Simplified Modelling Techniques for Dynamic Wireless Power Transfer

1. Introduction

The global effort to reduce carbon emissions has identified road transportation as one of the key contributors to CO2 emissions, as a result EVs have received increased adoption rate to allow a transition to greener energy sources and eliminating tail pipe emissions. The main issues for EVs stem from the use of a battery offering lower driving range (compared to Internal Combustion Engine (ICE) vehicles) and limited availability of charging solutions Some of the main issues facing EVs are the driving range (compared to ICE vehicles) and insufficient charging infrastructure[1]. DWPT aims to extend driving range by providing continuous charging opportunities which are accessible by multiple users. Wireless charging as a solution offers a better user experience and eliminates interoperability issues [2]. The stationary requirement placed on conductive charging is no longer present as power transfer no longer requires a physical connection. DWC offers an opportunistic charging approach, where any vehicle can be charged while driving over a DWC coil. Research on compensation topologies, coils design and control have been previously been covered to guide the design process [3,4,5,6,7,8,9,10]. Calculating the mutual inductance between coils depends on position, orientation shape and size of the coils [11,12,13,14,15,16,17,18,19].

With short high power energy bursts, the grid connections require more research to better understand the impact of adopting DWC. Further investigating the effects with larger adoption rates would require more simulation analysis, which becomes a computational issue as each set of coils runs at 85kHz. While a detailed simulation might be necessary to run for design and control purposes, this is less necessary for grid interaction studies, as such these more complicated systems are summarised by load profiles which aim to give a general idea of behaviour based on power consumption [20,21,22,23,24,25]. From these load profiles high level observations can be made as a first step, in order to gain a better understanding of the interaction and explore possible control strategies, as a load profile speaks only to the power consumption based over time which is not necessarily specific to the DWC system.

There are different ways to both design and control WPT systems depending on load and objectives for the system. Most commonly the series-series compensation is used for it’s power transfer capability while having efficiency independent of coupling factor. For the purpose of grid interaction, dual-sided control is recommended to allow control and ripple reduction of both grid and load current fluctuations [26,27]. For system equivalent models a series of equivalent models are used [28], while the coupling between two coils has previously been discussed, it remains a property specific to the designed circuit [11], either coupling factor or mutual inductance can be used. Updating this property achieves dynamic operation of the system. Grid connection studies are typically carried out on IEEE bus systems, for the purpose of this paper a scaled down version is used.

In Section 2 describes the switching system (and its operation mechanism) which is used was a baseline for the the proposed system in Section 3 which details the equations used for modelling the simplifying DWC system. Section 4 presents results of the proposed system and discusses implementation accuracy and speed trade-off.

2. Dynamic Wireless Charging Switching Circuit

2.1. Full Circuit

The basic system required for WPT for EV charging the minimum circuit required consists of an inverter, rectifier and inductive coils and compensation capacitors. Adding primary and secondary DC/DC converters helps improve efficiency and control of the system by decouple both the source and load from fluctuations in the wireless power transfer system.

The base circuit used for comparison in this research is shown in Figure 1. The sub-circuit function aims to control either the voltage or current at intersection or connection points between subsystems. Primary side DC/DC converter controls the DC bus voltage which is supplied to the inverter; The inverter controls switching frequency for WPT, based on system parameters results in a secondary coil current; controlled rectifier using secondary coil current maintains a reference DC voltage; secondary DC converter can be set to constant current or voltage based on load requirements (for EVs this is based on State-of-Charge (SOC)).

Note: The circuit is designed to be symmetrical around the inductive coils, hence can be configured to run in reverse mode for Vehicle-to-grid (V2G) applications.

2.2. DC/DC Converter

For both primary and secondary side of the circuit in Figure 1, a configurable bidirectional DC/DC converter is used. Based on the used of switches S1-S4, the converter can set to Buck, Buck-Boost or Boost mode in either forward or reverse direction, the possible configurations are shown in Table 1, where a PWM signal is used to track a reference signal using a PI controller.

Figure 2. Configurable bidirectional DC/DC converter.

Figure 2. Configurable bidirectional DC/DC converter.

2.3. Inverter

A fixed duty cycle inverter is used to drive the WPT circuit at resonant frequency. Figure 3 shows a basic implementation where a constant voltage sources and restive load are shown for circuit simplicity, these are later replaced with the circuits shown ins Figure 1.

2.4. Dynamic Wireless Power Transfer.

The DWPT model, relates the two inductive coils using the induced voltage which is a function of mutual inductance. Same as other mutual inductance systems such as fixed core transformers use the same approach. The mutual inductance is a parameter which describes the link between both primary and secondary sides, the coupling factor is a unit-less constant which describes the same relationship (coupling factor $k=M/\sqrt{L_1{L}_2}$).

The remainder of the circuit (Figure 4) uses series-series connected compensation for resonant operation and an equivalent load resistance is used for the diagram. The DWPT model only requires measuring $i_1$ and $i_2$ to generate the induced voltages (shown as $sM{i}_1$ and $-sM{i}_2$). Hence the source and load can be connected to the full circuit without changing the DWPT.

Finally, dynamic modelling is achieved by varying the mutual inductance variable. As the mutual inductance is a function of displacement between the two coils, hence changing the mutual inductance will model the dynamic changes in the system. Previous research discusses this in more detail [26] including the model used for mutual inductance [11].

2.5. Semi-Controlled Rectifier

For the function of a semi-active rectifier only switches S2 and S4 are used to control power flow. When S2=S4=0, this functions as a rectifier allowing maximum power flow; when S2=S4=1, a return path is created for the source current, effectively disconnecting the power flow.

The use of this circuit is recommended as secondary DC voltage source can keep rising throughout operation up to an unsafe level. This occurs when the WPT provides more power than is passed to the load, which occurs often when providing constant output characteristics and will also have an impact on output voltage/current ripple. The alternative is to introduce further control into the system where, typically at the primary side, where there is delay in response as secondary DC/DC voltage is decoupled from the primary converters by the WPT circuit. Furthermore the practicality of having primary/secondary side communication can be an issue.

Figure 5. Semi-controlled rectifier diagram.

Figure 5. Semi-controlled rectifier diagram.

2.6. System Control

The circuit used in Figure 1, is designed to have each sub system control variables within it’s circuit using a PI controller for duty cycle which is then used to generate a PWM signal supplied to the switches as described in each sub-circuit section above. Hence, the primary DC/DC converter controls primary DC bus voltage; the inverter is fixed duty cycle (hence uncontrolled); rectifier control secondary DC bus voltage; secondary DC/DC converter controls load charging characteristics (constant current used in this paper).

2.7. Connection of Subsystems and Additional Requirements

The interconnection of subsystems can be a challenge, as ideally these systems would connect directly to each other (similar to the switching circuit). However this is not always possible as each system is designed with ideal sources and loads, hence a solution can be achieved. Once multiple systems are chained together, the MATLAB Simulink solver has issues defining circuit parameters and views the system as a logic loop.

2.7.1. Primary DC/DC Converter

This main consideration for this circuit is the conversion of primary coil current from WPT to the equivilant DC current to be applied to the output of the DC/DC converter. The relationship between the magnitude of primary coil current ($I_p$) and primary DC/DC converter output current ($I_{2_{DC1}}$):

$$\begin{array}{cc}\hfill I_{2_{DC1}}& =\eta I_p/\sqrt{2}\approx 0.64I_p\hfill \end{array}$$

(1)

Where $\eta$ is the converter efficiency. This is an approximation assuming converter voltage drop is negligible, the remainder is converting the WPT models value to an RMS value.

2.7.2. Wireless Power Transfer

First, the implementation for WPT assumes a sinusoidal input voltage, as the voltage applied in this study is a square wave the approximate value for input voltage is given by:

$$\begin{array}{c}\hfill V_{i{n}_{WPT}}\approx 1.27V_{2_{DC1}}\end{array}$$

(2)

This implementation is shown in the appendix and code used as the WPT function.

Next, the implementation uses a small signal approach which has been converted to equivalent impedance. While the equations have preserved, note the scale factor (sf) applied in the code to scale up the state space implementation response time.

Approximating the load resistance (secondary DC/DC converter) is achieved by approximating the converter’s power consumption

$$R_L=\left\{\begin{array}{cc}4e3,\hfill & \mathrm{if}k<0\hfill \\ R_{DCDC2},\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(3)

Where,

$$R_{DCDC2}=0.5d_{rec}{\displaystyle \frac{V_{rec}}{i_o{V}_o}}$$

(4)

Where $d_{rec}$ is the duty cycle applied to the semi-active rectifier.

Finally approximating rectifier voltage ($V_{Rec}$) is done as a function of both the secondary coil current ($I_s$) converter equivalent resistance ($R_{DCDC2}$) such that:

$$C_{Rec}{\displaystyle \frac{d{V}_{r}ec}{dt}}=\sqrt{2}{d}_{rec}{I}_s-{\displaystyle \frac{V_{Rec}}{R_{DCDC2}}}$$

(5)

2.7.3. Secondary DC/DC Converter

Once the considerations above are implemented, the circuit can be used the same as shown in analysis by applying rectifier voltage ($V_{R}ec$).

3. Circuit Equations and Equivalent Circuits

3.1. DC/DC Converter

Figure 2 shows the bidirectional DC/DC converter. To achieve forward operation mode, $S_2=S_4=0$ leaving S1 as the buck control switch and S3 the boost control switch, allowing the use of buck mode ($S_1=PWM,S_3=0$), boost mode($S_1=1,S_3=PWM$), or buck-boost mode ($S_1=S_3=PWM$). The same is possible for reverse operation mode where $S_1=S_3=0$, buck mode ($S_4=PWM,S_2=0$), boost mode($S_4=1,S_2=PWM$), or buck-boost mode ($S_2=S_4=PWM$). Carrying out the circuit analysis gives the following:

3.1.1. Buck Mode

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-d{\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& =d{\displaystyle \frac{v_{c1}}{L}}-{\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& ={\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(8)

3.1.2. Buck-boost Mode

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-d{\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& =d{\displaystyle \frac{v_{c1}}{L}}-(1-d){\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& =(1-d){\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(11)

3.1.3. Boost Mode - Reverse

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-{\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& ={\displaystyle \frac{v_{c1}}{L}}-(1-d){\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& =(1-d){\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(14)

3.1.4. Buck Mode - Reverse

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-{\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& ={\displaystyle \frac{v_{c1}}{L}}-d{\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& =d{\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(17)

3.1.5. Buck-boost Mode - Reverse

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-(1-d){\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& =(1-d){\displaystyle \frac{v_{c1}}{L}}-d{\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& =d{\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(20)

3.1.6. Boost Mode - Reverse

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c1}}{dt}}& ={\displaystyle \frac{i_1}{C_1}}-(1-d){\displaystyle \frac{i_L}{C_1}}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& =(1-d){\displaystyle \frac{v_{c1}}{L}}-{\displaystyle \frac{v_{c2}}{L}}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{c2}}{dt}}& ={\displaystyle \frac{i_L}{C_2}}-{\displaystyle \frac{i_2}{C_2}}\hfill \end{array}$$

(23)

The equivilant model of the DC/DC converter can be seen in Figure 6 and it’s Simulink implementation in Figure 7

3.2. Inverter

For the purpose of this research, a fixed duty cycle is used to generate a square wave with amplitude $\pm V_{DC}$. The supply voltage/current can hence be related to it’s load voltage/current given by:

$$\begin{array}{cc}\hfill V_{source}& =|V_{load}-2R_{fet}\ast I_{load}|\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill I_{source}& =|I_{Load}|\hfill \end{array}$$

(25)

A more accurate representation takes into account MOSFET, practically this is <1% difference for analysis. Later this is connected to the WPT estimation which considers sinusoidal input instead of square hence:

$$\begin{array}{cc}\hfill V_{square}& =\sqrt{2}{V}_{sine}\hfill \end{array}$$

(26)

3.3. Dynamic Wireless Power Transfer

To replace the WPT circuit, a power estimate from the secondary DC bus is used to estimate load resistance ($R_L=P_{D{C}_2}/V_{REC}$), to incorporate the rectifier an equivalent resistance is used to account for the rectified power consumption ($R_{eq}=8R_L/{\pi }^2$). Like wise for inverter voltage, a sinusoidal input is assumed in AC analysis ($V_{in}=V_{DC}sin\left(f\right)$, where f is the operating frequency). The remaining variables represent compensation capacitors C1 and C2, coil inductance L1 and L2 and their resistance R1 and R2, the remaining parameter for mutual inductance links the primary and secondary sides. Note the mutual inductance is calculated from dimensionless parameter of coupling factor (k) where $k=M\sqrt{L_1{L}_2}$. With all of the above parameters, a steady state value can be estimated for both the input and output current using the impedance at operating frequency f for all passive components.

$$\begin{array}{cc}\hfill Z_1& =R_1+j\omega L_1+1/j\omega L_1\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill Z_2& =R_2+j\omega L_2+1/j\omega L_2+R_{eq}\hfill \end{array}$$

(28)

Voltage analysis of both sides gives:

$$\begin{array}{cc}\hfill V_{DC}& =i_1{Z}_1+jw\omega M{i}_2\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill j\omega M{i}_1& =Z_2{i}_2\hfill \end{array}$$

(30)

Hence the input and output current can be obtained:

$$\begin{array}{cc}\hfill i_1& =V_{DC}/(Z_1-{\omega }^2{M}^2/Z_2)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill i_2& =i_1\ast j\omega M/Z_2\hfill \end{array}$$

(32)

For dynamic charging, the steady state values will need to change as system parameters (such as coupling factor changes), hence we are more interested in the peak values of V/I. To achieve this we can use the total impedance of the system for the following differential equations:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\left(i_1\right)}{dt}}& ={\displaystyle \frac{Vin-i_1\Re Z_1-wM{i}_2}{\Im Z_1}}\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\left(i_2\right)}{dt}}& ={\displaystyle \frac{wM{i}_1-i_2\Re Z_2}{\Im Z_2}}\hfill \end{array}$$

(34)

Hence we can relate primary and secondary DC buses, based on input voltage and output equivalent resistance. The full circuit features a dual active bridge, where the primary and secondary currents values can be converted to the equivalent DC currents before the inverter and after rectification for i1 and i2 respectively.

Hence the WPT equivalent circuit implementation is shown in Fig Figure 8 and Figure 9 MATLAB Simulink implementation code used to implement the aforementioned equations shown in annex A and annex B.

3.4. Semi-Controlled Rectifier

A semi controlled rectifier is used to control rectifier voltage, switching off the input current from the WPT coil. For this function the two MOSFET’s are controlled on to disconnect the power flow to the secondary DC converter.

Analysing the switching behaviour of the controlled rectifier, the voltage before the rectifier is a function of the secondary coil current, rectified voltage and rectifier diode/MOSFET on resitance such that:

$$\begin{array}{cc}\hfill d=0:V_s& =V_{rec}\ast sign\left(I_s\right)+2R_{fet}{I}_s\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill d=1:V_s& =2R_{fet}{I}_s\hfill \end{array}$$

(36)

The current relationship is given by:

$$\begin{array}{cc}\hfill d=0:I_s& =I_c+I_L\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill d=1:0& =I_c+I_L\hfill \end{array}$$

(38)

Note, d denotes the duty cycle of the switching signal for the semi-controlled rectifier.

The approximation used for the proposed system is given by:

$$\begin{array}{cc}\hfill d{I}_s& =I_c+I_L\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill V_s& =I_L^2{R}_L/I_S\hfill \end{array}$$

(40)

4. Results and Discussion

4.1. DC/DC Converter

Operation of DC/DC converters is not a novelty, hence the main difference switching direction is of interest. For this reason a change from forward to reverse buck-boost mode is chosen to showcase the direction switching capability. Main working principle is the inductor current switching from positive to negative, to indicate the change of direction in current flow.

The results presented in Figure 10 show a comparison between the proposed model and switching model, where an accurate representation is achieved.

4.2. Wireless Power Transfer

Validation of the proposed WPT equivalent circuit is shown in Figure 11. First the system operation is shown at steady state with a step change in coupling factor, then the direction of power flow is reversed and the step change is reverted to show the system operating in reverse mode. Notably, the proposed system gives the peak value of primary and secondary currents, which normally remain 0, here however the magnitude is given as negative, to indicated direction of power flow.

4.3. Dynamic Wireless Power Transfer

Applying the WPT model to a dynamic scenario, Figure 12 shows the comparison of primary and secondary coil currents as a function of coupling factor where direction of power flow is switched at 50ms. The proposed system successfully estimates the magnitude of coil current which can than be used to connect to different sub systems (inverter and rectifier) for further analysis.

Table 2. Simulation system parameters.

Parameter	Symbol(s)	Value
Coil inductance	L1, L2	132.410$\mu H$
Coil internal resistance	R1, R2	1.3m$\Omega$
Compensation capacitance	C1, C2	26.477$\mu F$
DC/DC converter inductance	Lb	5mH
DC/DC converter capacitance	Cb	1.5mF
Inverter frequency	$f_i$	85kHz
DC/DC converter frequency	$f_{dc}$	2kHz

Table 2. Simulation system parameters.

Parameter	Symbol(s)	Value
Coil inductance	L1, L2	132.410$\mu H$
Coil internal resistance	R1, R2	1.3m$\Omega$
Compensation capacitance	C1, C2	26.477$\mu F$
DC/DC converter inductance	Lb	5mH
DC/DC converter capacitance	Cb	1.5mF
Inverter frequency	$f_i$	85kHz
DC/DC converter frequency	$f_{dc}$	2kHz

4.4. Full System

The main point of interest for grid interaction simulations is the input (source) and output currents which are shown in Figure 13. The results show very similar dynamic response with the main variance being a phase shift in input current. The proposed simulation captures the dynamic behaviour of the system at a much improved simulation time at the cost of a slight time inaccuracy.

Further exploring the accuracy of the system, the DC bus voltages are shown in Figure 15, again a similar response is shown, with the main reference voltages being obtained successfully. Notably the secondary DC voltage settles to a higher resting voltage which is due to the simplification of the system not including all internal resistances which would further reduce this value.

Comparing the energy consumption at the source gives a better estimation of accuracy, as it will settle down to an approximate value after the aforementioned time shift in results. Figure 14 shows this result, where it can be seen the main working area approximately between 5ms to 12ms, the error is higher due to the shift and lowers again after this period. Hence as a charging event, the proposed system would give an accurate representation of energy consumption with $\tilde{1}$0ms difference.

Figure 14. Full system power comparsion and running error estimation.

Figure 14. Full system power comparsion and running error estimation.

Figure 15. Full system DC bus voltage comparison in a dynamic scenario.

Figure 15. Full system DC bus voltage comparison in a dynamic scenario.

4.5. Simulation Time

The system used for simulation has the following specifications: CPU - Intel i7-9700 @3.00GHz; Memory - 16GB RAM; OS - Windows 11 64-bit (10.0 Build); Matlab - R2023a.

Table 3 shows a comparison of the simulation times for each circuit presented. Notably the main simulation showed $\tilde{3}$4x reduction in simulation time for the same simulation duration, confirming the effectiveness of the simulation techniques in reducing simulation time.

The increased simulation time is due the increased complexity of the full circuit model and computation required to obtain results. This is also true for the proposed equivalent circuit, however instantaneous values for voltage/current are omitted which in turn allows an increase in the required time step to obtain results.

5. Conclusions

The proposed modelling method successfully reduced simulation time for DWPT model, reducing simulation time by a factor of 30. The main circuit for WPT showed a speed improvement by a factor of 17 while maintaining accuracy within 2.5% error which is suitable for the application. Further expansion to a more complex circuit introduced compilations which reduced accuracy, however the modelling techniques used still proved useful in reducing the computation time. The proposed circuit will be able to connect to standard power factor correction circuits or grid connected converters.

6. Future Work

The main issue with the implementation is observed to be the interconnection of subsystems. In fact the issue of this interconnection is subsystem behaviour with low coupling factor (for WPT), this proved to be an issue with the controlled rectifier as at this time the power supplied to the output is approximately zero, meaning it’s resistance is effectively infinite. Hence further study of this interaction would be useful, alternatively expanding the circuit analysis as one whole system instead of separate sub systems is also possible.

Expanding the circuit to accommodate multiple coil systems, both primary and secondary, represents a significant advancement in research, as we move towards including multiple vehicles for charging and integrating multiple transmitter coils. This approach will allow for more comprehensive investigations into practical implementations of DWPT technology. Simulating multiple primary coils with a vehicle traveling between them is a logical next step, as it more accurately reflects future scenarios where road sections are electrified with several transmitter coils. This will also enhance the relevance of grid integration studies.