ModifiedPIControllerwithImprovedSteadyStatePerform

Chinese Journal of Electrical Engineering, V ol.5, No.1, March 2019Modified PI Controller with Improved Steady-State Performance and Comparison with PR Controlleron Direct Matrix ConvertersJianwei Zhang 1, 2*, Li Li 2, David G . Dorrell 3, Youguang Guo 2(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 controller1 IntroductionThe 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 MC* Corresponding Author, Email: zjwzachary@ Digital Object Identifier: 10.23919/CJEE.2019.000006Chinese Journal of Electrical Engineering, V ol.5, No.1, March 201954In 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 directMC.Fig. 2 PI controller with a voltage feedforward path andcombined feedforward and feedback controllerAnother possible solution for improving the steady- state 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 remain55Jianwei Zhang et al.: Modified PI Controller with Improved Steady-State Performance andComparison with PR Controller on Direct Matrix Convertersunder-explored for the three-phase direct MC. In theliterature, 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 feedforwardThe proposed combined feedforward-feedbackcontroller is shown in Fig. 2b, where the current reference is fed forward. The transfer function E (s )/R (s ) is given byf p c p 1()()()()1()()G s G s E s R s G s G s −=+ (1) where G c is the PI controller, G f is the feedforward controller (proportional controller), and G p is theknown resistive-inductive load plant. These are given by p i c ()K s K G s s+=, f ()G s K =, p 1()G s R Ls =+This is a two-degrees-of-freedom control systemin 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 isf p c p 1()()()()1()()G s G s E s R s G s G s −=+ (2)The reference R in the natural frame is usually a sinusoidal function R (t ) = I r sin ωc t and its frequency domain expression (Laplace transform) iscr22c ()R s I s ωω=+ (3)where I r is the reference amplitude and ωc = 2πf is the reference angular frequency. By substituting (3) into (2), it can be obtained as2c r 222p i c()()()Ls R K sE s I Ls K R s K s ωω+−=++++ (4)In order to derive the amplitude and phase responses of the error E (s ), s is substituted by j ω.Therefore, the amplitude and phase angle are obtained as(j )E ω∠=p 11c2i p 11c2i ()()tan tan ()()tan tan K R R K L K L K R R K K L ωωωωωωωωω−−−−+⎧−−<⎪−−⎪⎨+−⎪−+π>⎪−−⎩(6)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: K p = 10, K i = 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 theChinese Journal of Electrical Engineering, V ol.5, No.1, March 201956minimum 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 tocontrol the MC.Fig. 3 Bode diagram of E (j ω) for different values of K and proposed PI current controller with current feedforwardSimilarly, when the load is resistive-capacitive, i.e.,p 1()1/G s R SC=+ (7)The amplitude and angle of the error are(8)(j )E ω∠=p 11c i p11ci ()tan ()tan 1()tan ()tan K R C R K C K C K R C R K C ωωωωωωωω−−−−+⎧−−<⎪+⎪⎨+⎪−−+π>⎪⎩(9) 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 thedq -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 compensationThe 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 controllerIn 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 asR PR P 222()K sG s K s ω=++ (10)57Jianwei Zhang et al.: Modified PI Controller with Improved Steady-State Performance andComparison with PR Controller on Direct Matrix Converterswhere K P and K R are the proportional and resonantgains 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 byR c PR P 22c 2()2K sG s K s s ωωω=+++ (11)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 compensationIn order to compensate for harmonics, several PR controllers with different resonant frequencies are cascaded. The ideal PR controller with harmonic compensator is expressed asR PR P 221,3,5,72()()n n K s G s K s n ω==++∑ (12) and the nonideal PR controller with harmoniccompensator becomesR c PR P 221,3,5,7c2()2()n n K sG s K s s n ωωω==+++∑ (13) where n is the harmonic order and K R n is the individual resonant gain for the n th 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, K P = K Rn= 1 is used for simplicity.Fig. 4 Diagram with selective harmonic compensation and Bode diagram with harmonic compensation of ideal and nonideal PRcontrollers (K P = 1, K R n = 1, ω = 120π rad/s) The targeted resonant frequencies in the harmoniccompensator can be designed depending on specific applications. However, the 3rd, 5th, and 7th harmonicsare 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 appropriatelyChinese Journal of Electrical Engineering, V ol.5, No.1, March 201958to obtain the desired performance, and this can be carried out in a similar way to that in a PI controller.4 MC and SVM4.1 MC basicsAs shown in Fig. 1, a three-phase direct MC consists of nine bidirectional switches. These switches form a 3×3 switch matrix as shown ina Aa BaCa A A b Ab Bb Cb B B c Ac Bc Cc C C v S S S v S S S v v S S v v S v v v ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦S (14)A Aa Ab Ac TB Ba Bb BcC CaCbC a a b b c c c S S S i i i S S S S S S i i i i i i ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦S (15)A,B,C1a,b,c Xx X S x ===∑(16)where S (transpose S T ) is the switch matrix. The elements S Xx in 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 MCA 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 specifiedaccording 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 thiswork, the indirect approach is used.Fig. 5 Indirect SVM illustration with virtual DC linkIn 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 ()()R I sin sin 33t m θθγαππ=−− (17)()RIsin sin 3t m θθδαπ=− (18)()RIsin sin 3t m θθγβπ=− (19)R I sin sin t m θθδβ= (20)()0s t T t t t t γαδαγβδβ=−+++ (21)where t γα, t δα, t γβ, t δβ and t 0 are the time durations for active and zero switch states, and T s is the sampling interval. θR (θI ) ( where 0 θR (θI ) π/3) is the angle between the desired space vector I RS (V IS ) and the59Jianwei Zhang et al.: Modified PI Controller with Improved Steady-State Performance andComparison with PR Controller on Direct Matrix Convertersright-hand-side adjacent vector I R γ (V I α). m = m i × m v is 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.t γα t δα t δβ t γβ t 0t γβ t δβ t δα t γα S Aa 1 1 1 1 01 1 1 1 S Ba 0 0 0 0 00 0 0 0 S Ca 0 0 0 0 10 0 0 0 S Ab 0 1 1 0 00 1 1 0 S Bb 1 0 0 0 00 0 0 1 S Cb 0 0 0 1 11 0 0 0 S Ac 0 0 0 0 00 0 0 0 S Bc 1 1 0 0 00 0 1 1 S Cc11111In order to reduce the switching transitions between each switch state, a symmetrical switching sequence is employed in each sampling interval T s . This is achieved using the location information (k i and k v ) of the input current and output voltage space vectors. An example is given in Tab. 1 for k i = k v = 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 testsAccording 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 asshown 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 testedcontrollers are abbreviated as follows: open loop SVM (SVM), PI based SVM (PI-SVM), PI with currentfeedforward 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 MCTab. 2 Simulation parametersV i /V f i /Hz L A /mH r A /ΩC AB /µF L /mH R /Ω T s /μs 100504.830101420.3100Comparative simulation results for the PI controller with current feedforward are presented in Fig. 7. The parameters used in the PI controller are K p = 200 and K i = 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 K p and K i are 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 andChinese Journal of Electrical Engineering, V ol.5, No.1, March 201960current 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 amplitudescorrespond to different modulation indices.Fig. 7 Simulation resultsThe 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, theFig. 8 Simulation results under unbalanced conditions andinput variation61Jianwei Zhang et al.: Modified PI Controller with Improved Steady-State Performance andComparison with PR Controller on Direct Matrix Convertersoutput 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.The simulation results for the PR controller are shown in Fig. 9. The controller parameters are (unless otherwise specified): K P = 350, K R1 = 600, K R n (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 isinfluenced by the modulation index (voltage transfer ratio). With the PR-SVM and PRHC-SVM controlschemes, 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: K R4 = K R6 = 500, K R7 = 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-SCMMethod PI-SVM PICF-SVM PR-SVM PRHC-SVMError 0.381 0.075 0.127 0.13 THD(%)10.277.83.743.76 Experimental validationIn 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。