Jianwei Zhang , Li Li, David G. Dorrell, Youguang Guo

(1. College of Electric Power, Inner Mongolia University of Technology, Hohhot 010051, China;

2. Faculty of Engineering and IT, University of Technology Sydney, NSW 2007, Australia;

3. Howard College Campus, University of KwaZulu-Natal, Durban 4000, South Africa)

Abstract: This paper proposes a modified proportional-integral (PI) controller and compares it with a proportional-resonant (PR)controller. These controllers are tested on a three-phase direct matrix converter (MC). The modified PI controller involves current feedforward together with space vector modulation (SVM) to control the MC output currents. This controller provides extra control flexibility in terms of the current error reduction, and it gives improved steady-state tracking performance. When the coefficient of current feedforward is equal to the load resistor (K = R), the steady-state error is effectively minimized even when regulating sinusoidal variables. The total harmonic distortion is also reduced. In order to comparatively evaluate the modified PI controller, a PR controller is designed and tested. Both the modified PI and PR controllers are implemented in the natural frame (abc) in a straightforward manner. This removes the coordinate transformations that are required in the stationary (αβ) and synchronous (dq)reference frame based control strategies. In addition, both controllers can handle the unbalanced conditions. The experimental and simulation results verify the feasibility and effectiveness of the proposed controllers.

Keywords: Matrix converter, AC-AC conversion, steady-state error, PI controller, PR controller

1 Introduction

The three-phase direct matrix converter (MC)shown in Fig. 1 is an alternative solution for AC-AC conversion. It has several key features such as compact volume, bidirectional power flow, sinusoidal waveforms, and controllable input power factor[1-3].MCs have been proposed for various applications[4-7].In industry, manufacturers such as Yaskawa[8-9],Eupec[1,10]and Fuji[11]have produced MC products,e.g., the Yaskawa U1000, AC7, Eupec ECONOMAC FM35R12KE3ENG, and Fuji FRENIC-MX1S. The power ratings have reached 6.3 MV·A in the Yaskawa MX1S series MC[12].

Fig. 1 Three-phase direct MC1

In the literature, many control strategies have been proposed for MCs[3]. However, many are either too complex or impractical to implement. Model predictive control (MPC) is a widely researched strategy; however, it is not very practical to implement for MCs because of a high number of switch actions.Space vector modulation (SVM) is a developed and common control technique for MCs, but it is ineffective when the load is unknown since this method requires the output current in the modulation.Therefore, the development of the closed-loop SVM is important.

In order to form the closed loop,proportional-integral (PI) controllers can be used since they are simple and easy to implement. This approach has been widely used in power electronic converters and other industrial applications[13-14]. PI controllers,when used in a power electronic converter, usually adopt one of the three reference frames: synchronous reference frame (dq), stationary reference frame (αβ),or natural frame (abc)[4]. Both αβ and dq frame based strategies require multiple frame transformations leading to an increase in complexity, signal processing and implementation effort. The stationary frame PI controllers are considered as unsatisfactory controllers due to significant steady-state errors, especially when tracking AC variables. The reason is that the stationary controller only offers a limited gain at non-zero frequencies[5,15-16].

In contrast, synchronous PI controllers are commonly used because they can achieve zero steady-state error due to the infinite gain of the DC signal provided by the integral term. However, the synchronous frame controller is more complex than the stationary frame controller because several transformations are required (abc↔dq) to convert the controlled signals to DC quantities. The controllers are error-prone because of the noise in the synchronous reference signal detection which can introduce extra errors. In addition, the standard synchronous PI controllers are unable to handle the unbalanced conditions. Otherwise, the control needs to be implemented for both the positive- and negative-sequence components[13,17].

Therefore, it is important to improve the steady-state performance of stationary PI controllers.In this regard, some modified PI controllers have been reported in the literature. Some of these involve a feedforward controller which is simple in concept,robust and has good dynamic performance[18]. It is especially effective in handling a disturbance that can be measured. A stationary PI controller with a grid voltage feedforward path has been proposed and applied for grid-tied converters[19]. This control structure is shown in Fig. 2a. This strategy improved the transient and disturbance rejection performance.However, this scheme suffers from voltage background harmonics and stability problems[20].

A combined feedforward-feedback controller, as shown in Fig. 2b, was proposed in [4] in order to improve the overall performance of the whole control system. The control strategy described in Fig. 2b has been used in other fields[21-24]. However, it has drawn little attention in power electronic converter control. In this paper, a modified PI controller with a current feedforward path is proposed for a three-phase direct MC.

Fig. 2 PI controller with a voltage feedforward path and combined feedforward and feedback controller

Another possible solution for improving the steadystate tracking performance is the proportional-resonant(PR) controller. The PR controller has proved to be a good alternative solution to conventional PI controllers since it shows excellent steady-state performance and can provide specific-order harmonic compensation with reasonable computational burden[5,25-26]. It is difficult to achieve harmonic compensation with standard PI controllers. The PR controller has been investigated for some converters and good performance has been reported[27-30]. However, the PR controller and selective harmonic compensation remain under-explored for the three-phase direct MC. In the literature, only the resonant controller is investigated for a four-leg MC to regulate the output voltage[31]. In this paper, the PR controller is investigated for a three-phase direct MC to control the output currents and the performance is compared with the modified PI controller.

The contributions of this paper are summarized as follows:

(1) A modified PI controller with current feedforward in the natural abc frame is proposed and tested on an MC. The current feedforward provides extra flexibility in reducing the steady-state error.When the coefficient of current feedforward is equal to the load resistor (K = R), the steady-state error is effectively reduced even when regulating sinusoidal variables. To the best knowledge of the authors, this has not been reported in the literature. Unlike the traditional PI controller, the proposed controller does not require frame transformations because it can be implemented in the natural abc frame. The modified PI controller can also handle the unbalanced conditions.

(2) The PR controller and selective harmonic compensation are investigated on the three-phase direct MC to regulate the output currents. Selective harmonics can be compensated effectively. The authors believe that the PR controller has not been investigated for a three-phase direct MC (shown in Fig. 1).

(3) Simulation and experiment comparisons are conducted to evaluate and compare the modified PI and PR controllers. It is concluded that the proposed PI controller is relatively easier to implement and it demonstrates less steady-state error while the PR controller is more effective in reducing distortions.The proposed controllers not only can be applied to MCs, but also to other power electronic converters.

2 PI controller with current feedforward

The proposed combined feedforward-feedback controller is shown in Fig. 2b, where the current reference is fed forward. The transfer function E(s)/R(s)is given by

where Gcis the PI controller, Gfis the feedforward controller (proportional controller), and Gpis the known resistive-inductive load plant. These are given by

This is a two-degrees-of-freedom control system in which the closed-loop characteristics and the feedback characteristics can be regulated independently to improve the overall performance of the whole control system[32]. By rearranging (1), the error in the frequency domain is

The reference R in the natural frame is usually a sinusoidal function R(t) = Irsinωct and its frequency domain expression (Laplace transform) is

where Iris the reference amplitude and ωc= 2πf is the reference angular frequency. By substituting (3) into(2), it can be obtained as

In order to derive the amplitude and phase responses of the error E(s), s is substituted by jω.Therefore, the amplitude and phase angle are obtained as

According to (5) and (6), the error amplitude will have the minimum value when K = R and other parameters are fixed. The introduction of the feedforward controller offers extra flexibility to tune the steady-state error performance. The amplitude and phase responses for different values of K are shown in Fig. 3a. Here, the parameters used are: Kp= 10, Ki= 1,R = 20 Ω, L = 15 mH, ωc=120π rad/s. As concluded from the figure, the amplitude response for K = R is particularly distinct from others and it has the minimum level of the amplitude response for the whole frequency domain. This illustrates that an appropriate selection of K can help reduce the steady-state error. Fig. 3b shows the proposed PI controller with current feedforward for controlling MC output currents. The controller generates the voltage references which will be utilized in the SVM to control the MC.

Fig. 3 Bode diagram of E(jω) for different values of K and proposed PI current controller with current feedforward

Similarly, when the load is resistive-capacitive,i.e.,

The amplitude and angle of the error are

As seen in (8) and (9), we can obtain the same conclusion as that for the RL load.

Compared with the traditional PI controller, the proposed method has reduced steady-state error when regulating sinusoidal variables. This is achieved by involving a current feedforward path in the controller.This current feedforward path provides extra flexibility to reduce the steady-state error. The proposed method does not require frame transformations because it can be implemented in the natural abc (stationary) frame, which reduces the implementation demands. Although the proposed controller requires three controllers, it can easily handle the unbalanced conditions because the controller can be individually implemented for each phase. In this case, the computational burden of the proposed controller is reduced compared with the dq-frame based controller.

In this paper, only the known resistive-inductive and resistive-capacitive loads are considered. When the load is more complex or unknown, the load parameter identification and modeling techniques such as identification based on the hybrid learning algorithm[33], Kalman filter based estimation[34], and least squares methods[35-36], can be utilized. It is worth mentioning that the feedforward controller can be designed differently depending on the specific application to achieve the desired results. This topic remains open with future research work required.

3 PR controller with harmonic compensation

The PR controller can provide an infinite gain at the target frequencies to eliminate the steady-state error at these frequencies, and to compensate for selective harmonic contents by cascading multiple PR controllers[20,37].

3.1 PR controller

In principle, the PR controller can be derived by equivalently transforming a desired DC compensation network into an AC compensation network[38]. This is conceptually similar to transforming a synchronous reference frame based PI controller into a stationary frame based controller[20]. The PR controller is expressed as

where KPand KRare the proportional and resonant gains respectively. This controller can provide an infinite gain at frequency ω and cause no phase shift at other frequencies. However, it is not practical to implement an ideal PR controller because of the infinite quality factor[25]. Hence, the nonideal PR controller is usually adopted and it is expressed by

Here ωc≪ω is the cut-off frequency that affects the bandwidth around the targeted resonance frequencies.The nonideal PR controller has a wider bandwidth around the target frequencies, which leads to less sensitivity and more robustness. This means the nonideal PR controller is more robust to frequency variations around the targeted frequencies.

3.2 PR controller with harmonic compensation

In order to compensate for harmonics, several PR controllers with different resonant frequencies are cascaded. The ideal PR controller with harmonic compensator is expressed as

and the nonideal PR controller with harmonic compensator becomes

where n is the harmonic order and KRnis the individual resonant gain for the nth order harmonic. The diagrams of the ideal and nonideal PR controllers with harmonic compensator are shown in Figs. 4a and 4b respectively. Their corresponding frequency responses,i.e. Bode plots are shown in Figs. 4c and 4d respectively. It is worth noting that the gains at the resonant frequencies can be tuned as required. In Figs.4c and 4d, KP= KRn= 1 is used for simplicity.controllers

Fig. 4 Diagram with selective harmonic compensation and Bode diagram with harmonic compensation of ideal and nonideal PR

The targeted resonant frequencies in the harmonic compensator can be designed depending on specific applications. However, the 3rd, 5th, and 7th harmonics are usually considered since they are regarded as the most prominent harmonics in a typical current spectrum[37].

From Figs. 4c and 4d, it can be observed that the PR controller and harmonic compensator can provide large gains at the targeted frequencies. In comparison with the ideal PR controller, the nonideal PR controller has a wider bandwidth around the targeted frequencies, which results in better robustness to frequency variations.

The error of the MC output current passes through the PR controller to provide the voltage references. These references are then delivered to the SVM modulation stage to generate the gating pulses for the MC. It is worth mentioning that the MC input current can also be controlled by the proposed scheme and the input power factor can be regulated to unity. The PR controller parameters should be tuned appropriately to obtain the desired performance, and this can be carried out in a similar way to that in a PI controller.

4 MC and SVM

4.1 MC basics

As shown in Fig. 1, a three-phase direct MC consists of nine bidirectional switches. These switches form a 3×3 switch matrix as shown in

where S (transpose ST) is the switch matrix. The elements SXxin the switch matrix can be assigned with a value of one (for the “on” state) or zero (for the “off”state).

The constraints (16) are applied to exclude switch states that short-circuit the inputs (usually voltage sources) or that open-circuit the outputs (usually inductive loads). Otherwise, overcurrent or overvoltage will be generated. There are 27 switch states allowable in the matrix which are represented by 27 control actions (finite control set).

4.2 SVM of MC

A synchronous frame-based PI controller using SVM was proposed in [39] and [40] for an MC to control the power flow in a transmission system.However, the issues associated with synchronous PI controllers persisted. In this work, a modulation stage based on the indirect SVM is adopted to generate the firing pulses for the proposed controllers. There are two ways to implement the indirect SVM for the controllers: direct and indirect methods[5].

In the direct method, the output voltage and input current references are directly used in the SVM control to generate gating pulses for the semiconductor switches in the MC[40]. The output voltage reference is generated by the current controller that forms a current loop. The MC input current references are specified according to the system requirements.

In the indirect method, the SVM modulation is divided into virtual inversion and rectification modulation stages, as shown in Fig. 5. In the virtual inversion modulation stage, the designed current controller and SVM (for the inverter only) are used to generate gating pulses for the semiconductor switches in the virtual inverter. SVM (for the rectifier only) is used in the virtual rectification modulation stage to generate gating pulses for the semiconductor switches in the virtual rectifier. Then the two virtual modulation stages are combined to control the MC[39]. In this work, the indirect approach is used.

Fig. 5 Indirect SVM illustration with virtual DC link

In the indirect SVM control, the virtual DC link,shown in Fig. 5, is used to connect the virtual voltage source rectifier (VSR) and the virtual voltage source inverter (VSI). It is worth noting that the virtual DC link does not actually exist in the MC. It is only used to explain the modulation technique. By applying SVM to each stage, and combining them, the overall indirect SVM for the MC can be derived. As a result,the time durations (duty cycles) of switch states for the overall combined modulation stage are derived as

where tγα, tδα, tγβ, tδβand t0are the time durations for active and zero switch states, and Tsis the sampling interval. θR(θI) ( where 0≪θR(θI)≪π/3) is the angle between the desired space vector IRS(VIS) and the right-hand-side adjacent vector IRγ(VIα). m = mi× mvis the modulation index. More details on obtaining the time durations can be found in [4, 40-41].

Tab. 1 Duty cycles distribution for symmetricalswitching sequence.

In order to reduce the switching transitions between each switch state, a symmetrical switching sequence is employed in each sampling interval Ts.This is achieved using the location information (kiand kv) of the input current and output voltage space vectors. An example is given in Tab. 1 for ki= kv= 1.In this table, a value of “1” indicates that the corresponding switch is turned on for the designated time segments, and vice versa. As seen in this table,only one switching transition is required between each state in one sampling period. A switching look-up table can be obtained in a similar manner. Taking advantage of this, enhanced pulse width modulator(ePWM) can be implemented by assigning the compare registers appropriate values.

5 Simulation tests

According to above analyses of the proposed controllers and SVM, the proposed PI and PR controllers for the SVM modulated MC can be designed as shown in Figs. 6a and 6b respectively.These controller structures are simple and no frame transformations are required since they are implemented in the natural abc system. In order to control the input power factor, a phase-locked loop(PLL) is used to detect the location of the input voltages. The parameters of the simulation system are tabulated in Tab. 2. The output frequency is 60 Hz. In the tested RL load, the filtering effect of the inductor is limited, so the current waveform is not very good as shown in the following results. However, these comparative results still can verify the effectiveness of the proposed controller. The current waveform can be improved significantly by increasing the inductance.For simple and explicit figure captions, the tested controllers are abbreviated as follows: open loop SVM(SVM), PI based SVM (PI-SVM), PI with current feedforward based SVM (PICF-SVM), PR based SVM(PR-SVM) and PR with harmonic compensation based SVM (PRHC-SVM).

Fig. 6 Proposed PI and PR controller diagrams for the MC

Tab. 2 Simulation parameters

Comparative simulation results for the PI controller with current feedforward are presented in Fig. 7. The parameters used in the PI controller are Kp= 200 and Ki=10. The simulation results for the traditional PI controller(in the abc frame) are shown in Figs. 7a and 7c. The obvious steady-state error appears in the waveform. The fundamental reference current amplitude is 3.6 A while the regulated fundamental current amplitude only reaches 3.22 A with total harmonic distortion (THD) of 10.27 %. In contrast, the steady-state error performance is significantly improved with the proposed PI controller with current feedforward (K = R = 20.3 while Kpand Kiare the same as before), as shown in Figs. 7b and 7d. The steady-state current amplitude reaches 3.53 A and the THD is reduced to 7.80 % at the same time. The MC input voltage and current of phase A are shown in Fig. 7e. As can be seen, the MC input current is almost in phase with the input voltage.A small phase shift is caused by the input filters since the input current is controlled in the open loop approach.

The dynamic response of the proposed controller to the current amplitude change from 2.8 A to 3.6 A at 0.05 s is shown in Fig. 7f. This result shows the fast transient response of the proposed controller. It is worth mentioning that the modulation index of the MC also influences the controller performance. This can be observed in Fig. 7f where different current amplitudes correspond to different modulation indices.

Fig. 7 Simulation results

The results for the test under unbalanced conditions and input variations are shown in Fig. 8. For unbalanced conditions, the resistance of the output phase b is reduced to half of the other two output phases and the voltage amplitude of input phase A is reduced by 20% (as shown in Fig. 8a). As seen in Fig. 8b, the output currents can still be regulated effectively under unbalanced conditions.For the input variation, the sudden change in both the voltage (by -15 V) and phase (by 30°) of input phase A is simulated, as shown in Fig. 8c. As seen in Fig. 8d, the output current is slightly affected and the results are acceptable. If the unbalanced condition or input variation deteriorates, the results will become worse as what will happen to most of the other control methods.

Fig. 8 Simulation results under unbalanced conditions and input variation

The simulation results for the PR controller are shown in Fig. 9. The controller parameters are (unless otherwise specified): KP= 350, KR1= 600, KRn(n =1,2, 3, 4, 5, 6, 7) = 0, ω = 120π rad/s, ωc= 2π rad/s.

Fig. 9a shows the steady-state current of the PRHC-SVM controller. The simulated current for PR-SVM is not shown here because it is very similar to PRHC-SVM. As shown in Fig. 9a, the steady-state tracking error is not significant. Figs. 9b and 9c compare the FFT analysis results of PR-SVM and PRHC-SVM. As seen in Fig. 9b, the 4th, 6th and 7th harmonics are significant in PR-SVM; therefore these harmonics are compensated for in the PRHC-SVM.The FFT analysis results for the PRHC-SCM are shown in Fig. 9c. It is evident that the specific harmonics (6th and 7th) are suppressed appreciably and the controller is effective. The gain of each compensator determines the corresponding control effectiveness. The reduction of specific harmonics may result in an increase in other harmonics; therefore a compromise needs to be achieved when designing the controller. Fig. 9c presents a compromised result and this is why there is a slight increase in the 4th order harmonic distortions.

It is worth mentioning that the selective harmonic compensator does not necessarily improve the overall THD performance. In fact, it affects the harmonic spectrum because increasing the gain for other order harmonics essentially changes the relative gain of each harmonic. The THD of the MC output current is influenced by the modulation index (voltage transfer ratio). With the PR-SVM and PRHC-SVM control schemes, the input power factor can also be regulated and it can reach a unity power factor.

Fig. 9 Simulation results for the PR controller(for PRHC-SVM: KR4 = KR6 = 500, KR7 = 300)

In Fig. 9d, the dynamic response of the proposed controller to a step change in the reference current from 2.8 A to 3.6 A is evaluated. It can be seen that the controller exhibits a good dynamic response and the prescribed reference is tracked effectively.

The above steady-state simulation results are summarized in Tab. 3. As seen in this table, the modified PI controller shows an improved steady-state performance especially in terms of error while the PR controller has better performance in THD.

Tab.3 Simulation comparison of PI-SVM, PICF-SVM,PR-SVM and PRHC-SCM

6 Experimental validation

In order to verify the proposed strategies, an MC was built and the experimental work was carried out.The experimental set up is shown in Fig. 10. The bidirectional switches (IGBTs) were arranged in the common collector configuration. As a result, only six independent driver power supplies were required to drive eighteen IGBTs. The control card is a TI TMS320F28377D series DSP board. The real-time control implementation was carried out in Matlab/Simulink with C2000 hardware support packages. A serial communications interface (SCI)was used for the communication between the host computer and DSP card. The analog to digital conversion (ADC) and peripheral circuits were employed to process the signals from the voltage and current sensors. The carrier waveform used in the modulation stage was generated from the ePWM blocks, which facilitates the implementation of the symmetrical switching sequence.

Fig. 10 MC prototype

In the experimental tests, the parameters of controllers were set to the same value as those used in the simulation for each corresponding controller, unless otherwise stated. The steady-state experimental results are presented in Figs. 11 to 13 and Tabs. 4 and 5.

The experimental waveforms for SVM are presented in Figs. 11a and 11b. As can be seen, the SVM controlled MC can generate sinusoidal waveforms. However, the open-loop SVM becomes ineffective in controlling the output currents when the load is unknown.

Fig. 11 Experimental waveforms for SVM

Fig. 12 shows the experimental results for PI-SVM and PICF-SVM. As observed in the figures,the steady-state tracking error of PI-SVM is significant,as shown in Fig. 12a. However, for the proposed PICF-SVM controller, the fundamental current amplitude is improved from 3.34 A to 3.56 A when the reference is 3.6 A. At the same time the THD is reduced from 12.05% to 9.01%. These can be observed in Figs. 12c and 12d. The PI-SVM and PICF-SVM controllers have very similar performance in terms of the input current regulation, as shown in Figs. 12e and 12f. The input power factor is regulated to be close to one (unity).

Fig. 12 Experimental results of PI-SVM and PICF-SVM

Compared with the open-loop SVM, the output current in the closed-loop SVM is more distorted. This is because the modulation index keeps changing in the closed-loop SVM. In addition, some research works have investigated the effect of modulation index on the current regulation performance. A general conclusion is that the lower modulation index results in the higher current distortions[42].

In the rest of this section, the comparative experimental results for PR-SVM and PRHC-SVM are illustrated in Fig. 13 and Tabs. 4 and 5. The performance of the input current regulation is quite similar to that presented for PICF-SVM; therefore they are not presented here. The experimental waveforms of the PR-SVM controlled output currents and FFT analysis are shown in Figs. 13a and 13c respectively.Here, the 3rd, 4.2th and 6.8th harmonics are relatively more significant; therefore the 3rd, 4th and 6th harmonics are compensated for in the PRHC-SVM scheme.

Figs. 13b and 13d show the PRHC-SVM controlled output currents and FFT analysis. Tab. 4 compares the low-order harmonics between PR-SVM and PRHC-SVM. From these results, it can be shown that the PRHC-SVM can compensate the selective-order harmonics effectively.

Tab. 5 compares the steady-state experimental results. These results are consistent with the simulation results in Tab. 3. Comparing the modified PI controller with PR controller, the modified PI controller is especially effective in reducing steady-state errors while the PR controllers have less THD. In this paper,these controllers are tested on an MC, but they can be readily extended to other converters.

Fig. 13 Experimental results (for PRHC-SVM: KR3 = 500,KR4 = 200, KR6 = 300)

Tab. 4 Comparison of low-order harmonics between PR-SVM and PRHC-SVM

Tab. 5 Experimental comparison of PI-SVM, PICF-SVM,PR-SVM and PRHC-SCM

7 Conclusions

In order to improve the steady-state tracking performance, this paper proposes a combined feedforward-feedback PI controller and investigates the PR controller. These controllers are tested on an MC. Both controllers demonstrate reduced steady-state tracking error and THD. Firstly, a modified PI controller with current feedforward is proposed. The introduction of the feedforward control provides extra freedom to improve the steady-state error performance.The analysis indicates that the minimum tracking error can be achieved when the coefficient of current feedforward is equal to the load resistor (K = R). Then a PR controller with the capability of compensating selective-order harmonics is investigated for the MC.Both proposed controllers are simple and easy to implement. They do not require reference frame transformations because they can be implemented in the natural abc frame in a straightforward manner. As a result, the implementation effort is reduced. The ability to handle the unbalanced conditions is an advantage of both the modified PI and PR controllers.A constant switching frequency is achieved because of the SVM modulation stage. Based on the comparative study, it is concluded that the proposed PI controller is relatively easier to implement and it demonstrates less steady-state error while the PR controller is more effective in reducing distortions.

Simulation and experimental results corroborate the effectiveness of the proposed controllers. These controller schemes can be readily extended to other power electronic converters and applied in many relevant applications such as grid synchronization and microgrid.