here the values

of the gain parameters become and .

However, this gives unsatisfactory results, as the rules proposed

by Ziegler-Nichols only give a rough estimate of the controller

gains. The proportional gain was therefore decreased by 1.5

times to give and .

B. Optimised PID Controller Design

Results from the Ziegler-Nichols tuned controller indicate

that the controller is not optimally designed, as there are large

overshoots and oscillations around the desired output power.

Therefore, a more analytical method of determining the

controller gains needs to be used to improve the response of the

system.

The optimised controller is designed by matching the closed

loop response of the system and controller with a reference

transfer function. This reference transfer function can be

determined from the desired response of the system. A PID

controller was selected to control the power in the bidirectional

IPT system and the transfer function of a PID controller can be

given by

( ) [

] (13)

where kp is the proportional gain, Ti is the integral time and

Td is the derivative time.

The closed loop transfer function can be determined by

( ) ( ) ( )

(14)

where Gc(s) is the transfer function of the system relating the

pick-up’s input voltage to the input pick-up current, which can

be determined analytically from the dynamic state-space model.

An analytical method was chosen to determine the PID gains

based on the response of the system from the dynamic state-

space model. The PID tuning method has a target phase margin

of 60 degrees and automatically determines the PID gains to

ensure the best performance between response time and

stability margins. In designing the controller for a bidirectional

IPT system, the robustness is important as overshoots and

oscillations can cause instability in both sides of the IPT

system. This can be seen in the results for the Ziegler-Nichols

tuned PI controller. For the controller proposed in this paper a

response time of 700 μs was chosen and this gives the following

PID parameters kp = 0.0182, Ti = 27.17 μs and Td = 1.352 μs.

Results of the Ziegler-Nichols tuned PI controller and optimised

PID controller are given in section V.

IV. RESULTS

In order to determine the performance of the optimised

controller, the step response of the optimised controller was

compared with the step response of the original Ziegler-Nichols

PI controller. The two controllers were compared using

simulated results from PLECS a simulation package for

MATLAB. The parameters of both the simulated systems are

given in Table I. In each control technique, the phase shift

between and was set to 90 degrees and the duty cycle

was adjusted to control the power flowing between the two

sides of the system.

Fig. 3 shows the operating waveforms of the bidirectional

IPT system using the non-optimised PI controller, with a step

change in the reference output power occurring at 0 μs. At this

time the reference power level is changed from 0 to 1.5 kW in

the forward direction (from primary to pick-up). The primary

converter is controlled to produce a constant output waveform,

which produces a nearly constant sinusoidal current in the

track . The output of the pick-up side converter is

controlled to produce a varying duty cycle waveform that in

turn regulates the power flowing from the primary to the pick-

up. As evident from the controller output, and the output

power, there are significant oscillations and overshoots, and

these correspond to the oscillations seen in the pick-up’s input

current and primary track current. Oscillations in the primary

converter are not ideal for IPT systems especially when

multiple pick-ups are connected to the primary, and any change

in the primary will affect all the pick-ups.

Fig. 3 Primary track current, pick-up input current, pick-up controller output

and output power of the pick-up converter. With a + 1.5 kW step change in

output power results from the non-optimised controller.

The step response of the system with the same conditions as

in Fig. 3 is shown in Fig. 4; however, in Fig. 4 the system is

controlled using the optimised controller. Again, the primary

converter produces a constant sinusoidal track current and the

pick-up’s input voltage is controlled to regulate the power flow.

In this case, it is evident that only very small oscillations in the

power flow exist. This is further shown in the track current and

pick-up side input current being much steadier, without the

large spikes and oscillations.

Fig. 4 Primary track current, pick-up input current, pick-up controller output

and output power of the pick-up converter. With a + 1.5 kW step change in

output power results from the optimised controller.

Fig. 5 Primary track current, pick-up input current, pick-up controller output

and output power of the pick-up converter. With a - 1.5 kW step change in

output power results from the non-optimised controller.

In a bidirectional IPT system, the response of the system to

power flow in the reverse direction (from pick-up to primary) is

also important. Fig. 5 and 6 show the response of the system to

a step response from zero to 1.5 kW in the reverse direction,

with the non-optimised and optimised controller respectively.

From the plots, it can be seen that the non-optimised controller

produces large overshoots and oscillations when compared to

the optimised controller. It is also evident that the system

behaves differently in the forward and reverse directions;

therefore, any controller optimisation needs to consider the

operation in both the forward and reverse directions.

Fig. 6 Primary track current, pick-up input current, pick-up controller output

and output power of the pick-up converter. With a - 1.5 kW step change in

output power results from the optimised controller.

The output power and controller duty cycle are shown in Fig.

7, for 4 different power levels +2 kW, +1 kW, –1 kW and – 2

kW, from the non-optimised controller. As can be seen there

are large overshoots and oscillations in the power flow.

Especially with the power reference set to +2 kW, where the

system after 5 ms starts to become unstable. Conversely, Fig. 8

shows the operation of the optimised controller at the same

power levels as in Fig. 7. As can be seen there are no large

overshoots or oscillations and that the system does not become

unstable. It is also evident from both of the plots that the

bidirectional IPT system behaves differently depending on the

directional and magnitude of power transferred.

Fig. 7 Pick-up output power and controller output for various power

references results obtained from the non-optimised controller

Fig. 8 Pick-up output power and controller output for various power

references results obtained from the optimised controller

TABLE I

Parameters of the Prototype Bidirectional IPT System

Parameter Value

V. CONCLUSIONS

A control has been designed, which enables the power flow

in a bidirectional IPT system to be optimally controlled. A

dynamic state-space model of a bidirectional IPT system has

previously been developed and analysed. From this analysis,

the RGA matrix of the system could be determined. The RGA

matrix indicates that a decentralised controller can be used to

control the power flow. Therefore, a decentralised PID

controller was designed and optimised using the dynamic state-

space model. Simulated results of the controller’s operation are

compared with simulated results of a PI controller designed

using the Ziegler-Nichols tuning method. Results indicate that

the optimised controller preforms better with much lower

overshoots and oscillations when compared with the Ziegler-

Nichols controller, in both the forward and reverse directions.

Thus, the optimised PID controller is a much more suitable

controller for a wireless V2G IPT system when compared to the

Ziegler-Nichols PI controller.

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